HISTORY OF THE THEORY OF NUMBERS
VOLUME II
DIOPHANTINE ANALYSIS
Бт Leokabd Еиожкк
MOt of МаНетаНсй гл ffe Untaertiig of Chicago
CHELSEA PTJBLIsmNG COMPANY
NEW YORK, N-Y.

the present work is a textually unaltered reprint of
a work first published in 1919, 1920, and 1923 by the
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number 256 of the carnegie institute of washington
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preface:
Diopb&ntine analysis was named after the Greek Diophantus, of the
third century, who proposed many indeterminate problems in his arithmetic.
For example, he desired three rational numbers, the product of any two of
which increased by the third shall be a square» Again, he required that
certain combinations of the sides, area, and perimeter of a right triangle
shall be squares or cubes» He was content with a single numerical rational
solution, although his problems usually have an infinitude of such solutions»
Many later writers required solutions in integers (whole numbers), so that
the term Diophantine analysis is used also in this altered sense» For the
case of homogeneous equations, the two subjects coincide. But in the
contrary case, the search for all integral solutions is more difficult than that
for all rational solutions. In his first course in the theory of numbers, a
student is surprised at the elaborate theory relating to the equation which
in analytic geometry represents a conic; but it is a real difficulty to pick
out those points of the conic whose coordinates are rational and a greater
difficulty to pick out those points whose coordinates are integral»
Our subject has appeared not only ш works on arithmetic and geometry,
but also in algebras; to it was devoted the larger part of Euler's famous
Algebra. Some of its topics, as the theory of partitions, belong equally
well to analysis» Although most of the problems in this domain may be
stated in simple language free of technical mathematics, their investigation
has quite often required the aid of many branches of advanced mathe-
mathematics. A mere reference to the extensive subject index will show how
frequently use has been made of elliptic functions and integrals, infinite
series and products, algebraic and complex numbers, covariants, invariants,
and semmvariants, Cremona and birational transformations, geometrical
methods, matrices, gamma and theta functions, cyclotomy, linear differ-
differential and difference equations, integration, approximation, limits, minima,
asymptotic and mean values.
Following the plan used in Volume I, we proceed to give an account in
untechnical language of the main landmarks in the successive chapters.
If a reader will not pause to read this entire introduction, let him sample it
by selecting the account of the final chapter» This introduction is followed
by an explanation of the author's point of view in producing a work quite
different from conventional histories.
The notion of triangular numbers 1, 3,6, . . . goes back to Pythagoras,
who represented them by points arranged as are the shot in the base of a
triangular pile of shot. The number of shot in such a pile is called a tetra-
hedral number. In an analogous manner we may define a polygonal
number of m sides (wi-gonal number) and a pyramidal number. Simple
theorems concerning these numbers occur in the Greek arithmetics of
Theon of Smyrna, Nicomachus (each about 100 AJX), and Diophantus
B50 A.D-), who wrote also a special tract about them» They were treated
ш

iv Preface.
two centuries later by Roman and Hindu writers. The most important
theorem on the subject is that first stated by Fermat: Every positive integer
is either triangular or a sum of 2 or 3 triangular numbers; every positive
integer is either a square or a sum of 2, 3, or 4 squares; either pentagonal
or a sum of 2,3,4, or 5 pentagonal numbers; and similarly for any polygonal
numbers» Throughout his half century of mathematical activity, the great
Euler was engaged on the subject of polygonal numbers and solved many
questions concerning them, but was* able to prove Fermat's above theorem
only for the case of squares, and noted that the theorem for the case of
triangular numbers is equivalent to the fact that every positive integer of
the form 8n + 3 is a sum of three squares. This fact is a case of the
theorem that every positive integer, not of one of the forms 8n -f- 7 and 4n,
is a sum of three squares, which was proved in a complicated manner by
Legendrein 1798 and more clearly by Gauss in 1801, by means of the theory
of ternary quadratic forms. Gauss showed how to find the number of
ways in which a number N is a sum of three triangular numbers, by means
of the number of classes of binary quadratic forms of determinant — 8ЛГ — 3.
Cauchy gave in 1813-15 the first proof of Fermat's theorem that every
number is a sum of m m-gonal numbers (all but four of which may be taken
to be 0 or 1). Legendre immediately simplified this proof and showed that
every sufficiently large number is a sum of four or five m-gonal numbers
according as m is odd or even. In 1892 Pepin gave another proof of
Cauchy's result. In 1873 Realis proved that every positive integer is a
sum of four pentagonal or hexagonal numbers extended to negative argu-
arguments. In 1895-96 Maillet proved that every integer exceeding a certain
function of the relatively prime odd integers a and 0 is a sum of four numbers
of the form §(ах2+#г); also, if ф(х)^а^-\- . . . +<u, where the a's are
given rational numbers, is integral and positive for every integer x suffi-
sufficiently large, then every integer exceeding a fixed function of the a's is а
sum of at most v positive numbers ф(х) and a limited number of units,
where v = 6, 12, 96, or 192, according as the degree of ф is 2, 3, 4, or 5.
From formulas in his treatise on elliptic functions of 1828, Legendre
concluded that the number of ways in which N is a sum of four triangular
numbers equals the sum of the divisors of 2iV-f-l, and found the number of
ways in which N is a sum of eight triangular numbers. In 1918 Ramanujan
obtained expressions for the number of representations of any number as a
sum of 2s triangular numbers.
In 1772 J. A. Euler, the son of L. Euler, remarked that, to express every
number as a sum of squares of triangular numbers, at least twelve terms
are required, and stated that, to express every number as a sum of figurate
numbers
(n+l)(n+2o) (n+l)(n+2)(n+3a)
*■ n+a> Ь2 ' Ь27з ' ■"'
at least a-f-2n—2 terms are necessary. About the same time, N. Beguelin
stated erroneously that at moat a+2n—2 terms are sufficient. In 1851

Preface» v
Pollock stated that 5, 7, 9, 13, 21, 11 terms are needed to express every
number as a sum of tetrahedral, octahedral, cubic, icosahedral, dodeca-
hedral, and squares of triangular, numbers, and related facts. In 1862-63
LiouvUle proved that the only linear combinations of three triangular
numbers Д which represent all numbers are Д+Д'+сД" (c^ 1, 2, 4, 5) and
" W234)
Chapter II opens with an account of the method of solving ах+Ъу = с
given by the Hindu Brahmegupta in the seventh century» It was based on
the mutual division of a and b, as in Euclid's process of finding their greatest
common divisor. Essentially the same method was rediscovered in Europe
by Baehet de Meziriac in 1612, and expressed in the convenient notation
of the development of ajb into a continued fraction by Saunderson in
England in 1740 and by Lagrange in France in 1767. The simplest proof
that the equation is solvable when a and b are relatively prime is that given
by Euler in 1760, who noted that, on dividing c—ax (*==(), 1, . ♦ ., b—l)
by bt we obtain b distinct remainders which are therefore 0, 1, . . ., b—l
in some order, the remainder zero leading to a solution. Since the same
principle underlies the most elegant proof of Euler's generalization a"=l
(mod b) for /?— ф{Ь) of Fennat's theorem, it was a simple step to solve our
equation, or—what is the same thing—the congruence ax^c (mod b),
by multiplying its members by a*~l. This step was made about 1829 by
Binet, Libri, and Cauchy. Or we may evidently employ Wilson's gener-
generalized theorem, which states that the produet of the positive integers less
than and prime to b is = ±1 (mod b). In 1905 Lerch expressed the solution
of ax^l (mod b) as a sum involving the greatest integer function»
In the Chinese arithmetic of Sun-Tsu, about the first century, occurs
the problem of finding a number having the remainders 2, 3, 2 when divided
by 3, 5, 7, respectively, with a rule leading to the answers 23H-3-5-7n.
The same problem and answer 23 occur in the Greek arithmetic of Nico-
machus, about 100 A.D. The rule is essentially the following, given
centuries later by Beveridge, Euler, and Gauss: To obtain a number x
having the remainders rh r2,. ♦ .when divided by mlt m2,. . ^respectively,
where mh m2, * . . are relatively prime in pairs, find numbers al7 *b . . .
such that <*»=1 (mod m,), <*i=0 (mod m/mi), where m is the product
Wj7tt2 . . .; then x—<*ir1+«27'2+ * * . is an answer» In the seventh cen-
century, the Chinese priest Yih-hing extended this rule to the case in which
wb m*T. . . are any integers: express the least common multiple of mt} m^t
. .p ♦ as aproduct т=рцх2 . . . of factors relatively prime in pairs (some of
which may be unity), such that pi divides *n,-, and find al9 a^ . - - such
thattt^=l (mod pi), cti^Q (mod mjfii); then гг=а1г1+а2гз+ ....
The Hindus Brahmegupta and Bhascara found the correct answer 59 to
the *' popular problemJ' of finding a number having the remainders 5,4, 3,2
when divided by 6, 5, 4, 3, respectively; Leonardo Pisano m 1202 added
the condition that the number be a multiple of 7* He treated the problem
of Ibn al-Haitam (about 1000 A.D.) of finding a multiple of 7 which has
the remainder unity when divided by 2, 3, 4, 5 or 6, a problem occurring

vi Preface»
in many later books. This subject of the Chinese remainder problem found
application in questions on the calendar; for example, to find the year x
of the Julian period when the solar cycle, lunar cycle, and Roman indiction
are given numbers n, r$, гз, we seek a number which has the remainders
n, **, r9 when divided by 28, 19, 15, respectively, these being the periods
of the solar, lunar, and indiction, cycles.
The problem of finding the number of positive integral solutions of
ах+Ъу = с, where a, bt с are positive integers, was treated by Paoli in 1780,
Hermite in 1855-58, and many others» There is the corresponding question
for a system of such equations.
Systems of equations of the type x+y+z=m, ax+by+cz=n, where
mt n, a, 6, с are given positive integers and the unknowns are to have
positive integral values, occurred in Chinese and Arabic manuscripts of
the sixth and tenth centuries respectively, in Leonardo Ksano's writings,
and in many of the early printed books on algebra and arithmetic» The
usual method of solution, which began with the elimination of one unknown,
was called regula Coeci, or the rule of the virgins, a term later applied to a
system of any number of linear equations in any number of unknowns with
positive integral coefficients» The most important papers on general sys-
systems of linear equations or congruences are those by Heger A858), EL J. S»
Smith A859, 1861, 1871), Weber A872, 1896), Frobenius A878-79), Kro-
necker A886), and Stemitz A896).
Chapter II closes with a series of modern theorems, such a* the fact that,
if ш is irrational, there exist infinitely many pairs of integers x, y7 for which
y^ux is numerically less than the reciprocal of V5s; and Minkoweki's
theorem (of prime importance for the theory of algebraic numbers) that,
tf /1, • - -> /« are linear homogeneous functions of xh . . ., xn with any real
coefficients whose determinant is unity, we can assign integral values not
all zero to xlt . . ,t x*t such that each /< taken positively does not exceed
unity.
Chapter III treats of partitions, which have important applications
to symmetric functions and algebraic invariants. The first investigation
was that by Euler in 1741, who discussed the two problems of finding the
number of ways in which a number n (as 6) is a sum of a given number m
(as 2) of distinct parts F=5+1=4+2), and the number of ways n is a
sum of m equal or distinct parts (so that also 6=3+3 is counted}» The
numbers in question are the coefficients of x* in the expansions of xjn(M+1> I2JD
and x^/Dy respectively, into series of powers of л, where
D(lxH.x) . . . Aя)-
Functions like these which serve to enumerate all the partitions of a specified
kind are now called generating functions. In his more attractive exposition
in his Introductio in Analysin Infinitonim of 1748, Euler noted that \jD is
the generating function giving the number of partitions of n into parts ^m
which need not be distinct. For n=5, m = 3, these partitions are 3+2,
3+1+1, 2+2+1, 2+1+1+1, 1+1 + 1+1+1. Similarly, the reciprocal

Preface. vii
of Л£ГA~"ЖО *8 ^е generating function for the number of unrestricted
partitions of л, where now also 5 and 4+1 are counted. Again, the number
of partitions of n into m or fewer parts ^t is the coefficient of xn in the
expansion of
О.-х"*)A-х"*) . . . A-**♦*)/*),
where D is the above product. Euler stated empirically the important
fact that
2
which has since been proved by many writers, in particular by Jacobi in
his Fundamenta Nova of 1829, where he made important applications of
elliptic functions to the theory of partitions. As noted by Legeudre in 1830,
the last formula implies that every number, not a pentagonal number
Cn*-±n)f2t can be partitioned into an even number of distinct integers as
often as into an odd number, while (Зл2±л)/2 can be partitioned into an
even number of parts once oftener or once fewer times than into an odd
number of parts, according as n is even or odd» Jacobi in 1846 extended
this result to partitions into any given distinct elements.
In 1853 Ferrers gave a diagram which establishes a reciprocity between
the partitions of the same number. The partition 3+3+2+1 is repre-
represented by four rows of dots containing 3, 3, 2, 1 dots, respectively, such that
the left-hand dots are in the same vertical column. Reading the diagram
by columns, we get the partition 4+3+2_
Sylvester stated in 1857 that the number of partitions of n into given
positive integral elements a\,. . ., Or with repetitions allowed is 2И%, where
the "wave" Wq is the coefficient of l{t in the development in ascending
powers of t of
the summation extending over the various primitive gth roots p of unity.
Proofs were soon given by Battaglini, Brioschi, Roberts, and Trudi;
Sylvester published his own method in 1882. Cayley wrote several papers
on the theory and its applications.
During the years 1882-84, Sylvester and his pupils at Johns Hopkins
University published many papers on partitions, in particular on their
graphical representation, with the aim to derive the chief theorems con-
constructively without the aid of analysis.
Beginning with his paper of 1886 on perfect partitions, Major MacMahon
has made numerous contributions to the thoery of partitions and the more
general subject of combinatory analysis, culminating in his treatise in two
volumes published in 1915-16 (see the report, pp. 161-2).
Vahlen proved in 1893 that, among the partitions of & into distinct parts
the sum of whose absolutely least residues modulo 3 equals a given integer h,
there occur as many partitions into an even number of parts as into an odd
number of parts, except only when * is the pentagonal number (Ш—A)/2

viii Preface.
for which there exists an additional partition into an even or odd number
of parts according as Л is even or odd. Ibis implies the corollary of Legendre
mentioned above» Analogous theorems were obtained by von Stemeck in
1897 and 1900.
Mention should be made of the various papers by Glaisher of 1875-76
and 1909-10, that of Csorba of 1914, and the asymptotic formulas obtained
by Hardy and Ramamijan jointly in 1917-18.
Chapter IV reports on the extensive, mostly old, literature on rational
right triangles, a subject which was the source of various problems treated
in later chapters. Diopbantus knew that if the sides of a right triangle are
expressed by rational numbers they are proportional to 2wm, то1—л2,
mz+n\ and referred to the right triangle having the latter sides as that
"formed from the two numbers m and nS' Pythagoras and Plato had
given special cases. Among the many problems on rational right triangles
treated by Diophantus, Vieta, Bachet, Girard, Fermat, Frenicle, De Billy,
Ozanam, Euler, and others, are the following: Find n(n^S) rational right
triangles of equal areas; two whose areas have a given ratio; one whose
area is given or becomes a square on adding a given number or a certain
function of the sides; one whose legs exceed the area by equares; one whose
legs differ by unity or by a given number; right triangles the sum of whose
legs is given; or with a rational angle-bisector.
Chapter У deals with rational triangles, whose sides and area are ra-
rational, and rational quadrilaterals, having also rational diagonals. By the
juxtaposition of two rational right triangles with a common leg, we obtain
a rational triangle. During 1773-82, Euler wrote a series of four papers
on triangles whose sides and medians are all rational, while Bachet in 1621
bad been content when a single median or single ang)e-bisector is rationaL
The Hindus Brahmegupta and Bhascara showed how to form a rational
quadrilateral by juxtaposing four right triangles with pairs of equal legs
such that the right angles have a common vertex and do not overlap. In
1848 Kummer showed bow to obtain all rational quadrilaterals. Euler
gave (p. 221) a construction for a polygon of n sides inscribed in a circle of
radius unity such that the sides, diagonals, and the area are all rational.
No mention will be made of the 160 further papers reported on in this
chapter, which closes with the papers on rational pyramids, trihedral
angles, and spherical triangles.
Chapters VI-IX deal with the specially interesting literature on the
representation of numbers as sums of 2, 3, 4, n squares. Diophantus knew
how to express the product of two sums of two squares as a sum of two
squares in two ways:
He knew that no number of the form An — 1 is a sum of two squares. But
Girard in 1625 and Fermat a few years later were the first to recognize
that a number is a sum of two squares if, and only if, its quotient by the
largest square dividing it is a product of primes of the form 4n+l or the

Preface. ix
double of such a product. Fermat also knew how to determine the number
of ways in which a given number of the proper form is a sum of two squares*
He stated that he could prove that every prime 4n+1 is a sum of two squares
by the method of indefinite descent, i.e., if a prime 4n+l is not a sum of
two squares there exists a smaller prime of the same nature, etc., until 5 is
reached. Enler wrestled with this theorem for seven years before he
succeeded in finding a complete proof in 1749. He published more elegant
proofs in 1773 and J783. In the meantime, L&grange gave several proofs
in 1771-75- An expression for the number of representations of an integer
as a sum of two squares was given by Legendre in 1798 and by Gauss in
1801, while a more elegant expression was deduced by Jacobi in 1329 from
infinite series for elliptic functions .and proved arithmetically by him in
1834 and by Diricblct in 1840. In a posthumous paper, Gauss left a
formula for the number of sets of integers xt у for which x2 + у2 ^ A,
Le., the number of lattice points inside or on the circumference of a given
circle; the same subject was studied by Eisenstein in 1844, Suhle in 1853,
Cayley in 1857, Ahlbom in 1881, and Hermite in 1884 and 1887, while
asymptotic formulas were proved by Sierpinski in 1906, Landau in 1912-13,
Hardy in 1915-19, and Szilysen in 1917.
Diophantus stated in effect that no number of the form 8^+7 is a sum
of three squares, a fact easily verified by Descartes. Fermat gave in effect
tbe complete criterion that a number is a sum of three squares if, and only
if, it is not of the form 4n(8m+7). For many years Euler tried in vain to
prove this theorem, nor did Lagrange find a proof for all cases. In 1798
Legendre gave a complicated proof by means of theorems on the quadratic
divisors of 1?-\-cu2. In 1801 Gauss published a proof which also expresses
the number of ways a number n is a sum of three squares in terms of the
number of classes in the principal genus of the properly primitive binary
quadratic forms of determinant — n. Other such expressions were obtained
by Dirichlet in 1840 by means of his formulas for the number of classes of
binary quadratic forms; also by Kronecker in 1860 by use of series for
elliptic functions and in 1883 by means of the number of classes of bilinear
forms in two pairs of cogredient variables» In 1850 Dirichlet gave an
elegant proof of Fermat's criterion by means of reduced ternary quadratic
forms. Many writers have discussed the solution of z2-\-y2-\-z2—n2; a
simple expression for the number of solutions was given by A. Hurwitz
in 1907. The problem of the number of integers Шх which are sums of
three squares was investigated by Landau in 1908, while he (in 1912) and
Sierpinski in 1909 found asymptotic formulas for the number of sets of
integers uy vy w for which vP+tf+vPt^x.
In the three problems in which Diophantus employed sums of four
squares, he expressed 5, 13, and 30 as sums of four rational squares in two
ways without mention of any condition on a number in order that it be a
sum of four squares, although he gave necessary conditions for representa-
representation as a sum of two or three squares in the problems where the latter
occur» Hence Bachet and Fermat ascribed to Diophantus a knowledge

x Pbeface.
of the beautiful theorem that every positive integer is a sum of four integral
squares. In 1621 Bachet verified tMs theorem for integers up to 325. The
theorem was stated to be true by Girard in 1625 and as an unproved fact
by Descartes in 1638. Fermat stated that he possessed a proof by indefinite
descent.
This theorem engaged the serious attention of Euler for more than
forty years, as appears from his life-long correspondence with Gold bach;
in vain did he convert the problem into an equivalent, but squally baffling,
question, Not until twenty years after he began the study of the theorem
did he publish in 1751 some important facts bearing on it, including his
formula, which expresses the product of two sums of four squares as such
a sum. The first proof published was that by Lagrange in 1772, who
acknowledged his indebtedness to ideas in Euler's paper» The next year
Euler published an elegant proof, which is much simpler than Lagrange's
and which has not been improved upon to date. Gauss noted in 1801 that
the theorem follows readily from the fact that any number having the
remainder 1, 2, 5, or 6, when divided by 8, is a sum of three squares; but
the latter fact has not yet been proved in so simple and elementary a
manner as the former. In 1853-54 Hermite gave two proofs by means
of the theory of quadratic forms in four variables and a- proof by means
of a Hermitian form with complex integral coefficients and two pairs of
two conjugate complex variables.
In 182£-29 Jacobi compared two infinite series for the same elliptic
function to show that, if p is odd and <r(p) is the sum of the divisors of pt
the number of representations of Tp as a sum of four squares is 8<r(p)
or 24<r(p), according as a=0 or <*>0, where in a representation the signs
of the roots and their arrangement are taken into account. In a similar
manner, he and Legendre proved simultaneously that there are exactly
<r(p) sete of four positive odd numbers the sum of whose squares is 4p.
For the latter theorem Jacobi gave an arithmetical proof in 1834, which was
simplified by Dirichlet in 1856 and by Pepin in 1883 and 1890. For the
former theorem on the representations of 2"p, elementary proofs have been
given by Stern in 1889, Vahlen in 1893, Gegenbauer in 1894, and L, Aubry
in 1914, while Mordell gave in 1915 a proof by means of theta functions,
Cauchy proved in 1813 that any odd number к is a sum of four squares
the algebraic sum of whose roots equals any assigned odd number between
V3A-2-1 and V5fc. In 1873 Reafis proved also that every number
AT=4w+2 is a sum of four squares the algebraic sum of whose roots is
any assigned one of the numbers 0, 2, 4, . , ,, 2/*, where p* is the largest
square <If. Mention should be made of papers by Torelli (p. 294),
Glaisher (p. 296, p. 301), and Petr (p. 300).
Many of the papers in this long Chapter VIII prove the existence of
solutions of the congruence а2?+&!/*+сз*=0 (mod p), in which a, 6, с are
not divisible by the prime p, while some determine the number of sets of
воЬгёсшв* The corresponding question for n unknowns is discussed in the
brief Chapter X

Preface. xi
In Chapter IX the material on representation as sums of n squares is
separated from the reports on the more elementary papers giving relations
between squares and mainly concerning ti squares whose sum is a square»
Following a bint by Jaeobi, Eisenstein stated in 1847 that the number
of representations of an odd number as a sum oi eight squares equals 16
times the sum of the cubes of its divisors, and theorems almost as simple
for six and ten squares» He also gave, without proof, formulas which
express the number of representations of m by 5 and 7 squares as sums of
Legendr<vJAcobi symbols of quadratic residue character modulo m. In
1860-65 liouville stated various theorems on representation by 10 and 12
squares, which he apparently deduced from series for elliptic functions, and
which have been so proved and generalized by Bell in 191Q ^^ci were proved
Ъу хит;»!!» vi LLeta functions* by Humbert anu IVtii In 1CC7, x±± 1S67
H. J. 3. Smith stated general results on representation by 5 and 7 squares.
This paper was unknown to the members of the commission whose recom-
recommendation led the Paris Academy of Sciences to propose for its grand prix
des sciences mathematiques for 1882 the subject of representation by 5
squares» .Frizes of the full amount were awarded both to Smith mid to
Minkowski (the latter being then 18 years of age), each of whom developed
the theory of quadratic forms in n variables and evaluated the number of
representations by 5 squares. There are further papers on the last topic
by Sfcieltjes, Hcnnite, Pepin, and Hurwitz (pp. 310-1)* Mention should
be made of the papers by Gegenbauer (p. 313), Doulygukie (p. 317),
Mordell, Hardy, and R&manujan (p. 318) on representation by n squares.
Chapter XI, which is closely related to the last topic, gives a summary
of Liouville's series of eighteen articles published in 1858-65, in which
he stated results (apparently found from expansion of elliptic functions)
which express many equalities between sums of the values of quite general
arithmeticel functions when the arguments of the functions involve the
divisors of two (or more) numbers whose sum is given. The chapter closes
with a citation of papers which together give proofs of all the formulas,
except only (Q) of the sixth article, besides proving a few related theorems»
The sixty pages of Chapter XII give reports on more than 300 papers
on азР-^Ъх+с^у1* Diophantus wag led to such an equation in **t least
forty of his problems. He was content with rational solutions, wJiich he
showed how to find if a or с is a square, or if b — 0 and one set of solutions
is known. It is a remarkable fact that the Hindu Brahmegupta in the
seventh century gave a tentative method of solving а2?+с=#2 in integers,
which is a far more difficult problem than its solution in rational numbers.
His method was explained more clearly by the Hindu Bhascara in the
twelfth century- Much еагБег, the Greeks had given approximations to
square roots which may be interpreted as yielding solutions of a2?+l=2/2
for a^2 and a=3. Moreover, the famous cattle problem of Archimedes,
which imposed nine conditions upon eight unknowns, leads in its final
analysis to the difficult equation os'+l^y2, where a=4729494, and has
been solved in modern times.

xii Preface»
Such an equation x2—Ay*— 1 has long borne the name Pellian equation,
after John Pell, due to a confusion on the part of Euler; it would have been
more appropriately named after I*ermat, who stated in 1657 that it has
an infinitude of integral solutions if A is any positive integer not a square,
and who stated in 1659 that he possessed a proof by indefinite descent.
He proposed it as a challenge problem to the English mathematicians
Lord Brouncker and John Wallis, who finally succeeded in discovering a
tentative method of solution, without giving a proof of the existence of
an infinitude of solutions. This theorem is really only the simplest and
first known case of Dirichlet's elegant and very general theorem on the
existence of units in any algebraic field or domain» The former theorem
is also of great importance in the theory of binary quadratic forms. More-
Moreover, the problem to find all the rational solutions of the most general
equation of the second degree in two unknowns reduces readily to that for
x2—Ay*=B, all of whose solutions follow from one solution and the solutions
y
In 1765 Euler exhibited the method of solving a Pellian equation due
to Brouncker and Wallis in a more convenient form by use of the continued
fraction for VZ and found various important facts, but gave no proof that
the process leads always to a solution in positive integers» This funda-
fundamental fact of the existence of solutions was first proved by Lagrange a
year or two later; while in 1769 and 1770 he brought out his classic
memoirs which give a direct method to find all integral solutions of
xz—Ay*=B, as well as of an equation of degree n, by developing its real
roots into continued fractions»
Of the further extensive literature on the Pellian equation, the most
notable papers are those by Legendre, Gauss, DiricMet, Jacobi, and Ferott;
limits for the least positive solution were obtained by Tchebychef in 1851
and by Remak, Perron, Schmitz, and Schur in 1913-18» Useful tables have
been given by Euler, Legendre, Degen, Tenner, Koenig, Arndt, Cayley,
Stem, Seeling, Roberts, Bickmore, Cunningham, and Whitford.
Chapter XIII treats of further single equations of the second degree,
including axy+bx+qj+d^O, х*—у**=д, ах'1+Ъху-\-су2^<кг or d, the most
general equation of the second degree in я, уу and its homogeneous form
aX2+bY*+cZ*+dXY+eXZ+fYZ=O. Criteria for integral solutions of
the latter were stated by H. J. S, Smith (p, 431) and proved by Meyer for
the case of an odd determinant, while its complete solution was given by
Desboves (p, 432) when one solution is known. Lagrange's method for
x2— Ay2=By cited above, was employed by Legendre in 1785 to prove the
important theorem that, if no two of the integers a, bt с have a common
factor and if each is neither zero nor divisible by a square, then
ax*+by*+a?=Q has integral solutions not all zero if, and only if, —be, —ac,
—ah are quadratic residues of a, &, c, respectively, and a, b, с are not all
of the same sign. Gauss gave a proof by means of ternary quadratic forms,
while a generalization was made by Dirichlet (p. 423) and Goldscheider
(p. 426). Meyer gave criteria (pp. 432-3) for integral solutions of /=0,

Preface. xiii
where / is any quadratic form in four variables, with simple criteria in the
case of ax2-\-by2+cz2+du*=0; and noted that, when there is a fifth term
ei^, the equation is solvable in integers not all zero if the coefficients are odd
and not all of the same sign» Minkowski (p» 433) proved the generalization
that zero can be represented rationally by every indefinite quadratic form
in five or more variables, and gave invariantive criteria for four or fewer
variables.
Chapter XIV reports on many elementary papers on squares in arith-
arithmetical or geometrical progression» While there is a simple, general,
formula for three squares in arithmetical progression, known by Vieta,
Fermat, and Frenicle, there do not exist four distinct squares in arithmetical
progression.
Chapter XV opens with a collection of the problems from Diophantus,
in which it is a question of finding values of the unknowns for which several
linear functions of them become equal to squares» Such problems were
treated by Brahmegupta in the seventh century, by Vieta in 1591, and by
Bachet, Fermat, Prestet, Ozanam, and others, in the seventeenth century.
One of the problems studied most frequently is that of finding three numbers
such that the sum and difference of any two of them are squares; it was
treated by Petrus in 1674, Leibniz in 1676, Rolle in 1682, Landen in 1775,
by Euler in his Algebra and elsewhere, as well as by various later writers
(all cited in note 28, p. 448),
The story of congruent numbers, given in Chapter XVI, is a long one,
beginning with Diophantus. If x and к are rational numbers such that
x*-\-k and x3—к are both rational squares, к is called a congruent number,
Diophantus knew that z*-\-y2*=z2 implies z^±2x2/ = (x±i/J, so that 2xy is a
congruent number. This topic was the chief subject of two Arabic manu-
manuscripts of the tenth century» Leonardo Pisano, in his Liber Quadratorum
of 1225, treated the subject at length and with skill, making repeated use
of the fact that any integral square is a sum of consecutive odd numbers
beginning with unity» In particular, he stated, but did not completely
prove, that no congruent number is a square, which implies that the area
of a rational right triangle is never a square and that the difference of two
biquadrates is not a square, results of special importance historically»
Although part of Leonardo's work was incorporated in the arithmetics of
Luca Paciuolo, Ghallgai, Feliciano, and Tartagna, the original seemed to
be lost and Cossali made a laborious, but unsuccessful, attempt to recon-
reconstruct it. The original was found and published by Prince Boncompagm
in 1854 and in the Scritti di Leonardo Pisano, II, 1862. The most important
later papers on congruent numbers are those by Euler, Genocchi, Woepcke,
Collins, and Lucas»
The related problem of concordant forms is to make xi-\-myt and
&+ny2 both squares and was studied by the same writers, especially by
Euler in several of his memoirs. The remaining problems of this chapter

xiv Pkefacb,
and those of Chapter XVII relate to special systems of two quadratic
functions or equations and do not possess sufficient general interest to
warrant mention here» The last remark applies also to Chapter XVIII,
which treats of three or more quadratic functions.
Chapter XIX begins with the history of the problem of finding three
integers x7 y, z such that x2+yi, a?+**, уг+г* are all perfect squares. Solu-
Solutions involving arbitrary parameters, but obtained under special assump-
assumptions, were found by Saundereon (who was blind from infancy) and Euler
in their Algebras of 1740 and 1770. The problem is equivalent to that of
finding a rectangular parallelopiped having rational values for the edges
and the diagonals of the faces. If we impose the further restriction that
also a diagonal of the solid shall be rational, we have a difficult problem
which has been recently attacked but not solved.
The problem of finding n squares the sum of any n—1 of which is a square
was treated at length by Euler for n=4, and for any n by Gill by use of
trigonometric functions. The problem of finding three squares the sum of
any two of which exceeds the third by a square was treated by four special
methods by Euler in a posthumous paper, as well as by Legendre and others.
The problem of making a quadratic form in x and yy one in x and zt and one
in у and z simultaneously equal to squares has received much attention during
the past hundred years. Beginning with Diophantus, there is an extensive
early literature on the problem of finding n numbers such that the product
of any two of them increased by a given number shall be a square.
Euler developed an interesting method (p. 522) to make several func-
functions simultaneously equal to squares. Бе selected a suitable auxiliary
function / such that solutions of /=0 can be readily found. For any set
of solutions, P2—/ is evidently a square, whatever be the function P.
Many further problems occur m this long chapter, which closes with an
account of rational orthogonal substitutions.
The nature of Chapter XX will be illustrated by means of an example
of considerable interest for the history of algebraic numbers. Fermat
stated that he had a proof that 25 is the only integral square which if
increased by 2 beeomes & cube» Euler, in attempting a proof in his Algebra
of 1770, assumed that $4-2=£ implies that each factor x±V^2 is the
cube of a number p+gV-2, where p and q are integers, although he knew
that a like assumption is not valid when 2 is replaced by other numbers.
The justification of his assumption in the first example is due to the fact
that for these numbers p+q^-2 factorization into primes is unique and
to the further fact that ±1 are the only ones of these numbers which
divide unity. Instead of this explanation by means of algebraic numbers,
we may employ the theory of classes of binary quadratic forms, as was done
byPepin(p.54I).
In the 69 pages of Chapter XXI report is made on about 500 papers on
Diophantine equations of degree 3. The method by which Diophantus

Preface. xv
expressed the difference of two given rational cubes as * sum of two positive
rational cubes was given in his Porisms, a work which has not been pre-
preserved. The formula (p. 550) which Vieta used in 1591 for this" purpose
is vaHd only when the greater of the given cubes exceeds the double of the
smaller. While also Bachet could solve ouiy this case, Girard and Fermat
showed bow, by employing Vieta's three formulae in turn, to solve the
remaining case as well as the problem to express a sum of two given rational
cubes as another such sum. The last problem had been proposed by Fermat
to the English mathematicians Brouncker and Wallis, who gave merely solu-
solutions derivad from known solutions by multiplication by a constant. The
general solution in integers of this problem was first given by Euier in
175&-57* His solution was expressed in a simpler form by Binet in 1841
and deduced elegantly by Hermite in 1872 by means of the ruled lines on the
corresponding cubic surface (a method extended to a certain equation of
degree n by Brunei, p, 556). Report is made on pp. 560-1 on Japanese
writings during 1826-45 on this subject. The related problem of finding
three equal sums of two cubes arose in the question of finding four integers
the sum of any two of which is a cube*
There are many minor papers of recent decades which give relations
between five or more cubes, or express a sum of three cubes as a square.
The problem of making a binary cubic form equal to a cube was treated by
obvious elementary methods by Fermat and Euler, and recently by bi-
rational transformation by von Sz. Nagy, and by covari&nts by HaentzscheL
To make a binary cubic form equal to a square, Fermat and Euler equated
it to the square of a linear or quadratic function, and Lagrange used the
norm of an algebraic number (p. 570), while Mordell in 1913 employed the
theory of invariants.
Since every rational number is a sum of three rational cubes (p, 726),
it is an interesting question to determine the rational numbers which are
sums of two rational cubes, or, if we prefer, the integers A for which
&+y*=Az* is solvable in integers» Reports on fifty papers on this subject
are given on pp. 572-8» Euler proved that the problem is impossible if
4 = 1 and A =4, and that z = ±y if A =2. Legendre erred in his statement
that it is impossible if A =6. In 1856 Sylvester stated that it is impossible
if A «= p, 2p, 4p*, 4g, $*, 2g*, where p and q are primes of the respective forms
181 + 5 and 182 + 11- In 1870 Pepin proved these and similar recruits.
Using also analogous facts proved by Sylvester in 1879, we can state whether
or not any proposed number, not exceeding 100, is a sum of two rational
cubes.
There are 42 papers (pp. 582-8) on the problems of finding numbers
in arithmetical progression the sum of whose cubes is a cube or a square.
If F(jtj y, z) ^ 0 is a homogeneous cubic equation with rational coefficients
and if P is a rational point (i.e., having rational coordinates) on the curve
F=0, the tangent at P outs the curve in a new rational point, called the
tangential to P. Similarly, the secant through two rational points on the
curve cuts it in a third rational point. Curiously enough, the analytic

XVI P ЛЕГ АСЕ.
equivalents of these facts were obtained by Cauchy in 1826 without their
geometrical setting. Levi in 1906-5 denned a configuration of rational
points on a cubic curve without double points to be the set of all rational
points which can be derived from one or more rational points by the opera-
operations of finding the tangential to a point of the set and of finding the third
intersection of the curve and the secant joining two points of the set. In
1917 A. Hurwitz called such a set of points a complete set and obtained
theorems on the number of rational points on the cubic curve. Mordell
made use of the invariants of Fm
The problem of finding n rational numbers the cube of whose sum in-
increased (or decreased) by any one of the numbers gives a cube was treated
for я — 3 by Diophantus and his commentators, hy Ludolph van Ceulen in his
Dutch work on the circle, by van Schooten, J, Pell and others—the simplest
answer being that by Hart (p. 611),
Chapter XXII devotes 57 pages to reports on 400 papers on Diophantine
equations of degree 4. Fermat's proof of his challenge theorem that no
rational right triangle has an area which is a rational square is of special
interest, as it illustrates in detail his method of indefinite descent; his proof
also shows that the difference of two biquadrates is never a square» Leibniz
left a manuscript giving a proof,
Fermat affirmed that the smallest rational right triangle whose hypote-
hypotenuse and the sum of whose legs are squares has its sides expressed by
numbers of thirteen digits. The problem is equivalent to that of finding two
numbers (for n numbers, pp. 665-7) whose sum is a square and whose sum
of squares is a biquadrate, and was proposed in this form by Leibniz and
treated several times by Euler, and at great length by Lagrange in 1777, who
found it necessary to solve several equations of the form ах?-\-Ъу*^cz*.
The extensive literature on the latter equation is reviewed on pp. 627-634;
some of the methods employed apply also when there occurs a term cb^y2
in the equation (pp. 634-9)»
Just as in algebra no general equation of degree exceeding 4 can be
solved by radicals, so in Diophantine analysis nearly all the problems for
which solutions have been found are those which reduce finally to the
question of making a given binary form / of degree ^4 equal to a square or
higher power. Among the methods (pp. 639-644) of making a quartic
function f(x) of special type equal to a square are the rather obvious
methods of Fennat; the method of Euler of reducing/to the form P*-f-Q#,
where Pt Q, R are quadratic functions of s, so that /= (P-\-QyJ becomes an
equation quadratic in x and in y; and the invariantive methods of Mordeli
and HaentzscheL Eider's method is similar to that employed by him in
the problem of the multiplication of an elliptic integral; Jacobi noted a
generalization by use of AbePs theorem (p. 641).
Euler, after solving A*-\-BA— №+£>* by several methods, stated (p. 648)
that it is impossible to find three biquadrates whose sum is a biquadrate,
and that he believed it possible to assign four biquadrates whose sum is a
biquadrate. But his investigation was incomplete and led to no example»

Pbefacb. xvii
ТЪе fiist example, 304+12СУ*-|-272*-|-315<^353<, was found by Nome Ш
1911* In the meantime various writers gave examples of five or more
biguadrates whose sum is a biquadrate and cases of equal stuns of bi-
quadrates.
Chapter ХХШ, on equations of degree >4, will doubtless be more
useful than any other chapter in the volume since it reports on the papers
which offer general methods of attacking Diophantine equations. Lagrange
showed how to use continued fractions to solve f—c7 where / is a binary
form of any degree» Runge and Maillet obtained conditions for the
existence of infinitely many pairs of integral solutions off(x, y) = 0, where
f is an irreducible polynomial with integral coefficients, Thue proved the
useful theorem that, if U(x, y) is an irreducible homogeneous polynomial
of degree >2 with integral coefficients and с is a given constant, U—C
has only a finite number of pairs of integral solutions. Maillet gave a
generalization (p. 675) to non-homogeneous polynomials U,
Hilbert and Hurwitz, in their joint paper of 1890-1, proved that any
homogeneous equation with integral coefficient which represents a curve
of genus zero can be transformed birationally into a linear or quadratic
equation. Рощсагё in 1901 proved the same theorem and found when
a curve of genus unity can be transformed birationally into a curve of
order p. The related later papers are cited on p. 677.
It is convenient to define at this point the product
F(xy y, . . ., z)^n(x-h^-h . . . -for*-1*),
extended over all the roots a, , „ -of any irreducible equation of degree n
with integral coefficients, to be the norm of the general number х+ау+ . , .
of the algebraic field determined by a. Dirichiet noted that F^\ has
infinitude of integral solutions except when the field is an imaginary quad-
quadratic field. If the field is real and if F can take a given value, it takes that
value for an infinitude of sets of integers x9 . . ,, z* Also Poincare (p, 678)
discussed this problem F=g. Lagrange (pp. 570, 691) proved in effect
that the norm of a product equals the product of the norms of the factors
and hence solved F(X, Yt . . ,,Z) = F~, where V^F(x, y, . # ,, a). This
method is of considerable power in seeking special solutions of various
types of equations. The particular case tf-\-ny*-\-r№—Snxyz occurs in
the papers on pp. 593-5* This case is also a special case of another type
of equations of general degree obtained by Maillet from the theory of
recurring series (p, 695). A, Hurwitz'e complete discussion (p, 697) of
the positive integral solutions of z\+ . , . +a£=sxi • . . x* furnishes a
model for thoroughness which may well be imitated by writers on Diophan-
Diophantine equations, too many of whom веет to be content with a special solution
of their problems.
Chapter XXIV deals with sets of integers with equal sums of like
powers. For example, a, b, с and a-\-h-\-c have the same sum and same
sum of squares as a+6, a+c, b-\-c. Of the seventy papers on thifl topic,
only five are prior to 1878. On pp. 714-6 is noted the connection of this

xviii Preface.
problem with the older one of rapidly converging series convenient for the
computation of logarithms, in which we desire two polynomials in x which
differ only in their constant terms and have exclusively integers as their
roots»
Chapter XXV furnishes a typical example in the theory of numbers of
the contrast between the ease with which empirical theorems are discovered
and the difficulty attending a complete mathematical proof. On the basis
of numerical experiments, Waring announced in 1770 the empirical theorem
tha*, every positive integer is a sum of at most 9 positive cubes, a sum of
at most 19 biquadrates, and in general a sum of a limited number of positive
tfith powers. The last fact was first proved in 1909 by Hiibert, although
his investigation does not determine the precise value of the number Nm
such that every positive integer is a sum of at most NM positive mth powers.
About the year 1772, J. A. Euler stated that Nm^v-\-2m-2t where v is
the largest integer <C/2)"\ Just before 1859, Liouville proved that
JV^53 by means of an identity equivalent to
and the fact that any positive integer n is expressible hi the form ?
-f sj+sj, so that 6n* is a sum of 12 biquadrates. But any positive integer
is of one of the six forms 6p, 6p+l, , . ., 6p+5, while p=n\+r%+r%+n\.
Thus 6p is a sum of 4X12 biquadrates. Since 1, . . ., 5 are sums of as
many units, each a biquadrate, we have iV4^4Xl2-f-5, Martlet was the
first to prove, in 1895, that JV* is finite, in fact ^21, Later writers suc-
succeeded in proving that ЛГЛ=*9, In his proof that ЛГЖ is finite, Hiibert
employed a five-fold integral, while later writers have given an algebraic
proof. Quite recently, Hardy and Littlewood gave a proof by use of the
theory of analytic functions and showed that #,»^(т—2Jm~1+5, which
gives 9 cubes, 21 biquadrates, 53 fifth powers, ete. Earlier papers (pp.
726-9) gave elementary proofs that every positive rational number is a
sum of three rational cubes and a sum of four positive rational cubes.
The final chapter devotes 46 pages to reports on more than 300 papers
on Format's last theorem, which states that it is impossible to separate
any power higher than the second into two powers of like degree, and the
more general trinomial equation азГ+Ьу^сг1, and congruence of the same
form. In letters and in annotations to his copy of Diophantus, Fennat
announced many interesting discoveries in the theory of numbers, usually
with the statement that he possessed a proof. All of these facts have since
been proved with the exception of his "last theorem" above, for which
he stated that he had found a truly remarkable proof» If there was an
oversight in his proof it was certainly not one of the foolish errors com-
committed in the past decade in the thousands of efforts to secure a large cash
prize. Fermat proposed the cases of exponents 3 and 4 (p. 545, pp. 616-7)
as challenge problems to the mathematicians of his time. The general
case has remained a challenge problem to the mathematicians of the sub-

Preface. xix
sequent three centuries» At intervals during the past century, leading
scientific academies offered one of their prizes for a proof. The dignity of
this famous theorem was injured by the offer of a very large prize ш 1908.
Since only printed proofs may compete, the gain thus far has gone to the
printers; in this history no mention will be made of the very numerous false
proofs called forth by this last prize.
Fermat's last theorem is not of special importance in itself, and the
publication of a complete proof would deprive it of its chief claim to atten-
attention for its own sake. But the theorem has acquired an important position
in the history of mathematics on account of its having afforded the inspira-
inspiration which led Kummer to his invention of his ideal numbers, out of which
grew the general theory of algebraic numbers, which is one of the most
important branches of modern mathematics.
Although Gauss had proved in 1832 that the laws of elementary arith-
arithmetic hold also for complex integers (numbers like 5-|-7V-vi) and made a
brilliant application of them in his investigation of biquadratic residues,
the theory of algebraic numbers was really born in the year 1847» For it
was then (pp. 739, 740) that the mathematical world became definitely
conscious of the fact that complex integers uo+^i^-h - « » -l-an-ir*, where
the a's are ordinary integers and r is an imaginary nth root of unity, do
not in general decompose into complex primes in a single manner, do not
possess a greatest common divisor, and hence do not obey the laws of
elementary arithmetic. This historical fact came to light through dis-
discussions of lacunse in the attempted proof by Lame that, if n is an odd
prime, zn-ty4=z* is not satisfied by such complex integers. Other errors
of the same nature were made in the same year by Wantzel and by so great
a mathematician as Cauchy. Curiously enough, Kummer himself made
the error, ш a letter of about 1843 to Dixichlet, of assuming that factoriza-
factorization is unique, so that his initial proof of Fermat's last theorem was
incomplete. But Kummer did not stop with the mere recognition of the
fact that algebraic numbers do not obey the laws of arithmetic; he sue-
oseded in restoring those laws by the introduction of ideal elements, this
restoration of law in the midst of chaos being one of the chief scientific
triumphs of the past century.
Although the theory of algebraic numbers appears to be a powerful
tool especially adapted to attack Fermat's last theorem, it has not yet led
to a complete proof of it» Numerous facts have been obtained by a variety
of more elementary methods» Until the theorem is actually proved, it will
obviously be unwise to attempt to weigh the importance of any particular
fact or method. Hence no further analysis will be given here of the con-
contents of the long Chapter XXVI which ia itself a condensed history of
Fermat's last theorem. Moreover this subject is one of those for which
the subject index gives a rather minute classification of the subject matter»
In the preceding summary mention was made of only the most im-
important of the upwards of 5,000 writings upon which report has been made

xx Preface.
in the text. While many of these papers are of minor importance, the aim
has been to give an exhaustive account of the literature on the subject
rather than a selective account reflecting the author's imperfect views as to
relative importance. This wort is intended as a source book not merely
for the fastidious professional mathematician, but also for the larger number
of amateurs who find endless fascination for the "queen of the sciences,1'
whose rule began centuries ago and has continued without interruption
to the present.
Unfortunately, following the practice of Diophantus, many writers on
this subject have been content with a special solution of their problem,
obtained by making various assumptions which simplify the analysis. A
report which would give merely the final formulas in such a paper, without
indicating also the restrictive assumptions, would be useless. Instead, there
is given here a summary of the essential steps in the proof, and this plan is
followed especially in the case of papers not to be found in the average
large library. These papers which give only special solutions of the problem
attacked have at least the value of showing that the problem is not im-
impossible. Moreover, an examination of many such papers reveals the fact
that there are a few constantly recurring types of auxiliary Diophantine
problems (such as that of making a quartic function equal to a square),
whose complete solution would permit the complete treatment of a very
large number of problems, and hence suggest specially useful subjects for
thorough investigation. Since there already exist too many papers on
Diophantine analysis which give only special solutions, it is hoped that all
devotees of this subject will in future refrain from publication until they
obtain general theorems on the problem attacked if not a complete solution
of it. Only in this way will the subject be able to retain its proper position
by the side of other virile branches of mathematics.
It was initially planned to give this work the title "topical history of the
theory of numbers"; but the word topical was omitted at the advice of a
prominent historian. It is inconceivable that any one would desire this
vast amount of material arranged other than by topics. Again, conven-
conventional histories take for granted that each fact has been discovered by a
natural series of deductions from earlier facts and devote considerable
space in the attempt to trace the sequence. But men experienced in
research know that at least the germs of many important results are dis-
discovered by a sudden and mysterious intuition, perhaps the result of sub-
subconscious mental effort, even though such intuitions have to be subjected
later to the sorting processes of the critical faculties. What is generally
wanted is a full and correct statement of the facts, not an historian's per-
personal explanation of those facts. The more completely the historian
remains in the background or the less conscious the reader is of the his-
historian's personality, the better the history. Before writing such a history,
he must have made a more thorough search for all the facts than is necessary
for the conventional history. With such a view of the ideal self-effacement
of the historian, what induced the author to interrupt his own investigations

Preface. xxi
for the greater part of the past nine years to write this history? Because
it fitted in with his conviction that every person should aim to perform at
some time in his life some serious, useful work for which it is highly im-
improbable that there will be any reward whatever other than his satisfaction
therefrom. Certainly, the eight mathematicians mentioned below, who co-
cooperated with the author, are justly entitled to enjoy the same satisfaction
from their work.
Concerning the various sources of references consulted and the various
libraries in America and Europe in which the material was collected, the
remarks made on page XI of the Preface to Volume I apply also to the
present volume. In particular, those references in the Subject Index of the
Royal Society Catalogue of Scientific Papers, Volume I, 1908, which relate
to Diophantine analysis were used not only in the preparation of the manu-
manuscript, but were checked on the proof-sheets. The references to Diophantm
follow the usual numbering and hence not that in the second edition by
Heath.
The reports in Chapters XI-XXVI have been checked by the original
papers in case they are to be found in Chicago. The computations occurring
in the reports in Chapters XXI-XX1V were checked by the author and
various errors in the original papers were detected. Moreover the reports
in four chapters were read carefully and critically by an authority on the
subject of the chapter as follows: Chapter III on partitions by Major
P. A. Mac Mahon, Chapter XXIV on sets of integers with equal sums of
like powers by E. B. Escott, Chapter XXV on Waring's problem by A. J.
Kempner, and Chapter XXVI on Fermat's last theorem by Я. S. Vandiver.
A high degree of accuracy and clearness for these Chapters III, XXI-XXVI
was especially desired since they are the ones which will be most frequently
consulted. Also Chapters I-XII were read minutely by Kempner, thanks
to whom various imperfections and errors have been removed. Further*
more, the proof-sheets of the entire volume were read by R. D. Carmichael,
A. Cunningham, E. B. Escott, A. Gerardin, and E. Maillet, each of whom
has written extensively on Diophantine equations and made very valuable
suggestions on the present work. To these eight experts, who gave so
generously of their tune to perfect this volume, is due the gratitude not
merely of the author but also of every devotee of Diophantine analysis who
may derive benefit or pleasure from this history.
Miss Minna X Schick read the proof-sheets of the first eleven chapters
and compared them with the original manuscript, for which purpose the
authorities at the University of Chicago considerately relieved her of the
duties connected with her fellowship in mathematics. Mrs. Louise M*
Swain, who had just completed a year of postgraduate studies in mathe-
mathematics at the University of Chicago, read the proof-sheets of the last
fifteen chapters, checked the many cross-references throughout the volume,
constructed and checked the author indexes, helped to check the references
with the Royal Society Catalogue, and checked the page-proofs with the
galleys and separately for various types of faults. The author is under

xxii Preface.
great obligations to these gifted young women for the many improvements
in the book due to their accuracy and alertness. In addition to all this help,
the author has devoted a large part of hie time for fifteen months to the
proof-sheets, comparing them with his original notes, checking computa-
computations, comparing reports and readers' suggestions with the original papers,
adding reports on current papers, repeating the work done on the manu-
manuscript of examining minutely all the reports for results needing citation
elsewhere by cross-reference, and inspecting every change made in the proof.
Readers are requested to supply, for insertion in a concluding Volume
III on quadratic and higher forms, residues, and reciprocity laws, notices of
errata or omissions, as well as abstracts of the few papers marked by the
symbol * before authorsJ names to signify that the papers were not avail-
available iUl lt^iJil.
L. E. Dickson
April, 1920.

TABLE OF CONTENTS.
СВЛРТКЙ. PAGE.
I. FoLYGONAb, PtHAMIDAI, АТЯЪ FlGURATE NUMBERS .... - . . . 1
II, Lintah Diofbahtinz Equations and Congruence 41
Solution of ax+6y=e . . ...... 41
Solution of fL£=b (mod m) without Fermat's theorem ...►■♦ 54
Solution of ox =b (mod m) by Fermat'a or Wilson's theorem ., . 55
Chinese problem of remainders..., ..»,..♦ ..♦,.. 57
Number of positive integral solutions of ax-\-by — n...... T., r. T ... 64
One linear equatioD in three unknowns ., ♦ . ♦., 71
One linear equation in n>£ unknowns. .. , .,. 73
System of linear equations .. . . ♦. ♦ 77
One linear congruence in two or more unknowns. 86
System of linear congruences .. ._,... 88
Linear fortns, approximation .„.,.,. „ 93
III. Partition» . . 101
IV. Rational Right Triangles .,. . 165
Methods of solving jt'-j-^^z' .... r. ,. 165
Sides divisible by 3, 4, or 5 ♦ 171
Number of right triangles with a given side ,.,. 172
Right triangles of equal area. k t ....... 172
Right triangles whoee areas have a given ratio 174
Other problems involving only area. 175
Problems involving area and other elements 176
Right triangles whoee legs differ by unity Ш1
Difference oraum oMega given r..,,,,. 183
Two right triangles with equal difference* of legs, and larger leg of one equal
to hypotenuse of the other,.,, » 1&4
Problems involving the sides, but not the area ...........►-, 186
Right triangles with a rational angle-bisector 188
Tables of right triangles with integral sides. ■.. 139
V» TnMNGLtfs, Quadrilaterals, and Тетванепва, 191
Rational or Heron trianglee ♦ ^ 191
Pairs of rational triangles 201
Triangles whoee sides and medians are all rational. - 202
Triangles with a rational median and rational aides, parallelograms wtth
rational sides and diagonals 205
Heron triangle* with a rational median, Heron parallelograms ■.. 207
Triangles with one or more rational angle-bisectors. . 209
Triangles with a linear relation between the angle* 213
Tti&ngies whoee area need not be rational. .. .... r, 214
Rational quadrilateral^. 216
Rational inscribed polygon^. 221
Rational pyramids and trihedral angle* ,, 221
VI. Sum of Two Squakeb .,. 225
VII. Sum of Three Squared 259
VllL Sum of Fora Sqpahb» 275
IX, Son of n Square» ,.,, 305
Representation &* a sum of 5 or more squares.. . ►- 305
Relations between equares 318
X, Number or Solutions of Quadratic Congruences in n Unknowns. ...,.., 325
XI. Гюипчл'й Нкшез от EiflHTErd* Articles 329
XII. Pkll Equation; ox'+bje-j-c mad* a Square т 341
XIII. Further Simole Equations or твв Second Degree: ........... .., 401
Equations1 linear Ы one unknown,., „ T 401
Solution of z?—if-4ft, ,, 402
i ^ 404
........r ,. 407
♦,.♦> ,.„♦ » 408

xxiv Тлвье of Contents.
Сяапхв*
tf 412
W-O texeept *-Hf=2*) 419.
Farther angle quadratic equations in n^3 unknowns . 428
XIV. Squab» in Аяггнмжпсль on Geometrical Progression. 435
Tbreeaquarainarithmeticalprogression . 435
Four aquaree in arithmetical progreeBion ,....,. r ^. r...... 440
Numbers in arithmetical progression all but оюе being squares.,.,..,.,... 440
вошиев m geometrical progression , . - -. 441
XV. Two or mobs Linear Functions mads squares ....... 443
XVI. Two Quadratic Рожгпоад от Owe or Two Unknowns hadb Squares 469
Congruent numbers; z*d=fc both squares.. 459
Concordant forms: л?-И»Л a^+ntf1 both squares. Related problems...... 472
**+* and *+f* both squares , 478
x'+H-l and 4*-?>~l both square* 479
3*+#z?+V «^ ^4-^^+*^ both squares 480
Two functions of one unknown made squares .......,..,,.,......- 4Я2
Othe- pairs of quadratic functions made squares. .. 483
XVII. Stbtxub or Two Euuatjonb or Dboxes Two , ,. ♦♦.,.,... ASS
Two quadratic equation» in two unknowns - - - ..*.._*.» 485
Problems of Hertm and Flaaude; i^nczalixatiiine 4&5
System ^-2^-1, x"-2i»-l 487
Number and ite square both ■шив erf 2 consecutive squama 488
ЫиоеЦапеош fysteme of two equations 480
XV1H. Т^ккк or morb Quadratic Fcnctionb or Ose ok Two Ukknowxs made
Squawbs 491
dV-hd^, W^+er, - . . mmdeaquarce 491
ТЪгее linear and quadratic functions of z, у made squares..,.,..,,... 462
Four quadratic functions of xt у made squares ♦ ■ 493
xix. ЗтЕгтв of Thbks or more Equatkw*s op Dbgrbr Two in Тнвжк cm mors
Unknowhb 497
rf+tf», *•+*■, if+# аП aquana 497
Four eqttaree wboee sums by threes are squaree. 502
Three equares whose differencee axe aquanefr . ..,,,....... 505
Three squaras, mm of any two les third a aquare 507
Further aete of three or more linear functions of three or more squaree made
squares „ , ♦♦..««,,,♦,,♦._♦,,♦♦,♦..,.., 50&
Quadratic fortuB in x and y, x and zf у and z7 made squares 511
xy+Oy xz+a, yz+a all squares ,...t4..,..,* 513
Problems related to the last one ♦ 515
Product of any two of 4 or 5 numbere increased by I a square............ 517
Product of any two of 4 numbers increased by n a square 519
Other product* of nombciB in pairs increased by linear functione made
squaree k. __,.,,..,♦,_..._,.♦ ^ ..<,... 520
further equations whose quadratic terms are sums of products 522
equares increased by linear functions made squaree. 524
Square of each of three numbers phis product of remaining two a square... 526
MisceOsneou» systems of equations of degree two 527
Rational orthogonal substitutions. 530
XX» Quadratic Fork mads an nta Power ,,...,.,.. 533
Binary quadratic form made a cube ..,.>.,.... 533
Binary quadratic form made an nth power. -,. - 539
ttiti^-J- * ■ ■ +вА.* made a cube or higher power. 543
XXI. Equations of Dbghxx Three 545
Impossibility of a4V=** 645
Two equal sums of two cubes ..».......<.....»,.». 550
Three equal sums of two cubes 56t
Solution of 2(?+&)=y»+P 562
Relations between five or more cubes , 563
Sum of three cubes made a square 566
Binary cubic form made a cube 566

Table of Contents. xxv
снартин. faoe.
Bin&ry cubic form made a square * . —.,. ♦ 569
Numbers the sum of two rational enbes: x*+y*;=.4z* 572
Sum or difference of two cube» a square 578
Sum of cubes of numbers is arithmetical progression a cube 582
Sum of cube* of numbers ш arithmetical progression a square. 5S5
Homogeneous cubic equation P{xt yf z) —0. ..., ♦ 588
Ternary cubic form made a constant _ r. ..,...,..,,.,,.. 503
Miscellaneous single equations of degree three. -*... - - • 595
Systems of equations of degree three in two unknowns , . 599
Syeteme of equations of degree three in three unknowns 602
To find r numbers the cube of whose sum increased (or decreased) by any
one of them gives a cube .,,.>.. . _. 607
Systems of equations of degree 3 in *^4 unknowns,,. . 612
XXII. Equations or DkqWSE Foub 615
Sum or difference of two btquadr&tee never a square; area of a rational tight
triangle never a square.,«.,./ ....._ ....... 615
2х*—у4^П; tight triangles whoee hypotenuse and вила of legs are squares;
*+•=£«, x+y-A*. Abo, ^-2^-0,^+8^ = 0 620
627
jf+y square ...^ .,..,......,..<. . . 634
Quart t с function made a square ...♦.,,,. 639
A*+B*=&+& 644
A*+hB*=C*+h& ,... 647
Sum of three btquadratea never a biquadrate. , 648
Sum of four or more biquadr&tee a btquadrate .....,..,..., 64S
Equal sums of biquadratic , ♦ ♦,. ♦ ,.. 663
Relations involving both btquadrates and square» 657
Miscellaneous Hingte equutiona of degree four ..,..,.... 660
To find n numbere whoee sum is a square and sum ofsquareeiea btquadrate. 665
Miscellaneous systems of equations of degree four 667
XXIII. Equations of Degree n 673
Solution of f-Cj where /is a binary form, ♦,,.,.,,. 673
Conotitions for an infinitude of solutions of /(jt, y) =0. ► 674
Rational points on the plane curve fix, у, г) -0.,,,,., ♦■♦►., 675
Equations formed from Lncar functions. , <. ► 677
Product of consecutive integers not an exact power 679
Further properties of products of ooneecutive integers. *« ♦ ..,..,. 680
Sum of nth powers an nth power. 682
Two equal sums of nth powers....,..,. .,.>.,.,,* 684
Miscellaneous results on eume of like power». • 684
Rational solutions of &=y* ., 687
Product of factors (дг -Н1)/дг equal to such a fraction.. ..> 687
Optic formula, l/xi+ - . . +lfx.^lfa 688
Miscellaneous single equations of degree n>4... r 691
Miscellaneous systems of equations of degree n>4 • • - - 700
XXIV. Sets op Integers wtth Ecual Sous of Likjc Powers r. r.. r. 705
An equivalent problem in the theory of logarithms 7l4
XXV, Waking's Problem ani> Relateo RssVLTb 717
Waring's problem 717
Numbers expressible as sums of unlike powers. -. - 725
Every number я вит of three rational cubes....,.,,.. 726
Every positive number а вит of 4 poeitive rational cubes.., 727
XXVI- FjauiATblMOTTHEttftEMjiLr+bAr'^c^ANDTrata 731
Author Indzx ,...,.... ,....<.....,♦ , 777
Susjkct Index 79&

CHAPTER I.
POLYGONAL. PYRAMIDAL AND FIGURATE NUMBERS.
The formation of triangular numbers 1, 1 + % \ + 2 + 3, ■ ■ *, and of
square numbers 1, 1-1-3, 1 + 3 + 5, - - -, by the successive addition of
numbers in arithmetical progression, called gnomons, is of geometric origin
and goes back to Pythagoras1 E70^501 B.C.): 9
• • • • I
• ♦ • • • •
If the gnomons added are 4, 7, 10, ■ - ■ (of common difference 3), the
resulting numbers 1, 5, 12, 22, ■ • ■ are pentagonal. If the common differ-
difference of the gnomons is m — 2, we obtain nhgonal numbers or polygonal
numbers with m sides.
In the cattle problem of Archimedes (third century B.C.), the sum of
two of the eight unknowns is to be a triangular number (see Ch. XII).
Speusippus,* nephew of Plato, mentioned polygonal and pyramidal
numbers: 1 is point, 2 is line, 3 triangle, 4 pyramid, and each of these
numbers is the first of it» kind; also, 1 + 2 + 3■ + 4 = 10.
About 175 B.C., Hypsicles gave a definition of polygonal numbers
which was quoted by Diophantus* in his Polygonal Numbers, "If there
are as many numbers as we please beginning with one and increasing by
the same common difference, then when the common difference is 1, the
sum of all the terms is a triangular number; when 2, a square; when 3, a
pentagonal number. And the number of the angles is called after the
number exceeding the common difference by 2, and the side after the
number of terms including I/1 Given therefore an arithmetical progres-
progression with the first term 1 and common difference m — 2, the eum of r
terms is the r-th *rc-gonal number8 p'^
The arithmetic of Theon of Smyrna* (about 100 or 130 A.D.) contains
32 chapters. In Ch. 15, p. 41, the squares are obtained from 1 + 3 — 4,
>F. Hoefer, НЙЫп» des ^tbematiques, Paris, ed. 2, 1879, <*L 5, 1903, 96-Ш; W. W. R.
Ball Math. G**ette, 8.1915, 5-12; M, Cantor. Gaechkhte Math., 1, ed, 3,1907, «ИЬЗ,
252,
*Thedogmnena arithmetical, ed. by P. Aet, Leipzig, 1&17, 61,62. For a French tran«l. and
notee, вее P. Tanoery, Pout I'hiatoire de U ecience Hellene, Paris, 1887, 386-390 C74).
* Denoted by f*"} in Encyc. Sc. Math., I, h, p. 30.
4 Theonls Stnyrnaei Platonlci, Latin transl. by Iem&el Buliialdi, 1644. Cf. Expoeitio rerum
matbemAticanun ad legendum Flatonem utiliumt ed., К Hiller, pp. 31-40.
1

2 History of the Theory of Ntjmbjges. [Chap. I
1 + 3 -h 5 ~ 9, etc. In Ch. 19, p. 47, the triangular numbers are denned
to be 1, 1 + 2, 1 + 2 + 3, • •». In Ch. 20, p. 52, the squares are obtained
as before and the pentagonal numbers are obtained by addition of 1, 4, 7,
10, "^ In Chapters 26 and 27, pp. 62-64, pentagonal and hexagonal
numbers are shown by dots forming regular pentagons (as ш the figure
on the preceding page) or hexagons. Ch, 28, p. 65, gives the theorem that
the sum of two consecutive triangular numbers is a square. In Ch. 30,
p. 66, is defined the pyramidal number P^ = p\> + p* 4- — - 4- P*«
Nicomachus* (about 100 A.D.) gave the same definitions and results
as did Theon of Smyrna and perhaps gave them slightly earlier. Ch» 12
gives the theorem on consecutive triangular numbers:
2 + 2 ~T'
also the corresponding theorem that the sum of the rth square and (r — l)th
triangular number is the rth pentagonal number, just as a pentagon is
obtained by annexing a triangle to a square. He gave the generalization
(apart from the notation):
These theorems are illustrated by means of the following table:
Triangles
Squares
Pentagons
Hexagons
Heptagons
1
1
1
1
1
3
4
5
6
7
6
9
12
15
18
10
16
22
28
34
15
25
35
45
56
21
36
61
66
81
28
49
70
91
112
36
«4
02
120
148
45
81
117
153
189
65
100
145
190
235
Each polygon equals the sum of the polygon immediately above it in
the table and the triangle with 1 less in its side ^triangle in the preceding
columnJ; for example, heptagon 148 is the sum of hexagon 120 and
triangle 23.
Each vertical column is an arithmetical progression whose common
difference is the triangle in the preceding column.
In Ch. 13 he remarked that just as polygonal numbers arise by suinming
the simple arithmetical progressions, so by summing the polygonal numbers
one obtains the like named pyramidal numbers,—triangular pyramid from
the triangular numbers, pyramid with square base from the squares, etc»,
the base being the largest polygon.
* Ixrtroductio arithmetic* (ed*, Hoche), 2,1866, Book 2, Cha. fr-2O. Cf. G. H, F> Nceselmann,
Algebra dcr Giiecbeo, 1843, 202.

Chap. II POLYGONAL, PyHAMIDAL AND FlGURATE NUMBERS. 3
Plutarch/ a contemporary of Nicomachus, gave the theorem that if
we multiply a triangular number by 8 and add 1, we obtain a square:
This theorem was given by Iamblichus7 (about 283-330 A.D.), who
treated at length (pp. 82-176) polygonal and pyramidal numbers,
Diophantus* (about 250 A.D.) generalized this theorem and proved by
a cumbersome geometric method that
A) 8(m - 2)P; + (« - 4)' = {(m - 2)Br - 1) + 2\\
and spoke of this result as a new definition of p equivalent to that of
Hypsicles. Diophantus gave a rule for finding r, equivalent to the solution
of A) for r, and a rule for finding p equivalent to
4 ' "• 8(m - 2)
but did not give the equal simpler expression
C) pi ~ §rB 4- (m — 2)(r — 1)J.
In fact, starting with B), he gave a long geometric discussion to find the
number of ways a given number can be polygonal, but made little headway
before the abrupt termination of the fragment» G. Wertheim9 gave a
lengthy continuation in the same geometric style which eventually leads
to the geometric equivalent to C) and remarked that we can readily find
from C) the ways in which a given number p can be polygonal: Express
2p as a product of two factors > 1 in all possible ways; call the smaller
factor r; subtract 2 from the larger factor and find whether or not the
difference is divisible by г — 1; if it is, the quotient is in — 2, and p is
a pr^. Since m — 2 equals 2(p — т)/[т(г — 1)], the latter must be an
integer ^ 1, so that
For example, if p = 36, then r ^ & Since r divides 2p = 72, we have
r = 2, 4, 8, 3, 6, of which г = 4 is excluded. We get
36 = P« = plf = p4 = Pl-
In the Roman Codex Axcerianus10 D50 A.D.?) occur a number of
special cases of the remarkable formula for pyramidal numbers
6
*Pklr>nicac quaestion., II, 1003.
J In Ntcomachi Gerueeni arith. introd.p ed.p S. Teimuliua, 1668, 127-
•Polygonal Numbers. Greek text by P. Tannery, 1893P 1895. Eng!. transL by T. L.
Heath, Cambridge, 18S5,1910; German trend, by F* T. Poeelger, 1810, J, O, L. Schulz,
1822P and G. Werthdm, 1890; French transl. by G. Maseoutie, Paris, 1911. Cf. Nessel-
marni, Algebra der Griechen, 1842,462-476; M. Cantor, Geechicbte Math., ed,3,1,485-7.
•2dtBchrift fur Math* Phyeik, HiHt. Lit. Abl. 1897, 121-6. Reproduced by T, L. Heath,
Diophantuep ed. 2,1910, 256, where doubt ie expressed aa to the validity of the nestora*
tion in view of the ease with which the geometric equivalent of C) can be derived geo-
geometrically from that of B).
10 Cf. M. Cantor, Die RGmiecben Agrimeneoren, Leiprig, 1875, 95-127.

4 Histohy of the Theory of Numbers. [Chap, i
It gave pi = iCr* -f- r), vl - $D** -f- 2r), where the plus signs should be
minus. M. Cantor11 suggested the following probable derivation* By
factoring the numerator of B), we obtain
As known by Archimedes (b. Syracuse about 287 B.C.),
i + a + ...+г-*^, i. + *+...+f.-£^-- .
Hence
The Hindu Aryabhatta" (b. 476 A.D.) gave the formula
i . о , « i Kr+1) r{r+l)(r + 2) (r + I)8 - (r + 1)
1 + 8 + 6+'-'+—2— = e = 
for the number of spheres in a triangular pile, and hence for the rth
pyramidal number PJ of order 3, called also a tetrshedral number. The
Hindus of his time knew13 also that P\ = p; -f- PJ'1, whence
The above general formulas relating to polygonal and pyramidal numbers
were collected about 983 A.D. by Gerbert1' (Pope Sylvester II).
Yang Hui16 gave in his Suan-fa, 1261, the formulas
hhHi(i)()
for the sums of triangular numbers and squares.
Chu Shih-chieh,36 in 1303, tabulated in the form of a triangle the
binomial coefficients as far as those for eighth powers. This arithmetical
triangle was known37 to the Arabs at the end of the eleventh century.
Such a triangle was published by Petrus Apiamis.18
Many of the early arithmetics mentioned (some with fuller titles) in
Vol. I, Ch. I, of this History, gave definitions and simple properties of
M Die Rdmiechen Agrimens^ren, 1875, 122; Geechichte'd^M^^ 1, ed. 2, 519; ed. 3, 558.
a, H. G. Zeutben, Bibliotheca Mathematics C), 5, 1904» 103.
"French tranel. by L. Rodet, Jour- Aaiatique, 13, 1S79; Le^oite de calcul d^Aiyabhatta,
p. 13, p. 35,
UE. Lucae, La Nature (lUvucdes 8с1епвда), 14,1886,11,282-6: L'Arithm&ique en Batonfl
dans Tlnde au tempe de Cbvia»
" Geometrie, Che, 55-65.
» Y. Mikami, Abh, Geechichte Math- Wlas., 30, 1912, 85.
»/Ш„ 90, Cf. K, L. Biernatrici, Jour, fur Math., 52,1856r 87; Stifel"
17 M» Cantor, Geechichte Aer Math,, 1, ed. 3,1907, 887.
11 Em newe . . . Kauffmftna Kechnung , . . f Ingototadt, i527, title page. The latter was
reproduced by Ю. E. Smith, Rara Anth4 1908,156, who remarked that ho knew of oo
earlier publication of this Faecal triangle.

Л POLYGONAL, pYRAMlDAb AND FlGUBATE NtTMBBRS. 5
polygonal numbers; for example, Boethius,1* G. Valla," Martinus,"
Cardan,*5 J. de Muris^ Willichiue," Michael Stifel,*4 who gave a table
of figurate numbers (binomial coefficients), Faber Stapulensis,28 and F.
Maurolycus,* who gave
P. - ЗяГ1 + г, pi - 2?Г1 + Л
a treated (pp. 32-74) polygonal joumbers of the second order or central
polygonal numbers (the pentagonal being 1, 6, 16, 31, 51, 76, •••, when
in the second are counted the vertices and center of a pentagon), as well as
central pyramidal numbers {the pentagonal being 1, 7, 23, 54, 105, - • -)-
Also I* Unicormia,27 and G. Henisehiib."
Johann Faulhaber" treated polygonal and pyramidal numbers.
Johann Benzius" devoted twenty chapters to these and figurate numbers,
J. Kudolff von Graffenried51 noted that
\
the final number being 666 for r => в.
С. G. Bachet35 wrote a supplement of two books to the Polygonal Num-
Numbers of Diophantm The most important ones of hia theorems (when ex-
expressed as formulas) are as follows:
I, 10. A*' = Pi + Pi + *r(m - 2), pi = К + (m - З)^1-
п, 21. s
П, 28. n» + ад + 1 - (n +
11 Aritbmetica boetij, 1488, et<?*, Lib. 2, Cbpa, 7-17,
w De expetendia et fygjeadie rebve opvs, Alduar 1501, Lib. Ш.
* Are Aritbmetlciir 1513,1514; Arithmetica, 1519, 15-18.
* Practica Aiith., I637r etc.
■■ Arith. Speculativae, 1538, 5^62.
* I. Vvillkhii RefleUiani, Aritb libri traf l54Or 95-111.
" Axith. IntegA, 1544. See inferences 16-18, 50-52.
4 StJupuIeuH«r Jacobi Fabrir Ar»th. Boethi epitome, 1553, 54-65.
M Arith. libri dvo, 1575t 6-S, 14-21. Hiatorical remarts on same by M4 Fontana, Memorie
deU7 Irtituto Naadoimle IiiJ^ Mat., 2, Pt. lr 1808, 275-296.
" De PAnthmetica Vnivermle, 1598, 67-70.
toArtth. Ferfecta et DemonatniiA С1605Д, 1609,133.
n Cubicon Uwtg&rt€nr 1604 (ftbo in part 2 of Petnim Rothen, Arithmciica Philoeophica,
Numba^ 1608); Neuer Math. Kunat^eed, Uhn, I612r which notes that 1335 (men-
(mentioned ш the Bible, Daniel, XHi 12) is л pentagonal number whctfe root 30 IB a prooic*
number witii the pentagonal root б wh<*eroot 2 is pttmic^ while 2300 (Daniel, VIП» 14)
is tetiadecagonal whose mot 20 ш pronie, etc.; Numerue FigiiraUie, 1614, 24 pp.;
Miiacula Arithmetic*, Augppuig, 1622, a book chiefly on arithmetical combination*
giving the "Wuader Zahl" 666, tl^ Apocalyptic number mentioned in the Bible,
Revelations, Щ 18; «£. Reminding A- G. Kietner, Geechichte Math., Ш, Ш-62.
m Munuductio ad Nrmenrm Geometricvm^ Kempten, 1621.
" Arith. Logjkica Popularis, 1613, 238, 627.
11 Diophanti Alex. Arith., W2L

6 History of the Theory of Numbers, [Chat, i
His П, 27, relates to the formula of Nicomachus5 (Cb, 20)
1 = P> 3+5-2», 7+9 + 11=3*, 13+15 + 17+19 = 4', -.-,
from which follows the above formula П, 25, by addition (as in the
Codex Arcerianus10). Fermaf1 generalized this proposition by introducing
"colonne": In the arithmetical progression 1,1 + (m — 2), 1 + 2(m — 2),
* > * leading to m-gonal numbers, the first term 1 gives the first oolonne;
the sum of the next two terms diminished by та — 4 times the first tri-
triangular number 1 gives the second oolonne 2m; the sum of the fourth,
fifth and sixth terms diminished by m — 4 times the second triangular
number 3 gives the third colorme Qm — 9; similarly, the fourth oolonne is
8Cwi - 4) and the rth is r* + r*(r - l)(m - 2)/2. It follows (as noted
by Editor Tannery) that the НЪ. oolonne is the product of the rth m-gonal
number by r, and for m = 4 ia A The term colonne was not coined by
Fennat, as Tannery thought, but*4 was used by Maurolycus.*
J. Remmelin34 noted that 666 (ef. Faulhaber*») is a triangular number
with the root 36, which is a square with the root 6, while 6 is a pronic
number [of the form n2 + n] whose base 2 is also a pronic number.
Later we shall quote Bachet's empirical theorem that any integer is the
sum of four squares, made a propos of Diophantus IV, 31, In this connec-
connection Fermat" made the famous comment: "I was the first to discover the
very beautiful and entirely general theorem that every number is either
triangular or the sum of 2 or 3 triangular numbers; every number is either
a square or the sum of 2, 3 or 4 squares; either pentagonal or the sum of
2, 3, 4 or 5 pentagonal numbers; and so on ad mfinitum, whether it is a
question of hexagonal, heptagonal or any polygonal numbers. 1 can not
give the proof here, which depends upon numerous and abstruse mysteries
of numbers; for I intend to devote an entire book to this subject and to
eflfect in this part of arithmetic astonishing advances over the previously
known limits/' But such a book was not published. Fennat3' stated
the theorem in a letter to Mersenne, Sept», 1636 (to be proposed to St.
Croix); to8» Pascal, Sept. 25,1654, and Digby, June 19,1658. The theorem
was attributed to St. Croix by Descartes39 in a letter to Mersenne, July 27,
1638, Descartes40 gave an algebraic proof of Plutarchrs6 theorem that
8Af + 1 = Br -(- IJ. We shall often write Af or Л(г) for the rth tri-
triangular number т(т -f- l)/2, A or Ar for any triangular number, D for
any square, И, GB or Ш for a sum of two, three or four squares.
" Oeirvieij I, 34L ~ ~
« Werthcim, Zeitachr. МдШ, Vbye. 43, 1Ш, Hisfc.-lit. Abt., 41-42.
«Jobmne Lvdovico Remmelino, Structure Tabularvm qvadratarvm, 1627, Preface. The
book treata magic squares at length*
* Oeuvree, I, 305; French trwwl*, III, 252. E. Braseiiiiie, Fr&ie des Oeuvres Math» de P.
FermM, Mem. A<iad. Imp. Sc ТооЪшэе, D)r 3, 1853, 82.
17 Oeuvree, П, 1894, 65; Ш, 287.
11 Oeuvres de Fermat, II, 313, 404; Л1,315.
^Ocuvree de Iteecartea, II, 1808, 256, 277-^8 (editore' oommente); Xr 2»7 {statement of
the theorem in a posthumous MS.).
«Oeuvres, X, 298 (рмЛЬ. MS.).

Chap. IJ PoLYGOtfAb, PlRAMIDAb AND FlGURATE NUMBERS.
The rth figttrate number of order n is the binomial coefficient
Л V n
Thus fl is the rth triangular number pi, while fl is the rth pyramidal or
tetrahedral number PJ. In a comment on the Polygonal Numbers of
Diophantus, Fermat41 stated a theorem which, in the present notation, is
and called f[ the rth triangulo-triangular number.
In April, 1638, St. Croix proposed to Descartes the problem: "Trouver
un trigone ^triangular number] qm\ plus un trigone tetragone, fasse un
tetragone [square} et de rechef, et que de hi somme des c6tes des tetragones
resulte le premier des trigones et de la multiplication d'elle par son milieu
le second. J'ai donne 15 et 120. J'attends que quelqu'un у satisfasse par
d'autres nombres ou qu'il montre que la chose est impossible." The
problem, without the example, was proposed to Fennat (Oeuvres, II, 63)
in 1636, who did not solve it.
Descartes42 understood a trigone tetragone to be the square A* of a
triangular number, and proved that 15, 120 is the only solution if the
problem is understood to require two triangular numbers such that, if
either be added to the same A2, the sum is a square; while if one is per-
permitted to add both A3 and a new Л'2 to the second required triangular
number, the two triangular numbers may be taken to be 45 and 1035r since
45 + 62 = 0*, 1035 + 6*H-152 = 362, 36+9 = 45, 45 46/2-1035.
St. Croix did not admit the validity of Descartes' solution, and probably
meant a trigone tetragone to be a number both triangular and square (like
1, 36). The question would then be to find two numbers of the form
n(n -f- l)/2 such that, if a number both triangular and square be added
to eaehr there result two squares; further, the sum of the square roots of
these squares must equal the first required triangular number and must
also be the first factor n used in forming the second triangular number.
If, as seems intended, the numbers to be added to the triangular numbers
are to be identical, the only solution is 15, 120. Cf. G^rardin.220
Fermat41 proposed to Frenicle the problem to find a number which
shall be polygonal in a given number of ways* Neither gave a solution.
[Cf. Euler," end.]
John Wallis*4 derived by summation the expression for the general
triangular number (p. Ш), pyramidal number with triangular base P;
(p. 143), the sum (called trianguli-pyramidal number) of the latter for
r — l,2t * ,1 (p. 145), and the sum (called pyramidi-pyramidal number)
of these last for I *= 1,2, ♦. •. The values found are the expanded forms of
«Oeuvra, I, 341; Ftench trand., 11^^273, Abo, II, 70,"8M; French ttausl., ^
lettera to Mersennc, Sept., 1636, and to Robcrval, Nov. 4, 1636.
e Oeuvree, П, 18ВЙ, 158-I65r letter from Desc&rte» to Mcreenne, June 3, 1638.
« Oeuvrea, 11, 225, 230,435, June and Aug., 1641, Aug., 16S9.
** Arithmetic» Infiztitorvm, Oxford, 1666.

8 History of the Theory of Numbers.
the figurate numbers flt flt f[t fit so that his work amounts to a verification
of cases of
/;+1=/+/'+ ■■■+!:■
Frans van Schooten** quoted three of Bachet's rules, proving one.
On certain hexagons whose sum is a cube, see Frenicle* of Ch. XXI.
Fermat* proposed that Brouncker and Wallis find a proof of the pro-
proposition (which he himself could prove): There is no triangular number,
other than unity, which is a biquadrate.
Diophantus, IV, 44, desired three numbers which if multiplied in turn
by their sum give a triangular number, a square, and a cube» Let the sum
be x7* Then the numbers are a{* + l)/Bz*), 07**, i*fz\ Thus
Take 0 = x* - L Then A* = 2s* - У - 1. But
8дж + i = Ba + IJ = 16л2 - 87* - 7 = Dx - 5}*,
if x = (gy _|_ $* + 7)/(8a). Take у = 2, Й = 1; then x - 9 and the
desired numbers are 153/81, 6400/81, 8/81.
Bachet convinced himself by trial that д must be unity in order that
a = (8Ta H- 7 - F - 25)/{4S) be integral.
Fermat remarked that "Bachet's conclusion is not rigorous. Indeed,
let 7 be any number of the form 3n -(- 1* say 7 = 7. To make
2x2 - 7* - 1 = A
and hence 16s* — 8-71 — 7 = П, we may take the latter to be the square
of 4x — 3 [whence x = 115, Ь = 3} Nothing prevents us from gen-
generalizing the method, taking instead of 3 any odd number and making a
suitable choice of 7,"
G. Loria" remarked that the solution becomes evident if we replace
x* by x\ the problem did not require that the sum of the numbers be a
square.
Bachet*2 (p. 274) proposed the problem to find five numbers which if
multiplied in turn by their sum give a triangular number A, a square, a
cube, a pentagonal number, and a biquadrate. The sum of the latter
shall be x*. Let the square be (x2 — 1)*, the cube 8, the pentagonal number
5, and the biquadrate 1. Then A = 2x* — 15. Thus
8A + 1 = 16л2 - 119 = □,
say Dx - 1)\ Hence x = 15.
Ken6 R de Sluse" A622-1685) employed the triangular number?, the
square Ъ* and cube 21. Then q + b* -f- z3 « □ = (& + *}*, whence
* Exerritationnm M*tb., 1667, lib. V, 442-5?
u Oeuvresr Ш, 317, letter to Вфу, June, 1U58.
47 be scienxe eaatte ndl' &ntica Grecia, libra V, 13S.
"Ren&ti Francied Sfriaii, MesoUbum, -*, acceaait pare alter» de analyst et
Leodii ЕЫтшипц 1668,175.

■n Polygonal, Fyramidal and Figtjrate Numbers. 9
we may assign any values to g, #% n and find Ъ. Likewise for
T*fdiet's generalization, we may assign any values to all five products
other than the square 62, and find b.
A G6raxdin4fl noted that the simplest solution of Diophantus' problem is
by the three numbers (s2 + l)/2, P, x7 with a - **, 0 ~ хв, у = xt
Seta? 2Я-К Ь Then 6* - 2H2 = - 1, with the solutions (tf, $) - A,1),
/5 7) /29 41), etc., giving the numbers 5, 1,3; 61,49,11; 1741,1681,59.
' Rene F. de Sluse60 gave the table [cf, Stifel24]
0 11111
12 3 4 5
1 3 6 10
1 4 10
1 5
1
in which the numbers (like 1, 3, 3r 1) in a diagonal are binomial coefficients,
those in the third column are triangular numbers, those in the fourth
column are pyramidal numbers with triangular base, those in the fifth are
triangular pyramids of the second order,
B. Pascal*1 gave the same table and noted that any number in it is the
sum of the numbers in the preceding column and hence (p. 504) is the sum
of the number above it and that immediately to its left» He noted (p. 533)
that n(n + 1) — * (n + к — 1) is divisible by Л I, the quotient being a
figurate number.
G. W. Leibniz" gave a table formed by the diagonals (as 1, 2, 1) of the
above table*
J. Ozanam** found pairs* of triangular numbers 15 and 21, 780 and 990,
1747515 and 2185095, whose sum and difference are triangular» Their
sides are 5 and 6, 39 and 44, 1869 and 2090- Polygonal numbers are treated
in the English translation by C. Hutton, London, 1803, pp. 40-47, p. 60,
Pierre Remond de Montmort64 cited special cases of A), due to Dio-
Diophantus* *
F. С Mayer* defined "generalized figurate" numbers
11 SpMnx-Oedipe, в, 1Ш,42.
" MS. 10248 <Ь foads Utm, BMotheque Nationale de Paris, f. 187.
* TWt6 du triangle arith., Parisr 1665 (written 1654); Oeuvn», Ш, 1908, 466-7»
•Leibnb Math. Schriftenr ed.p C. L Gcrhardtr Vll, 101,
Others are 171 and 105, 3741 and 2145* Gdratfdin gave a general discussion in Sphinx»
Oedipe, 1914,113.
Racreatione math, et phys.r 1, 1696, 20; new eda,, 1723, etc,
Меш. Acad. Roy. Sc., 1701. Easai d*Analyse aur les Jeiucde Hazuxda, 170S;ed, 2,1713,17,
Maieno, Comm. Acad. Petrop., 3, ad annum 1728 Cl726], 52.

10 History of the Theory of Numbers.
which for n = 2, 3r 4 include the polygonal numbers and the pyramidal
numbers of the fiist and second kind, the number of sides being a-f2.
К Euler6* investigated polygonal numbers which are also squares.
The problem ia a special case of that to make a quadratic function a square.
The triangular numbers equal to squares are those with sides 0, 1, 8, 49,
288,1681,9800, --• and equal the squares of 0,1, 6, 35, 204,1139, 6930, —,
The Jth polygonal number with I sides is {(I — 2)x* — (I - 4)xj/2, To
make it a square, set 2A - 2)p2 + 1 = g2. Then the product of the
polygonal number by 4 is the square of 0, (I - 4)p, 2{l - 4)pq, — if
Euler gave a law for the derivation of any solution x ш terms of two solu-
solutions. It remains to make the expressions D) integers» For I — 5, q is
to be chosen from 1, 5, 49, - • • and hence p from 0, 2, 20, «•«. The first
fraction D) is here* A - g)/6 and is an integer for q — 49, whence x = — 8.
But Euier had previously stated thatr for I > 4, q was to be taken negative»
The value q = — 5 gives x =* 1 and the pentagonal number 1*
Euler67 proved Fermat's theorems that no triangular number except
unity is a cube (since x* =Ь y* is not a square), and no triangular number
x{x + \)j2 > 1 is a fourth power, According as x is even or odd, xj2 or
(x + l)/2 must equal a fourth power m\ if the Д is to be a fourth power.
Thus 2w4 i 1 = n4. But he had just proved that 2n* zb 2 « □ only
when n — 1, whence m = 0 or 1, x = 0 or L
Abbe Beidier58 gave the simplest properties of polygonal numbers and
derived central polygonal numbers as follows: adding unity to the products
of the triangular numbers 0, 1, 3, 6, 10, - * by 3, 4 or 5, we get central tri-
triangular, square or pentagonal numbers, respectively.
We shall now quote from the correspondence6' between Euler and Gold-
bach remarks on polygonal numbers, reserving for later use the comments
in which the interest is chiefly on sums of squares» June 25, 1730 (p. 31),
Euler noted that (x2 + x)/2 equals F/7L for x = 32/49, but said this does
not disprove FermatJs assertion that no (integral) triangular number is a
biquadrate. Aug. 10, 1730 (p. 36), Euler noted that if
Ь = C
the square of (a—&)/D ^2) is a triangular number with the side (а+Ъ—2)/4
[evident since ab = 1]. Chr. Goldbach stated April 12,1742 (p. 122) that
4t?!« - m - n' # A. Euler remarked May 8, 1742 (p. 123) that
4mn — m — n is not a heptagonal number» June 7, 1742 (p. I26)r Gold-
«Comm. Acad. PfetropV^ 1732-3, 175; Comm. AritK. Coll., I, 9. Cf. Euler.31
* Thus q - 1 - ftr so that Bp« + I - g* become» p* = - 2x + Gz*. Hence p « 2P,
P1 — Ex* — x)f2, and we h*ve returned to the problem from which we started-
*7 Comm, Acad. Petrop., 10, 1738, 125; Comm. Arith. CoU,r I, 30^ 34» Proof republiahed
by E. Waring, Modit. Algety, ed. 3, 1782, 373»
tt Suite de rarithmfitique dee gtotoetresr Paris, 1739, 352-365.
•■ Correapondance Mathematique et Physique (ed.f P» H. Fuee)^ St, Pe4erabourgr 1, 1843.

р. I] Polygonal, Pyramidal and Fighratb Numbers. 11
jjach inferred that every number is of the form 2A d= O, and incorrectly
(Euler, p. 134) that every number is a sum of three triangular numbers.
Euler June 30, 1742 (p. 133) noted that every number is of the form
у, + у _ xx - 2Д, - x\ April 6, 1748 (pp. 447-9, 468)r GoMbach stated
that every number can be expressed in each of the eight forms
П + 2П' + A, n+2D' + 2A, П + D'H-2A,
2D + A -(-2 A', etc,
June 25, 1748 (pp. 458-^60), Euler gave the identity
Hence CFermat's* theorem} every n ia a sum of three A's implies
n = Q + 2A + Д'.
Euler expressed his belief that every number of the form An -(- 1 is a sum
Ш of three squares, whence n = П + П' -(- 2A. Replacing n by 2n, we
все that every n = П + П' -{- A* Euler gave fourteen such formulas.
June 9, 1750 (p. 521), Euler remarked that an algebraic discussion of the
theorem that any number n is a sum of three triangular numbers is of no
help, since the theorem is not true if n is fractional (unlike the theorem on
H]). Dec» 16,1752 (p. 597), Euler noted as facts, of which be had no proof,
that every prime Sn + 1 or 8л + 3 ia of the form x1 + 2jf, whence if
n ф П + A, 8ti+ 1 * prime, and if n Ф 2 A + A', Sn + 3 Ф prime* Also
(p. 680), if n Ф П + 2Д, 4n + 1 Ф prime,
April 3r 1753 (pp. 608-9), Euler treated the problem [of Fermat^J to
find a number z which is polygonal in a given number of ways. Let ft be
the number of sides of the polygonal number, % its root. Then
= (n - 2)# - (n ~ 4)x, я = 2 _^
-1
Thus 2г must be divisible by x, and 2z — 2 by x — I. Hence we desire
two numbers differing by 2 which have divisors differing by L For example,
450 and 448 have such divisors 3 and 2, 5 and 4, 9 and 8, 15 and 14. "ITius
225 is a square, 8-gon, 24-gon, and 76-gon*
Euler40 noted that, if 4n + 1 is a sum of two squares, 8n -f- 2 ia a sum
of two odd squares Bs -f- IJ, By + 1)J, whence n = A* + Ar S. Realis"*
noted that conversely this expression for n implies
4n+l = (z + y + iy+(x- y)\
In Ch. Ill are cited Euler'*1 theorem ЦA - xk) - £(- l)*x*, where
P = Cy*=fcj)/2 ia a pentagonal number, and theorems by Legendre,21
Vahlen, and von Sterneck,JI* on the partitions of N, m which an excep-
exceptional r6le is played by the Nya which are pentagonal or triangular*
M Novi Comm, Acad. Fetrop., 4, 1752-^3 A749), 3-40, § 34; Comm. Aritb. Coll., Ir 1в4.
ш Nouv. Ann. Math., C), 4,1885, 367-8; Oeuvree de Fermai, IV, 218-20.

12 History of the Theory of Numbers. £Cha*. i
G. W, Kraft61 and A. G- Kastner*2 proved that
24*-n _ 2*» - 1 _ BN)BN + 1) _
<f " 2 " Л'
since Bй" — 1)/3 is an integer N.
M. Gallimard6* obtained "central polygons" by multiplying each term of
0p 1, 3> 6, 10, 15, ■ ■ - by the number n of angles of any polygon whatever
and adding unity to each product» Given a central polygon, he treated
the problem to find the number of angles if the side be given, or vice versa»
L. Euler64 proved that a number not the sum of a square and a triangular
number Д is composite; one not A H- 2 A' is composite.
Nieolas Engelhard65 treated Plutarch's* questions on triangular numbers.
Elie de Joncourtw gave a table of triangular numbers N(N -(- l}/2, N
up to 20000, and showed how the table may be used to test if a number
less than a hundred million is & square or not, and to extract square roots
approximately.
L. Euler*7 noted that, \£ N — ab = Др + A^ ~f At and p — q = a — b,
then N = Ар+ь + A^-* + Ar. N. Fuss, I, (pp. 191-6) also gave an
incomplete argument to show that N is a sum of three triangular number?
if every integer < N is. Let N - p = дл + A* + A« and
p = (b — a)n H- n2
£a restriction]; then N = A«~» + Abf* + Ac* He gave a similar in-
incomplete discussion of the problem to express N as a sum of m ш-gonal
numbers, given that evsry integer < N is such a sum» He noted (p. 201)
that 9n + 5, 8; 49n + 5, 19, 26, 33, 40, 47; 81» + 47, 74; etc., are not
sums of two triangular numbers; thusr 49л -(- 19 = Ae -f~ Аь would imply
Ba + IJ + Bb + iy ^ 8D% + 19) + 2, whereas the factor 7 of the
Latter is not a divisor of a sum of two squares. L. Euler (p, 214) noted that
ЛгА^ Д, is satisfied* if px(y + 1) = 2gsrgy(x + 1) = p(z -\-1); the results
ing values of z are equal if [Bq* — p?)x -\- 2q*)y = p*x + 2pq< L. Euler
(pp, 264-5, about 1775) noted that
9Aa + 1 = Азн-1, 49 Aa + 6 = Ата+i,
25 Aa + 3 = Аьн-2, 81 Aa + 10 =
J» A« Euler*5 (the son of L, Euler) stated that to express every number
as a sum of terms of Is, З2, б2, UP, 15*, «• •, at least 12 terms are required.
11 Novi Comm. Ac»d» Petnop., 3, ad annum 1750 et 1761^ 112.
* Comm, Soc. Sc. GottrngeneiB, 1^ 1751^ 198- Cf. T. Pepin. Atti Acced. Nuoyi Iinceir 32,
l878-9r 29S,
tt I/Alg&re ou U Science du Caicul litteral, Parifi^ 2f 1751,131-143.
* Novi Comm. Acad. Petrop., 6, 1756-7 [17541 185^ Comm. Aritn, СоУ., Ir 192.
"Veriiaadd. Hollaocbe M*ftt«chappy Weetenschsppe be Harlem, 3 Bed, 1757, 223-230;
4 Deel7 1758, 21 (correction to p. 224).
M De Natura et Praeclaro Usu Simpltci»imae Speciei Numerorum Trigonalium, Hagae
Comitumr I7fi2, 267 pp.
" Opeia poeturoa, lf 1862, 190 (about 1767),
* The leant solution ina:-2,y-5,*-9r Spbin*X)edipe, 1913, 90; l»Hr 145»
«1Ш., pp. 20Q-4, about 1772.

Polygonal, Pyramidal and FmmiATB Numbers. 13
To express every number as a sum of figurate numbers
n + a
1 > 12' 123
12 3 4 ' "'
at least a -f 2n - 2 terms are necessary. Cf, Beguelin,71 Pollock,11'
Millt1"-2
Euler" remarked that Fermat's* theorem that every integer is a
gum of rn tfi-gonal numbers would follow if we could prove that every
integer occurs among the exponents in the expansion of the with power of
1 -{- x + xm + я2*""* + *• • > whose exponents are the w-gonal numbers.
Format's theorem that every integer is a sum of three triangular numbers
would follow if it were shown that in
all integers occur as exponents of x in the series for R,
Euler™ found squares which are triangular or pentagonal. If дж «= x*t
then y> = Sx2 + 1 for у = 2z + 1. If (Зг* - z)/2 = x\ y* = 24x* + 1 for
у = 6* - 1. If (Зд2 - <?)/2 = A,, Fg - 1)* = 3x* - 2 for * = 2p + L
Special solutions of the three equations y* = ax1 + & are found by his
general method of treating the latter (Ch. XII, below).
Euler71 admitted that he had no proof of Fennat's assertion that every
number is a sum of three or fewer triangular numbers and noted that
this is true only of whole numbers, since no one of Jr I, S, i, etc», can be
resolved into three triangular numbers. There are no rational solutions
z, y, z of
i »*+.* i У1 + У . ^ + <
Nicolas Beguelin7* attempted to prove Fermat's theorem that every
integer is a sum of s polygonal numbers of s sides» For $ = d + 2, the
latter are 0 and 1, A = d + 2, В - 3d + 3, С - fc? + 4, D = lOrf + 5,
- • *, a series whose second order of differences are d. Let t be the number
of terms > 0 needed to produce a given sum e. For 1 < e < A, evidently
«^ A - L Fora = A + ь where 1 ^ t ^A - \tt ^ A; fore = 2A + «r
0 ^ 6 ^ d - 2, t ^ A Next, let В < 6 < С For e « Я + «, 1 ^ «
S4-U<i; for e = В + A + «, 0 < e ^ A - 2, t s A; for the
"doubtful case" e =B-{-^ +A - 1, we replace В by its equal 2A +d - 1
and have g = 4A+d~2, t~d + 2; finally, for e = В + 2A + *,
n Novi Comm, Acad. Petrop., l4r I, l7№, 168; Comm. Arith. Coll., Ir 399-Ш0,
19 Algebra, 2, 1770, tf 88-91; French tmneb, 2,1774, pp. 10^-9 (Vol. I, Ch. Vr pp. 341-354r
for definitions of polygonal numbered Opera omnk, A), I, 373-5, 159-&4.
n Acta Eruditoram, lipe., 1773, 193; Acla Acad, Pfetrop., I, 2, 1775 A772), 48; Comm.
Arith. ColL, I, 548.
n Kouv. M6m. Aced, So. Berlin, шшбе 1772,1774, 387-413,

14 HlBTTORY 09 THE ТнЕОВТ OF NUMBERS, [Chat. I
0 ^ « ^ d — 4, t^ d — 1. After this expansion of the argument by
Beguelm, we are ready to admit that if e ia in one of the intervals 1 to A,
Л to £, 2? to C, it is a sum of d + 2 or fewer terms 1> A, B. He treated four
more intervals with a rapidly increasing number of "doubtful cases" fot
which linear relations between the polygonal numbers were employed,
and found in every case that t ^ d + 2. But he finally admitted (p. 405)
that this method does not lead to a proof of the general theorem of Fermat.
On p. 411y Beguelm stated without proof the erroneous generalization
[cf. J. A, Euler," L. Euler*] that any number is the sum of at most
t = d + 2n — 2 terms of the series
t «_!_л (n -h 1)(» -h 2d) (n-
^a' 2 ' 12-3 *
a series whose nth order of differences are constant and equal to d. For
n = 2, we have the case of polygonal numbers just considered. For n =* 3,
we have the pyramidal numbers P^ for г ^ 1, 2f 3f - • ■; for n ^ 4, their
sums, etc* For n = 4, d =* 1, the series is I, 5, 15, 35, 70, --* and the
theorem gives t *= 7, whereas 8 terms are evidently required to produce the
sum 64 (since 4 terms must be unity), as expressly mentioned on p. 412.
Thus Beguelm contradicts himself in bis generalisation of Fermat'e theorem
to pyramidal and figuratc numbers.
L. Eiiler™ probably overlooked the last remark, since he stated that the
unproved generalization merits great attention* He extended Beguelin's
tentative process to any series 1, Л, B7 - - *. We muet employ Л-Ьп-2
eummands 1, A to produce nA — 1, Thus if
nA - 1 < В < (n -h l)A - I,
we need A -h n — 2 summands 1, A to produce all numbers 1, 2, -* -, B.
Then
Л — 1 Ч д < А -\- п — Л ^ А —
Denote by |xj the least integer > x, and by *i the number of terms 1. A
needed to produce 1, • • •, B. Hence
Bringing in also the summand B} let b be the least positive integer such
that В -h b requires tt + 1 summands 1, At B. If С < В + Ь, we need
only tt summands to produce the numbers ^ С But if С ^ В -f 6, let
(to -h 1)B + Ь > C> mj? -h Ъ.
To produce the numbers ^ С from I, Л, В, we need
_ , 1С-П-Ъ\
~h + \ в 1 -
^ Оршс. Anal., 1,1783 A773)» 296; Cotnjn. Aiith. Coll.. II, 27v

t} Polygonai,, Pyramid ль and Figurate Number», 15
pii Next, bring in the summand С and let с be the least posi-
positive integer such that С + с requires U + 1 summands from 1, A7 B7 C. To
produce the numbers ^ Д we need
summands, etc. In the case of an infinite series 1, Ay By • ■ -, the process
furnishes a lower limit to the number t of summands. Euler showed that,
for series whose nth order of differences are constant, Beguelin's rule is
often quite erroneous, but did not treat the series 1, n + d, - * • of polygonal
and pyramidal numbers.
K. Beguelin74 made a puerile illogical attempt to prove that every num-
number is the sum of three triangular numbers. Admitting the last theorem,
Beguelin75 deduced Bachet's theorem that every integer is a SI. For,
E) n ** 2(as + a)/2 implies 8n + 3 = 2Ba + 1)'.
Adding 1, we conclude that 8« + 4 is a Э). But it is known that the half
or double of a 31 is a SI ♦ Hence 2n + 1 and its product by any power of
2 are 3). Since Lagrange had given in 1770 an independent proof of this
theorem of Bachet, Beguelin next attempted, but failed completely, to
deduce from it the result that every integer is a sum of three triangular
numbers Д. On p. 338, he gave the equivalent formulas
q = ~-? + ИГ^' 4? + 1 = (a - ЬJ + (« + Ь + 1)>.
He concluded without adequate proof that every number is a sum of a A
and two squares, and also is Л + 2л' + 2a" (p. 345); further, that
every integer = 1, 2, 3, 5 or 6 (mod 8) is a SI, later proved by Legendre1*
of Ch. VII. A fitting sample of the lack of insight of Beguelin is furnished
by his final theorem* (p. 368): If any number 4m -|- 1 is a sum of three
squares [[each Ф 0], it is composite [but 17 = 9 + 4 + 4 is prime].
Curiously enough, he supposed he had verified the theorem for all numbers
< 200; but his tables (pp. 363-4) imply that he assumed that a number
can be expressed in a single way as a sum of squares. On this he based a
new "proof" that every prime 4m -h 1 is a El.
L. Eulcr7* noted the result E).
Euler77 noted that | is not the sum of three fractional triangular numbers
(s2 + x) /2, since 7 is not the sum of three odd squares Bx + 1)*. But every
71 Nouv. M&u. Acad. Berlin, «шёе 1773,1776, 203-215.
rt Nouv. Mem. Ac»d. Berlin, annfe 1774, 177^ 31Э-3№.
* One error is that if the виш of three &% each + 0, is of the form 3* + 2t then v is
not divisible by 3, assumed to follow from the conserve in f 50. Bat 45 + 10 + 1 ■ 2
(mod9).
^Ас4аАс*а.Реггор.,4,ПД780[1775138;Со1пт,Апи1.С<Л1т11,137- Eater" of Ch. VH
77 Opoec AnaL, 2> 1785 A774), 3; Oxnm. Axfth. Coll., П, p. 92.

16 Нвдгокт of тнв Theory of Numbers.
number N is the sum of four fractional pentagonal numbers (Зх1 —
since
2AN + 4 =* 2a> = 2(te 1)\ x =
о
To prove the theorem that any number ia the sum of three integral tri-
triangular numbers A, it would be sufficient to show that the coefficient of
every term xk in the expansion of
A + * + ** + ^+'-'+:^+—)'
is not zero; similarly for squares, pentagons, etc. [Euler68]. Let the
polygonal numbers with r Bides be 0, a = 1, ft = *, у = Зт — 3,
and denote by [n] the coefficient of x* in A + x* + x* + • • ■}"•
proved by logarithmic differentiation the recursion formula
F, W. Marpurg78 treated (pp. 185-257) polygonal numbers, giving special
cases of formula (!) of Diophantus, pyramidal numbers and central poly-
polygonal numbers, viz,, unity more than the number of the angles and division
points on wi-gons drawn about a common mid point. Also (p. 307) poly-
polyhedral numbers, the rth hexahedral, octahedral, dodecahedral and icosa-
hedral being
Euler79 proved that (z2 -h x)/2 is a square y1 only when
= ^±f~2, у = ^£, a = C + 2^)-, 0 = C - 2V2)".
For n = 0, 1, 2, we get x = 0, 1, 8; у ^ 0, 1, 6. We have the recursion
formulas
x* = Ox^j - x^j -h 2, yn = бЧ/^i - y^i.
Certain squares x2 which exceed (y8 -|- y)/2 by unity are given by
l)g- Bл/2- l)/3 1
For n = 0, z = 1, j/ ^ 0; for n = 1, з = 4, у = 5. The recursion formula
is
xrt = fte,,! - z^3j ул = 64/»-i - y^7 + 2.
A second series of solutions is obtained by use of these formulas for negative
n's. Thus x_i - 2, jf_j s= — 3; x_? = 11, y_a = — 16. Since the tri-
triangular number A_m equals л„«~ь we replace у = — mhy m — 1 and get
the sets of positive solutions 2, 2; 11, 15; etc.
7* Anfan^grthde dee Ptt*re«iioii*l Calcub, BerU», 1774, Book 2.
« Mem. Aced. St. Feteaabouis, 4,1811 E1778}, 3; Comm, Arith. CoIL, II, 287-4.

j Polygonal, Pyramidal and Figurate Numbers. 17
To find triangular numbers whose triples are triangular, Euler proved
that 3(я* -h x) = у* + У has only the solutions
x^^J-\, r - CV3 + 5)B + V3)-,
4V3 2
y-—-^-^, 5 - {3V§- 5)B-V3)»,
for n = 0, db 1, dh 2, - - -♦ Examples are s =* 1, 1/ *= 2; ж *= 5, # *= 9.
It was proposed as a prize problem in the Ladies' Diary for 1792
to find n (n > 1) such that I- + 2й + - - + n2 *= D. The sum is
n(n + l)Bn + l)/6. T. Leybourn80 took 2n + I = a*, whence («* - l)/24
is to be a square y*. Thus г* = 24y* + 1 *= □ =* B^ - IJ, say. Thus
у «= 2х/(я2 - 24) > 0. It is stated that j; - 5 or 6. Since s = 6 is
excluded, n = 24. C. Brady took л - 6V2. Then the condition is
FH + l)A2r* -f 1) ** □. Thus (9т2 + IJ - Cr*J - □, so that 9»-f + 1
and Зг2 equal the hypotenuse and one leg of a right triangle. Thus the
other leg is 9т2 — 1, whence r — 2, n — 24.
A. M. Legendte*1 proved FermatJs theorems that no triangular number
x(x + l)/2, except unity, is a fourth power or cube. For, in the first
problem, x or z + 1 is of the form 2ти4, whence either 1 — nA — 2w4,
contrary to 1 + 2wi4 + □, or 1 *= 2/«4 — n4, m8 — «4 = (m4 — l)?r con-
contrary to ря — n* Ф □ unless w = 1 - x. In the second problem, one of
1 + x, x is a cube and the other the double of a cube, whence n3 ± 1 = 2m*,
which b impossible if » H= 1»
С F. Gauss82 proved by means of the theory of ternary quedratic forms
that every number n — SM -f 3 is a sum of three odd squares, so that, by
E), M is a sum of three triangular numbers» The number of ways M
can be so decomposed depends in a definite manner on the prime factors
of n and the number of classes of binary quadratic forms of determinant — n.
G. S. Kliigel8' gave an account of figurate, polygonal, polyhedral, and
pyramidal numbers Prm of the first order, those of the second order being
John GoughM attempted to prove Fennat's theorem that every number
ifl a sum of m m-gonal numbers» P. Barlow*5 noted that the first three
propositions by Gough are correct, but are not used in his defective proof
of Fermat's theorem, while various points are not proved, as the Cor. 2
to Prop. 4: every number is a sum of a limited number of polygonal num-
numbers. As to Gough'S reply (pp. 241-5), Barlow88 stated that the defense
80 Ladies' D»ry, 1793, p. 45, Quest, 953. T. Leyboiim'e Math» Quest- jm»po®cd in the
Ladies' Diary, 3, 1817, 256-7. Cf. Lucae, papers 130-8.
«Theoricdea потЬпя, 1796, 406, 409; ed, 2P 1808,345,348; «L3,II, 1830, arte. 329, 335;
pp. 7,11- German tranel by Maaer, 1S93, II, 8, 13.
M Diaqais. Arith., 1801, art. 293; Werke, 1,1863,348; German tranaL by Maaer, 18S9, p, 334.
" Math. Wfirterbuch, 2, 1805,245-253; 3, 1808r 825-8, 931.
* Jour, Nat. Phil., Chem,r Art* (ed,, Nu-hobon) 20, 1808, 161.
■/Mi., 21, 1308, 118-121.
*/btd, 22, 1809, 33^35.

18 HlBTORT OF THE THEORY OF NmOtEBS- [Cm*. I
is on grounds not proved. As to the revised version by Gough** Bar-
low noted (p. 44) that the argument is correct and trivial to within 12
lines of the end; the proof is valid for numbers ё 3m, bat not for those
E. Barruel" noted that sums of 1, 2, 3, • • • give the triangular numbers
1, Z, 6, • • •, whose sums give the pyramidal numbers 1, 4, 10, 20, —, etc.
Forming these sums, we get the general triangular and pyramidal numbers
n(n -Ь 1)/2, п(п -Ь l)(n -h 2)/6, eto. Application ia made to prove the
ordinary rule for deriving a binomial coefficient from the preceding coeffi-
coefficients.
F. T, Poselger" gave (pp. 19-31) various properties of numbers from
the writings of Theon of Smyrna, and (pp. 32-60) gave algebraic expres-
expressions for polygonal and figurate numbers, with a discussion of arithmetical
series of general order.
P. Barlow» noted that, if N is a sum of &ve pentagons Cws - w)/2,
and M a sum of six hexagons 2x* — xy then
24ЛГ -f 5 = Z iptii - l)\ &*f + 6 = £ Dx> - l)K
In general, if P is a polygonal number of a + 2 sides, Fermat's* theorem is
equivalent to
SaP + (a + 2)(« - 2J = g
He erred100-ш (p. 258) in saying that no triangular number> 1 is pentagonal.
J. StruveM discussed figurate numbers (binomial coefficients).
J. D. Gergonne*2 noted that the number of terms of a polynomial of
degree minn unknowns is (m + n)! -*■ (ml nl). If the latter be designated
(m> n), then (m, n) « (m — 1, n) -h (то, л — 1).
A. Cauchy93 gave the first proof of Fermat's theorem that every number
is a sum of m m-gonul numbers. The proof shows that ail but four of the
m-gons may be taken to be 0 or 1. The auxiliary theorems on sums of
four squares will be quoted in Ch. VIII. In the simplified proof by
Legendre,*4 the case m — 3 is not presupposed, as was done by Cauchy.
Moreover, Legendre proved (p. 22) in effect that every integer > 2S(m — 2)*
is a sum of four wa^gonal numbers if m is odd; while, for m even, every
integer > 7(m — 2)aisasumof five w-gonal numbers one of which is 0 or 1.
" New Sen» of the Math. Repository («L, T. Leyfaourn), 3,1814, H, 1-7.
« Correspondence виг 1'Ecole Imp. PtOytechniqae^ Pari^ 2,1800-13, 220-7.
•• Diophantua uber die PolygoruAhlen udberaeUtA mit Zueatseo, Leipzig, 1810.
^ Theory of Numbers, 1811, 219- Minor appbcatione in рарегв 17-19 <rf Cb, IX
* Uber die gewohnlichen fig. Zahlen, Ptt>gr. Altoaa, 1812.
«AnnaJee de Math, («d., Gergonne), 4, 18134,115-122.
* Mem. Sc MaUu et Fhys. de 1'Inetitut de France, A), 14, 1813-15, 177-220; ваше m
ExercicesdeMath., Paris, 1,1826,265^-296. Reprinted in Oaivree de Cauchy, B)F У1Л
320-353. J. dee Мшев, 38r 1815, 995. Report by Caudiy, Boll. Scp^Soc РЫЬ-
matique de Paris, C), 2,1815,10&-7.
*Tb6oried«jtombres, let supplemeftt, 1816, to the 2d edition.1808, 13-27; 3d ed., 1830,
I, 218; П.340; German taiuL Ьу Маввг, П, 332.

Polygonal, Pyramidal and Figurate Numbebs. 19
Cauchy* denoted the xtii polygonal number of order m + 2 by
and proved that if A, Bt - -, F are integers, not divisible by the odd prime
pf there exist integers xlf - - -, **, such that
A2: + B2;+> + E2: + F = Q (mod p),
where ft = mif miaeven, and я = 2m if mis odd The casern - 2 shows
that there exist integral solutions of [Lagrange,* etc., of Cb YIIIJ
Ax\ + Bj> + С = 0 (mod p).
L. M. P. Coste96 showed that the problem to make two integral functions
of one variable equal to polygonal numbers of a given order can be reduced
to the problem to make two functions equal to squares. Let
P{Z) = (pZ* + qZ)j2t }X~A* + A'z + P{a), /, = №* + B'z + P(b).
Then to make fx and /s equal to numbers P(Z), take Z = az -h a and
Z — Дг + 6 in the respective cases. We obtain a quadratic equation for a
and one for & each linear in z. Solving for a and ft we require that the
quantities under the radical signs be squares, viz-, Spfi -b ff1 - D,
Sp/i + fl* = D. Next, if /i and /± are of the form 2а*р& + Az + A', use
Z = 2az -h cu We can make two quadratic functions equal to P(Z) if a
particular solution is known.
Several solvers97 readily found two pentagoual numbers Рл ^ Cz* — x)f2
and Py whose sum and difference are triangular by solving
A. M. Legendre" concluded from the formula
that every integer/^ is a sum of four triangular numbers in *BДГ 4- 1) ways,
where <r(k) denotes the sum of the divisors of it. He gave an identify which
shows the number of ways N is a sum of eight triangular numbers,
Cauchy" gave F); it was attributed to Jacobi by Bouniakowsky (see
VoLI, Ch. Xй-» of this History), a. Plana."»
Several100 found numbers > 1 which are simultaneously triangular,
pentagonal and hexagonal Let %m(m + 1) = £Cл* - я) - 2p* ^ p.
Then m = 2p - 1, n - A -Ь й)/6, where IP = 48p^ - 2ip + 1. Thus
1 + R = 6kp, whence p = B - *}/D - 3**). Take * = Ь/o. Iben p is
integral if 4«^ — 36* = 1. By the continued fraction for ^ we get
y^ 16, VoL S, 181Д, 11<J^123;
de Mmth. («L, Geigaime), 10,1819-20,101-122.
n The Geatkmui'a Mmlh. Companicm, Loodoo, 5, No. 30,1827, 658^9.
« Timite dee fbocUntf effipdquee, 3,1828,133-1
«CompteeTtendiwPAri», 17, 1M3, 572; Оешгпя, (l), VIII, 64.
m Ladies' Dingy 1182^ 3^-7, Quart. 1408L

20 History op the Thkobt or Numbers. [Chap, i
Ba, h) ** B, 1), B6, 15), C62, 209), E042, 2911), • > >, whence p =* I, 143,
27693, • -, so that aDswers are lf 40755, 1533776801, - - -.
J. Whitley1(a found pairs of pentagonal numbers p, q whose sum and
difference are pentagonal. The conditions are
+ q) + 1 - Л
Н |, » ,
Then г = n -\- m, v -= n -~ m. Take n = т* — s*, тл = 2ra, whence
s s* r* -h s2. There remaine the condition
4г*(т* - ^ -h 1 = D = y».
This is said to hold if г =- §(^5 - ф), s = J(^fi - Зф), which lead to larger
DUniber8 Шап those found by trial, using (r, &) =» C, 2), @, 1), (8, 5),
A3, 2), A3, 8), A9, 14). [But the resulting numbere p - 7, 37, 330, * - -
art not pentagonal.] See Gill.10*
C, G. J. JaoobP of Ch. VII gave in 1829 the result
j xi (-
^e exponents on the left being pentagonal numbers for m negative, and
those on the right triangular. Polygonal numbers appear incidentally in
jacobi1» paper of 1848 [see Ch. Ill j.
J. Huntington,101 given a pentagonal number P = r(Sr — l)/2 of n
digits, found another number p also of n digits such that if p is prefixed to
p there results a pentagonal number. Let x be the root of the latter.
Then shall 10*p -f P = sCs — l)/2. Taking p = z — r, we get
x = |<2-10* + l) -r.
>г example, let г =* 1; then n = lf p = 6, p = 5 and 51 is pentagonal.
A- Cauchy108 defined triangular and pyramidal numbers as binomial
J, Baines1" found two squares ж2, у2 whose sum and difference are
hexagonal Take Six3 - y*) + 1 = \2{x + y) ± 1JJ, whence x *=Sy±l.
Then 8B? + f) + 1 = 86y> ± 48j/ + 9 = (ny ± 3I determines y.
A- Bemerie106 gave a table of triangular numbers.
A. Casinelli10* noted that every triangular number is of one of the three
forms
(9m» + 3m)/2 - A^i -b 2 д1ж, (9m> + 9m + 2)/2 = Дж+ Д^+ Ai^+i,
(9mJ -h 15m + 6)/2 - Д^! + 2Д^Ъ
^leoasum of four A^s, and hence a sum of any number of A's. By adding
"«■ Ledics^ Dbiy, 1829, 39-40, Quart. 14^ ~~ ~~
v* Ladies1 Diary, 1832, 36-7, Queat. 1630,
*• Resumes Analyt^ T^rin, 1,1832, 5.
J^Tbe Gentleman's Diary, or Math. Repoeitoiy, London, 1835, 83, Quest. 1320.
i» Nmiv, table dee tnaogdaiiw, Bordeaux, 18S5.
t"Novx Comm. Acad. Sc- Imrt, Вошмпепш, 2,1836, 415-34.

сидр.
Polygonal, Pyramidal and Figukate Xumbeks, 21
the first two or the second and third equations, we get
Cm + IJ = m3 + Bm + IJ -f 2Д*»,
Cw -f 2)* = (m + IJ + 2AW1 + Bm + IK.
Also C"» + 3)s = (m + IJ + 2Bwi + 2)\ Hence every square is a sum
of three squares or a sum of two squares and two A's. Further, every A
is a sum of a square and two equal A's. Next,
дв+ , Am + A« -h (tfi + 1)(л + 1) - Aw^n+ъ
and similarly for three or more A's- Also, Am + Д„ — ro(rc + 1) = A*-*,-
C. GiIIia7 found numbers both ш-gonal and #-gonal, and the generalisa-
generalisation
T = ax* - a'x * V - bV,
where a, a', &, &f are g^ven integers with no conxmon factor. Take
ax - a' = yplq, % ^ (by - b')q/p,
so that
* - q{Vp + a'bq)/Np у - q(a'p + Vaq)flff - N = p8 - abq\
Let p', ^' give a particular solution of the last equation such that
A = (ay + йЬУ)/ЛГ, 5 = (&У + fteVJ/AT
are mtegers. Take p ^ pft + ebq'u, q ^ q't -h p'u. Then
where F « i2 - a£u2, and дг = ^(^ + Abu)/Ff у = q(At + Bau)/F. From
the initial solution ^ = 1, «o ^ 0, of F = 1^ we get as usual the solution
It remains only to find a solution p', q' of p2 — obg* — — N. While oae
пшу employ the continued fraction for VS5, it suffices for our initial problem
to note the solution p' *= a — a^, g' = 1, for the case a — a' — Ъ — b';
then N = ab' + baf — аЪ% А ^ В «= L First, if m and n are both odd,
we may take
a — m — 2f a' = m — 4r & = л — 2, 6' = n — 4,
which have no common factor. Then a — a' = b — bf =■ 2. For Pi = \Tif
Л = 1, Pi = 2UVi - Pi~t + Brf - 1)(* - 1)#
, (tn + n - 4)(mn - 2m - 2n + 8)
16(*i- 2)(n- 2)
But if m and n are both even, take a = \m — 1, af «= Jm — 2r Ь *= Jn — t,
^' = \n ~- 2, whence a — a' = & — V = 1, and /\ ** Ti satisfies the same
recursion formula. Also,
P i(U + 1) + rf(< 1) +

22 Hibtoby op the Theory of Numbers. [Сна* 1
where e = 8 in the former case and e = 16 in the present case. For ex-
example, 1, 210, 40755 are both triangular and pentagonal, whereas Barlow80
stated that this is true only for unity.
GUI100 found n-gonal numbers whose sum and difference are n-gonal,
i. e., P9 -b P, = Рш, P*-P,~ P., where Px = (n - 2)z* - (n - 4)z.
As a generalisation, take Рж = тз? *- m'x, where m and mf are relatively
prime. The first condition ш satisfied if
z — i/ = - {mz — m*), m(? -fri-w'»»rx,
а о
Each of these linear equations is solved separately and the resulting х*в
equated. The second of our conditions is treated similarly and the two
sets of values of x and у are compared. But the resulting solution does
not lead to "convenient numbers.19 Another method is to assume that
x = aw — h, у = bwfz = cw — h, where
a* + & = <•*, 2mh(c - a) = m'(a + Ъ-с),
whence our first condition is satisfied. Thus take
a = 2Ы, Ь = k> - P, с = V + Pf l = mh, k = mh + mr.
The second of our initial conditions now becomes
4т'(* - 2mT)v? - 4mBmh + m')dw + BтЛ + т')г = Bmv - m')\
where d *= a — m*. Take 2mv - m' = 2wt/u -h 2mA -h m'. We get w
and then v rationally* By choice of the denominator f1 — (d3 — 2m*t)m1u^
we get integral answers unless mf = 0.
Many10* found two squares z*, jf such that & ± &* are pentagonal
Let24(a?-yt) + 1 » И(яг + у) zfc IIs, whencex = &y^ 1. Then
24C^ + Й + 1а в24у» =F 24Oy -f 25 ^ E - flr/e)8
determines |/. Again, to find pentagonal numbers p, q whose sum and
difference are squares x?, y2, take I2(x* - \?) -h 1 ~ {3(tr + ^) ± I }s and
12(s* -f ^ + 1 =* G - yr/sy, whence x = ly =b 2.
О. Тегф1етш proved tiiat no triangular number > 1 is a biquadr&te.
The ordinary definitions of polygonal and ngurate numbers as sums of
series were repeated by F. Stegmann,m George Реасоск,ш A* Transon,111
H. R Th. Ludwig,m Albert Dilling,114 and V. A, Lebesgue^
F* PoUockU7 stated that every integer is a sum of at most 10 odd squares,
and a sum of at moat 11 triangular numbers 1,10,28,55, »»< of rank 3n-H,
'" M*Ul Mi*dUnj, Fluebiii& N. Y^ 1,1836, 225^230.
м ТЪе UdfB Mtd Gentleman'B Diary, Loodoo, 1842, 41-3, Quart. 1677.
™ Nott. Ajul Mvtlb, b, 1846, 70-78.
ш AirfHF Mfttb. РЬуа., б, 1844, 82^8».
ш EncyciopaedU Metropolitmna, London, i, 1846,422.
"• Now. Ашц M*Ub, 9, 1850,257-9.
ш Ueber fi^ Zehkn a. uhh. Rctbca, Progr. Cbcnmiti, Leipzig, 1853.
ш Die Ftotpwfmefijfe.il-polys. &,Ftogr. MtridheuBen, 1865.
"• Ехеккм d'anmlyBe mim&iqae, 1850, 17-20.
^PK.Soc London, 5, IS61, 922-^. Of. Euler,»- * Beguelm."

l] Polygonal, Pykaiodal and Figttratb Numbers, 23
whU 5, 7, 9, 13, 21, 11 terms are needed to express every number as a sum
of tetrahedral, octahedral, cubic, icosahedral, dodecahedral, and squares of
triangular, numbers. Legendre bad proved that 8n + 3 is a sum of three
odd squares, each being SA + 1* Pollock gave the generalization that,
if fs is any figurate number of order x, 8FX -h 3 is a sum of 3 or 3 -h $, - - -,
or 3 + Sn terms of a series whose general term is $FV -f 1.
V. Bouniakowsky11* employed A) and (9) of Vol. 1, Ch- X, of this
History, to prove tbat every odd pentagonal number can be expressed as a
sum of another pentagonal number and either a square or the double of a
square; every odd square not a triangular number is a sum of double a
triangular number and either a square or the double of a square. Similarly,
A, 2)<i* - Ax + A, 2)«* Ax = A, + A, 2)u*,
tbe factor A, 2) denoting 1 or 2.
F. Ро11оскш stated without proof that any integer between two con-
consecutive triangular numbers is tbe sum of four triangular numbers the sum
of whose bases is constant.
J. B. Sturm120 gave the relations
Bn + IJ + DA»V = <4A. + II,
Bn + 1)*Bт + IJ + DA. - 4Л01 - DД. + 4A* + II.
V. A» Lebesgue1*1 gave two proofs of the final theorem under Wallis.44
J. Liouville proved readily that the only forms a A + &A' -h cA"
which represent all numbers, where tbe Д'в are triangular numbers and
в, &, с are positive integers, are Д-1-ДЧ cA" (c — 1, 2y 4, 5) and
A + 2Д' -h dA" (d = 2,3,4). That conversely each of these seven forms
represents all numbers is proved by use of Legendre's theorem that a
number = 1, 2, 3, 5, 6 (mod 8) is a GO, The case с - 1 was treated by
Gauss.83 Next,
2Bn + 1) = Ыг -f B1 + I}2 -f B? + I}2
Ы + 4 ** Bu + 71 + 1)» + B1 - 2u + 1)* -h 2Bг + IJ,
n = AtH-e + Ai-. + 2Aj,
proves tbe case с = 2. Next,
8* -h в = Bar + 1J+ By + iy + 4B* + 1}', n = A, + A, + 4Ae,
8n -h 5 ** Bx + IJ + 4B* + 1)* + Ш1
= Bx + II + 2B* + 1 + 20s + 2B* + 1 - Щ\
n ~ Ax
or case d =* 2, Next, as shown by Gauss,
8n + 7 = D + □ -h 2П = Bx -h 1У + 4Bг + IJ + 2By + II,
n = А. + 2Д,+ 4Д..
Reproofs for the remaining cases с = 5 jnd <f = 3 are longer.
M&n. Acad. 8c. St. Petei8boar& F), 5, \B5/
tPha. Тмвд. Ноу. Soc. London, 144,1854, 31b
• Ardiiv Blatlb Fbye>, 33,18», 92^3.
1 Introduction & la tWorie dee nombrea, Fan*. 18в2,17-20B6-8).
Jot», de Math., B), 7,1862, 407; Ь, 18СЗ, 73.

24 History op the Theory of Numbers, [Chap, i
X Plana1* wrote £* for the left member of F). By expanding the second
member as a power series in q and examining the earlier terms, he verified
that
where <r(k) is the sum of the divisors of к. Hence any integer n is a sum of
4 triangular numbers in <rBn + 1) ways. Give to £f the notation of a
power series in q, multiply it by f and compare with the above series for ?;
we get a recursion formula for the coefficients of £4 He states without
proof that the coefficient of every power of q is not zero, and so concludes
that every integer is a sum of three triangular numbers.
F. Pollock124 verified for small values that any number may be expressed
in the form s — л', where s and *' are sums of two triangular numbers.
Now a ie always the sum of a square and the double of a triangular number.
Thus the theorem is that
G) a* + a + 6* - (m* + m + n2)
represents any number* Take p1 — <? — с + q as the number. Then
Double and add unity. Thus A = M -f* 2q, where
A « 2a* + 2a + 1 + 26* + ie + 2c, M - 2m2 4- 2m + 1 + 2n* + 2p*.
Since q is arbitrary, it is concluded that any odd number can be represented
by either of the forms A or M. But M is the sum of four squares.
Again, represent p1 — ifc1 + с) + q by G). As before,
represents any odd number 2n -f- L But a* + a + 5* is the sum of two
triangular numbers. Hence n is the sum of three triangular numbers.
Pollock" of Ch* VIII noted that the theorem that every number 4n + 2
is a sum of four squares implies that every integer n is a sum of four Д'в.
J. Iiouville1* considered the partition of any number into a sum of
ten triangular numbers.
S. ВШв1*8 solved A* + A, = Ao by setting у =* a — xrfs and finding
x rationally.
E. Iionnet stated and V. A. Lebesgue and & RealisUT proved that
every integer is a sum of a square and two A% also a sum of two squares
and а Д*
A. Kochheimm gave linear relations between polygonal and polyhedral
numbers.
. Acad. Turin, B), 20, 1863, 147^
"* Fkoc Hoy, Soo. London, 13,1864, 542-4.
" Compttt Rendos Рапв, 62, 1B66, 771.
ш M&tlL Quest. Edue. Times, 6,1866,18.
w7 Nquv. Ann. Math,, <2)f 11, 1872, ftfr-6 616-9* B), 12, 1878,2l7.
"•Archiv Math. РЬув., 55,1873, 18^ 192.

i} Polygonal, Рткамшаь an» Figttratb Numbers. 25
S. Realis1* proved that every integer is a sum of four numbers of the
form Cz* rfc г)/2 and also of four of the form 2г* ± z, L c, pentagons and
hexagons extended to negative arguments. Use was made of the theorems
that any odd number a is a sum of four squares the algebraic sum of whose
roots is 1 or 3, and its double 2c* is a eum of four squares the algebraic sum
0{ whose roots is zero. Further, every odd number divisible by h, or the
double of every odd number divisible by the even number h, is a sum of
four polygonal numbers of order A 4- 2, extended to negative arguments.
E. Lucas1*1 stated that \j&L Leybourn*3 Is 4- • • * 4- nz is a square only
vrhen n — 24 [and n = 1], and is never a cube or fifth power. A triangular
number [> 1] is never a cube, biquadrate or fifth power [TSuler^J. No
pyramidal number is a cube or fifth power, or a square with the exception of
12-3' 12 3
Hence except for these and for the pile 24 25 49/6 — 70* with a square
base, no pile of bullets with a triangular or square base contains a number
of bullets equal to a equare, cube or fifth power.
Lucas1*1 stated and proved incompletely that the [pyramidal^ number
#(x4~l)Br4~l)/6 of bullets in a pile, whose base is a square with x to a side,
as a square only when x = 1 or 24 {see papers 130, 132, 137-8).
T. Pepinm noted that one case of Lucas' proof of the last result leads
to an equation 9т4 — 12/^ — 4f4 = #*, not treated by Lucas when / and
R are divisible by 3. Pepin found an infinitude of solutions in this case.
G. N. Watson*1" noted the solution r = 5T / = 3, R = 51, and132* proved
Lucas'181 theorem by use of elliptic functions.
Lucas1" stated that the number of bullets in a pile with a square or
triangular base is never a cube or fifth power. Moret-Blanc1*1 gave a proof.
Moret^Blanc1** notsd that the tetrahedral number n(n -fl)(n + 2)/6
is a square for n = 1, 2, 48* Lucas stated that it is a square only then,
a fact proved by A. MeyL1*
E. Fauquembergue117 and N. AUiston1* proved that 1* + • - + n* + Q
if n > 24. Cf. Lucas111 and the papers cited on p. 26.
m Nouv. Ann. Math,r B), 12, 1873, 212; Noii^ Cb^ep. Math., 4/1878, 27-30,
"° Rccherchee but l'an&Iyse indetemuo£er Moulin», 1873, 90; extracted from Bulletin de la
eoctftf ^'emulation Dept, de PAllier, Sc. Belt Let., 12, 1873, 530.
ш Nauv, Ann, Math., B), 14, 1875, 240; B), 16, 1877, 4294132. The pnwf by Moret-
Bknc, B), 15, 1876, 4^8» ifl incomplete (us noted p. 528).
ш Atti Accal Pont. Nuovi Lincei, 32,1878-9, 292-8.
*■ PWkj. London Math. Socv Record of Meeting, МагЛ 14, 1918.
1X9 Messenger of Math., 48,1918,1-22.
ш Nouv. Ann. Math., B), 15,1876,144 (Nouv. Соггеяр. Math., 2,1876, 64; 3> 1877, 247-8,
433, and p. 166 for incomplete proof by £L Broeard).
ш IbuL, B), 20, 1881, 330-2.
тШ.,{2), 15,1876,46.
Ш1Ш., B), 17,1878, 464-7.
w L'mtermWiaire dee math., 4,1897, 71.
111 Math. Quest. Educ. Time», 29, 1916, 82-3 (for « < 19ю by J. M. ChOd, 26, 1914, 72-3;
forn < 10" by G. Heppd, 34,1881,106-7).

26 Нштоях of the Тнвдвт of Nrnmnxa.
For analogous theorems en sume of consecutive squares or the sum of
the squares of the first n odd numbers see papers 70, 76, 81, 86, 87, 100,
and 103 of Clu IX, and ВимяпчР1 of Cb XXIIL
W. Goring111 proved by use of infinite series that 2 A + 6A' i- 1 «an
always be represented by the form a1 4- 3&*<
J. W. L. Glaisner1" noted that every representation of ал odd number я*
a sum of an even square and too triangular numbe&s corresponds to a
representation in which die square is odd,
for p, q both even or both odd, with a similar identity if one is even and
the other odd.
Glaisher stated that every triangular number is a sum of three
pentagonal numbers.
D. Marchand" noted the relations
Marchand14J gave identitiee Uka
(x 4- «• - ^ -
where i/ = x* -f *» and (p, 105) diseossed triangular numbers which are
squares.
E. Lucas144 asked when (Д*Ч -f A*)/(AiH + Am) isasquare
S. Realis, Б. Catalan and others"» investigated numbers simultaneously
equares and tiianguba*. 8. KeaUs stated and E. Cesaro1* proved that the
square of every odd multiple of 3 is a difference of too AJs prime to 3,
9Bn + II = A(9n + 4) - ACn + 1)- D. Afarchand*" gave the gen-
generalization tbat the square of any odd number m the difference of two
relatively prime triangular numbers (with sides Zx 4- lands). C.Henry*"
proved a like result for the product of any odd square by any number.
S. R&Uis"» stated that die theorem that every integer n is a sum of
three Д's mpbefi that л is a sum of four A'sof which two are consecutive
and that n is a sum of four A's two of which are equaL
"' Math. Annafeu, 7, 1874,386. "~~^ ~ " "" " "
i«« PhiL Mft&r Loodon, E), 1,1870,4&
1Л Meae^igpr Math., Sr 1876,1M-5.
ia Les Moodc^ A 1877, iOir-170.
i" La Science dee nomfrm, 1877.
i« Noav. Соггеяр. M«Ul, 3,1877, 433.
«■ЛИ, 4,1878/167; fi, 1878, 2B5-7; Mftth. Qua*. Educ Timee, 30,187ft, 37.
" Nout. Соггеяр. M*tb^ 4, 1878,156.
w Nauv. Ann. MaUu, B), 17,1878, 483.
»*/W«t, B), 19,1880, 517.
"•/ЫЛ, B), 17,1878,38b

ф Г} РбЪТООКАЬ, РТВАВСШАЬ AND РюТШАТЖ КцКЩЕВв* 27
Б. 1дю1&ш listed vahies of Л far which xy(x f у) = Лз* bad no distinct
rational solutions Ф 0. Taking у = 1 and # — x + 1, we obtain theorems
od triangular numbers and numbers x(x + l)£2r -f 1).
Lucas stated and Moret-Blane1*1 proved that I1 -f* - - - ■fx5 = ky1 and
д^ + ... + Л* = Ay* are impossible if к = 2, 3, 6\
S. Roberts1» proved byuseof^-a^^l [Enter*1;] that the A'a
which are equates are
J. Neuberg stated and E. Cesaro1" proved that the шип of the equares
of n + 1 consecutive integers, beginning with the 2nth triangular number,
equals the sum o£ the squares of the n succeeding integers, each being
divisible by 1* + - - + nJ. Cf. Dostor* of Ch. IX.
E. Lionnet154 stated that unity is the only triangular number Л which
equals the aum of the squares of two consecutive integers; 10 is the only A
equal to the sum of the squares of two consecutive odd Integers; when л
is a product of two consecutive Integers of which the least is double a
triangular number, then 4 Л -f- 1 (and its equare root) is a sum of squares
of two consecutive integers.
Moret-Blanc proved the preceding theorems stated by liopnet.
E. Cesaro"* noted that no triangular number ends with 2, 4, 7, 9.
S. Realism noted tnat
AEp + 1) = ADp-hl) -Ь
A(Jb + a) = Л№) + ABap + a), к = 2ap* + Ba + l)p.
Ж JAomie%m noted that 0, 1, 6 are the only A's whose squares are A's.
He stated and Б. Cesaro16* proved that there is at least one and at most
two A's between any two consecutive equares Ф 0; at most one equare be*
tween two consecutive a's; if there are exactly two A's between (a -f- II
and (a -J- 2)\ where a >0, there is just one A between a* and (o + IJ, and
just one A between (a + 2)* and (o + 3L
E. Cesaro110 denoted by V(n) the number of the first 2n triangular
numbers which are relatively prime to n. Let Яг(п) he the number of
products 1 2, 2-3, 3-4, -. -, n(n + 1) which are prime to n. Then if у
is the Jarge&t odd divisor of «>
» Nouv. Ann. MMh^ B), lT, 1S78,513.
/btf., 527; B)f 18,1S79^ 4704.
Math. Quest. £du& TI™, 30,^
Noov. Conrwp. M*t^ в, 1880, 233.
Nout. Azul MmtiL, fl), 1 4
, C), 1, 1Я82; 357.
.tab d» math, врбе^ 1Я84, &
Now. Aim. Mj^, C), I, 188?, ЗЭ& Fraof by H, ftocudp 0), lflv 1ЗД 93-6,
/feid^ C), 2,1883, 432 (ющлтЫ); &> 1886,200-213.
Annafi di Bint, (?), 14,1886-7,150-3.

28 Bistort of the Tkkort of Numbers. £chaf. i
The mean value of V(n) is three times that of Яг(п). Не found that the
probability that two triangular numbers taken at random shall be relatively
prime is
Cesaro161 stated and E. Fauquembergue1*1 proved that 5 and 17 are the
only integers whose cubes diminished by 13 are quadruples of triangular
numbers.
G. de Rocquigny1** noted that, iffc^ (л*+Ь* + л + 6)/2,
Д* = Д*-1 + Д« + A*, («2 + II = 1 + A(e* + a) + A(a* - a).
S. Itealis1** used the known fact that, if p is a product of primes Sq + 1,
2s2 + y2 = ?? has integral solutions. Thus
3(8* + 1) = 2Ba + 1У + B6 + 1)»,
so that 3$ -= 2Л + A'.
Healis^ gnve various sums Hke
A, + A* + A* + ■ • • + Ai^i = fa(n + 1)D« - 1),
A2 + A< + Ae + •"" + А2ж = Мл + ЦD» + 5),
A> + A6 + A9 + """ + A»w = |л(» + II-
E. Cesaro1M noted that (nfi - l)/4 «Д,+ Д„«Ф 5, implies that
2p + 1 or 2q H- 1 is composite.
S- Tebay and otbers1" found that the least heptagonal number
j{5s2 — 3a;) which when increased by л2 is equal to a equate is given by
x «24A9»-9).
C- A. Laisant167 wrote o^ for the ath (a + 2)-gooal number j£+a and
gave
(a + Ь)л - aA + К + *ab, (« + - - - + *)* = 2св + a2o6.
E. Cesaro168 noted that the number of A's prime to n and < 2n(n + 1)
is к = пЩ1 — 2/p) or 2k according as n is even or odd, where p ranges
over the odd prime factors of n.
E* Catalan189 proved that every A > 1 is a sum of six pentagonal num-
numbers. For,1" 6B* + 1)* ~ Fx T II + Fy =F II + 4(вг ЯР 1)\ whence
\
w Mathesis, 6f 1886, 23; 7, 1J887, 257-9.
"^/bid^e, 1886,224.
ш Nouv. Ann. Mhth., C), 5r 1S86,113.
'« Jour, de math, apfe., 1888, 94.
» Mathetie, 8, 1888, 75.
111 Matb. Quest. Educ» Timee, 5O7 1889,84-5.
™ Bull, S<xj. PHlomathiqiie de Рлга. (8). 3,189СЫ, 29-30.
** Mathesie, B)f 1, 1891, 95-96.
» Ai»oc. frami. av. ec.r 1R91, II, 201-2.
l» Recherches eur quelqaes prod. ind6f.f M&o. And. Roy. Befejque, «t, 1873, Cl-
393.

. JJ POLYGONAL, FtRAICBAL AND FlGURATB NUMBERS. 29
fBut the denominator 2 on the left should be suppressed. Legendre" had
already proved a more general theorem.]
E. Lucas171 collected results, mostly algebraic, on triangular and 6gurate
numbers. ,
E. Catalan17* stated that every Д, not pentagonal, is a sum of fewer
7 pentagonal numbers. £Catalan.JW3
T. Pepin171 gave a proof of Cauchy's formulation of Fermat'e theorem
that every integer Л is a sum of m -f* 2 polygonal numbers £m(x* — x) -f* x
of order m + 2 of which m — 2 are 0 or L We are to prove that
Л = \m{a — Ъ)+Ъ + г,
where a = a* + ■ " 4~ $l, 6 ^ a + ■ • ■ + ^ О^г^тл — 2, whence
a ^ 6 (mod 2). Since r can take the values 0 and 1, we may take b odd,
whence
> у > я > 0. Determine integers a, - ■ ■, 5 so that
Then a ^ 2a1 is satisfied. The condition Ъ* < 4a is satisfied if Л > 110,
where A = mB -h c, 0 < с = т. Hence the theorem is true for all numbers
A > 110m. It was verified separately for all numbers ^ 120m -f- 16.
G. Musso174 proved, by use of geometrical representations, Bachet's**
second formula I> 10, and the generalizations
Р*ш—2~ A+ (*-*)!?*+ -2~PT Y~ (« odd),
» — 2 щ л . ^ » * t з — 2 ^t » — 2
Py = ^^^* + (ff - *)^ + "^— P5 + «- 2^~
also
p; = n^ - (» - 1У + (n - 2I - ^- - ± 1,
E. Catalan175 gnve a shorter proof of Bachet's same formula.
G. de Hocquigny17* noted thatf fta+b + c=*ct+0 + y = 0,
P ^ (&¥ + At 4- ДС)(Л. + Л,, + Дт), ГДР
6л^, 6л4 + lf Git4 + 2n* + 19 • • ■
are sums of three A's, while n1 + Bя — IJ +■ Bя -h 1)* is a sum of two.
171 TWorie dee гюшЬгац 1891, 52-63, $3.
TR Jour, de matb. apfe^ 1ЗД2, No. 353,
1»АШ Acc*d, Pont. Noovi Ibcd, 46, 1892-3, 110-13 L
™ СИопЫе di Mat., 31,1893,173-S: Hie /^ is «ir p"
I7»/but, p. 227.
1If Ustfaou, B), 4,1804,123 171, 211; B), 5,1895, 23,150, 211-2, C-

30 Кгдаовт of the Theory of Numbers. [Chap. 1
The sum of 2л -f- 1 consecutive A's equals the product of the middle one
by 2n + 1, increased by lf + •• ■ + n1- He asked when
Ai+ •■■ + Д. = Д.
E. Barbette177 noted that the sum Tk of the kth powers of the 6rst n
triangular numbers equals Sk(S -f* l)*/2* symbolically, where, after expan-
expansion, S* is to be replaced by St, the sum of the fth powers of 1, ■ ■ -, n.
The values of Tlf TS} Тш are given as functions of n and as functions of the
S'&. It is shown that Tl = T*r implies x = yt к = r.
A. Boutin1** noted that A, ~ pA9 has an infinitude of solutions if p
is not a square. Let 2x *? ft - 1, 2y - г - 1. Then A? - рг2 = 1 - p.
Let A = л + p&, z — fi -f* a. Then a2 — pfP = I, having an infinitude of
solutions if p is not a equare* If p = m*7 the problem has only a finite
number of solutions if any. It is impossible if m = 3, 4, 5, 7, 8, 9, 11,
12, 13, 15, 16, 17. If m = 4X + 2, г - 4X2 + 4X, у = X is a solution.
Several179 solved A* + Д„ = 2Д,, i. e.,
Bx + I)" + By + IJ =
See Ch. XIV.
H. W. Curjel,1M to prove de Rocquigny's17* first statement, took
г£ -Ь уч + хГ, ^ = yf + жч + «Г, and got
- Z)
E. Maillet1* proved the following generalization of FermatV6 theorem
on polygonal numbers: If at and 0 are relatively prime odd numbers,
a > 0, every integer Л exceeding a certain limit (function of a, P) is а зита
of four numbers of the form {ахг + fix) /2. We can assign an inferior limit
to A such that this decomposition can be made in any assigned number of
ways. A like theorem holds if a/2 is an odd integer and one of Л, 0/2 is
odd and the other even, provided a/2 and 0/2 are relatively prime; also if
a/2 is even and 0/2 and A both odd* He proved three complicated theorems
stating that every number with certain residues modulo 6 is a sum of at
most & < 59 (or В < 53) numbers of the form (ax4 + 0x?)l2. Later
Maillet182 proved that if ф[х) = a<#? H~ • • • + abl in which the a's are given
rational numbers, is integral and positive for every integer x ^ pt every
integer exceeding a fixed limit, depending on the a's, is the sum of at most
у positive numbers ф(х) and a limited number of units, where v — 6, 12,
96, or 192, according as the degree of Ф is 2, 3, 4 or 5. Every integer
^ 19272 is a sum (p. 372) of at most 12 pyramidal numbers (ж* — s)/6.
, B), 5, 1895,111-2.
"* Jour, de math. №m.t D), 4, 1895. 179-180.
"»Malb. QueeL Educ. Timee, 63,1895, 40.
^/M.,33^ Other рлюЬД2), 20,1911, 78^9.
" Bua Soc M*th, de France, 23, 1806. 41b».
"■ Jour, de Mmth., E), 2, 1806, ЗЯЫ80.

] Polygonal, Рувамгоаь and Figtjbatb Nuhbebs* 31
Several writers181 found the first six integers n making n(n -f l)/2 a
square. Several1*4 proved that the difference of the roots of two successive
triangular numbers, each a equare, equals the sum of two successive integers
the eum of whose squares is a square.
A. Boutin1*1 reduced я* = Л, -J- 1 to p1 — 2$* = — 1 by setting
2s = 3g=Fp, У = №- 1)A к = 3p =F 2q [Euler71}
G, de Rocquigny111 noted the identities
- Ь - 1)}
ab - 1)
{ЛGа-Ы)+ Л(а
= ЛGс + I) + Д(е - 1), с = 5оЬ + а +
and expressed п*у п + л1 -Ь л' + rt4, n1 Н- п* + п§ 4- л* and n 4- ■ • ■ +
аз sums of three triangular numbers, ete.
A. Boutin solved Д,^ -f A» = y* by setting ж = an — &, у = an —
Then
tf -Ы = 2*1, V + Ъ « 2JS1, 4^5 -j- 1 « aB6 + 1),
which are solved by means of recursion formulas.
A. Berger19* proved many relations and inequalities involving the rth
m-gonal number C) designated by P(m, r). If j x j < 1,
He evaluated 21/P(o, r), where r ranges over all integers for which P(a, r)
takes positive values and each but once. If a ^ 3, | z j < 1, < = dr 1,
П A - s^Ott
combinations of special cases of which give
^ па -«о ^
Let <r(k) be the sum of the divisors of k, and ф(к) the excess of the sum of the
odd divisors of к over the eum of the even divisors. Then
i« Amer. U«Ul Monthly, 3r 1896,8l 2; МдОъ Quest. Edoc ТЪшв, 65,188бг 63; 60,1808,
51.
1U Amer. Math. Monthly, 4,1897,187-9.
»• WatbeeiE, B), в^ 1896, 3»-29.
• МЛЦ B), 7, 18fl7, 217-221.
Not» Act* Soe. 8c. UpwOieDeie, C), 17, 1S98, No. 3.

32 History of the Theory of Numbers. [Chap, i
He studied {pp. 20-25) the number ф(а, к) of polygonal divisors of order a
of a positive integer h; every integer has in mean two triangular divisors,
*r*/6 square divisors, etc.
A. Goulard and A. Emmerich*89 found two consecutive integers of which
one is a equare and the other triangular. In x* — %y{y + I) = =b 1, set
2x *= z, 2y + 1 =t, whence 2z* — t* = 7 or — 9, which are reduced to the
Pell equations и* — 2& = — 1, or + 1, and solved.
P. Bachmann1™ gave an excellent exposition of CauchyJsM proof of
his modification of Fermat's theorem: every integer is a sum of m tn-gonal
numbers of which all but four are 0 or 1.
R. W. D. Christie191 noted two formulas of the type
J. W. West1" noted that if Aa - 6Л* + 1, Д« is not a square.
R. W. D. Christie1" proved that, if рГ is the mth r-gonal number,
BпУ(г - 2)p; + iDns - n)|f = (x - yY + (x - ЪуУ + (x - ВД» + ...,
x = 2?nn(r — 2), у = r — 4.
W. A. Whitworth and A. Cunningham1" noted that if N = Am + Д»,
4ЛГ + 1 =^ (ш + n H- II H- (m — n)*; conversely, if AN -\~ I has no prime
factor 4£ — lf it is a sum of two squares and hence N is a sum of two Axs.
Crofton1*5 noted that
9A* + 1 - ACfc + 1), 4A, + 4Л, + 1 = (k - J)> + (k + 1 + 1)*.
Christie employed Am + A^i = (m + IJ to get
- Л2я_1+ «* + £*+ •■■-
W. A. Whitworth1^ gave rules, depending on the convergent» to the
continued fraction for V2, to solve Д = D or Д = 2a', equivalent to
known rules to solve u2 — 2v* = ± 1.
E. Lemoine197 called a number N decomposed into its maximum tri-
triangular numbers A, and m the index of N, if ЛГ — Al + — • + Лж, where
i4j is the largest A ^ JV, Л2 the largest A ^ N — AXj At the largest
A ^ N — Ai — At, etc. If Fm is the least number of index m,
Ym ^ jY^Y^ + 3), 2^У„ = (Гх + 3)(У, + 3) - -(^^i + 3).
fiiBr B), 8718^8,52-4, Cf. Tite.™
1Ю Die Aritb. der Qundiatiscbeii Fonmen, 171898,154-162.
>n Math, Quest. Educ, Times, 68, 1898, 84.
™lbid.,Wt 1898, 114»
lw/fcrf., 70,1899, 119.
^IWI.,71, 1899,33,
1Я /Ьй., 69.
lM /WA, 73,1900, 32-3.
117 Aesoc. Ггалд, avanc. ec,, 1900p II, 72,

. I] PoLYGONAb, PYRAMIDAL AND FlGHRATE NUMBERS» 33
Б. Grigorief198 discussed Fermat's theorem that every number is a
sum of three A's.
L. Kroneeker*5* gave a brief history of polygonal numbers.
J. J, B^rniville200 evaluated series involvmg figurate numbers, such as
j, + Aj + 33J-i + (I3 + 3* + 63J^ + A» + 33 + & + 1OJJ + ■ ■ • =6416.
A.Cunningham201 noted that л!+д! = 2 Ay if Дх = ^Дг, Д„ = rjAiy
where (£, ч) = (lr 1), G, 5), D1, 29), B39P 169), etc.
Cunningham and Christie202 solved ^Д* — vAvt which is equivalent to
fiBx -h IJ "" K% + IJ ~ & "~ *j by use of a solution of £2 — /**V =? 1.
R. W. P- Christie203 argued that no Д is a cube > L
A. Cuimingham301 noted that A.A* = A^A* is equivalent to
АЛ(Х* - I) = Aa(Y* ~ 1),
whose solutions follow from the least solution of £2 — ДаЛа^2 = !•
Christie20* noted that N = А2л -h А2л Н- Ajc implies
2N + 1 = (a + 6 + с + II + (a - 6 - cJ + (a + b - cJ + (a - Ь + cJ7
and similar formulas in which some of 2a, 2b, 2c are replaced by odd numbers.
Cunningham** noted that, if x = §(№ - 1), A* - 2- -21 • • -1 (я
two's and n one's).
E. B. Escott207 proved that 55, 66 and 666 are the only triangular
numbers, with fewer than 30 digits, consisting of a single repeated digit.
F. Hrom&dko208 noted that if Ai, ■ * ■, Ал are any consecutive AJs,
Al- At = (At - ДО' + (Аз - A*M + ■ ■ ■ + (A. - A._i)'.
L. von Schrutka3^ proved that, if I ^ p* (mod A:), then
and conversely if ??i/2 — 1 ia prime to A;7 so that к is called regular. The
question of polygonal residues thus reduces to that of quadratic residues.
Irregular moduli к are treated on pp. 190-3»
J. Blaikie210 noted that \n(n + 1) is also a pentagonal number
im(Sm - 1) if Zy% — x* = 2, where x = 6wi - X, у = 2« + 1. From solu-
tions of the Pell equation p2 — 3q* = 1, we get solutions x = 123g ± 71py
у = 41p ± 7Ig of the former.
п bv. fiz. mat. obec. (=* Bull. Math. Phya. Socr Квдап), l\7 1901, No. 2y 64-69
(in Russian).
1M Vorieeuiigen uber Zahlentheorif, 1901, 17-22.
3M Math, Quest. Educ. Times, 74, 1901, 80.
3W /Ш 65G
Wa4f 87-8.
Ш.У 75, 1901,
Ы(Ы
"" /Wd,, B), 1, 1902, 94-5; 6, 1904, S5-6.
** /bwf.f 8, 1905, 25.
"" /Wd.t 33-4.
5CR Zeitschr- Math. Naturw. Untcrricht, 35, 1904, 306,
"f Monatsbefte Math. Phys., 16, 1905, 167-193.
110 Math» Quest. Edue. Times, B), 9, 1906,40-41.

34 Hibtobt of the Theory or Numbers,
It is stated*11 that every nth power is a sum of n д'э Ф 0; for example,
3< - 55 + 15 + 10 + 1,
5* = 2*50+210+ 45+ 2 10 = 3003 + 105 + 10 + 6 + 1.
G. Nicolosi211 gave an elementary proof of Cantor's result that
has one and but one set of integral solutions. Solving for у we see that
&c _|_ 8a + 9 must be a square u\ Thus x is integral only if иг — I = SB,
whence В = t(t + l)/2.
C. BuralbForti213 noted relations like
- 2),
A. Cunningham111 gave a method of expressing an integer as a sum of
three triangular numbers.
P. Bachmann*15 gave an introduction to polygonal and figurate numbers.
T. Hayashi*16 proved that the quadruple of a number a(a + &)(<* + 20)/6
and hence of a pyramidal number is not a cube, by making use of the known
impossibility of s1 + y1 = 3z*.
E. Barbette217 summed the pth powers of consecutive n-gonal numbers,
found sums of pth powers of n-gonal numbers equal to a pth power of an
n-gonal number> found cases with n^6 in which a sum of two n-gonal
numbers is n-gonal, and gave a table of the first 5000 triangular numbere.
Н. Brocard21* solved ШЛХ + A» = ^ for x and made the radical
rational*
h. Aubry21* noted that A*-iA*A*+i ** D if (x - l)(x + 2) = 2y*,
whence 1? — 8t£ = 1> where 2x + 1 = 3u, у = 3v. The solutions are
known to be и ~ 1, 3, 17^ . • -, «я « Qu*-i — u^.
A. G^rardin210 collected recent problems on triangular and pentagonal
numbers* He noted (p. 70) that
3*"s = Да - A>, а = 3*s + C- - l)/2, 6^3^- C"
Let a, b become c, d when x = y2; then
A(c)
pp, 7, 31, 46.
П Fitacon, Palermo, 15, 1908-9, 15-17. In Suppl. al Periodic^ di Mat., 1908, faac. &-6,
there ii a proof by triangular numbers
«"/Wi, 1^ 1909-10,135-6.
*" ММЬ. QoeeL Educ. Tlmee, B), 15,1909, 44-5.
» Niedne Z^entbeorie, 2t 19Ю, 1-14.
**N<mY. Amu M*Ul, D), 10r IS 10, $3.
"'Leeeommefl de p-tfemee puiesanoee dietinctee egalee h une p-i&me ршввапсе, Ii^ge, 19 Ш»
154 pp. Extant by Baifaette.»
» Sphini-Oedipc, 6, 1911, 29^30.
ш /Ш^ 137-& РгоЫеш of lionnet, Nouv. Aim. Math., C), 2, 1883, 310.
»• Sphmz-Oedipe, l»liF 40-3, 57-8, 81-6,11J2112*32

ChapЛ] POLYGONAL, PYRAMIDAL AND FlGtJKATE NuMBEBS. 35
He treated (pp. 97-101) the decomposition of various types of numbers
into a sum of three triangular numbers.
The ordinary definitions of polygonal and figurate numbers were repeated
by L. Tenca221 and E. A. Engler.222
L. Tits2" solved Emmerich's189 equation for y, made the radical rational
and was led in both cases to F — Sv* =* 1.
E. Barbette2" gave many numerical examples in which a sum of n-gonal
numbers equals an n-gonal number.
L. von Schrutka*2* found that ^{(|2*)* + XP\ is not expressible in *
single way as a sum of two numbers of the form T{x* + x)/2 + Ux unless
T/2 ~ 3 or 5. In the first case it is shown that, if p is a prime = 5 (mod
12), (p — 2)/3 can be expressed in one and but one way as a sum of two
S^gonal numbers 3x* — 2x. He gave an analogous theorem for 12-gonal
numbers Ъх2 — 4xt and one for numbers 5x? — 2xr
A. Gerardin225 solved пг + 2'n = ля for x by setting n — xplq. He
(p. 128) reduced A±AV — А^+я to 2A^ + 1 = Avand noted the solutions
Ax - 10,45, Av = 21,91.
L. Bastien22^ noted that x* — ^ ^ Л, X г = x2 + y* and x1 - 3y* = 1,
or if z — (x2 + у*)!\ г + 1 = 2\(x2 — y2) or vice versa, whence
BX2 - l)x2 - BX2 + l)y* = =h X,
G. Metrod225 noted that Au — Av - x2 if (u - v)(u + tf + 1) = 2s3,
whence 2s3 is to be expressed as a product of two distinct factors, one even
and one odd.
R Mariares229 noted that the sum of lf 2, * • *, n is n(n + l)/2 since the
sum duplicated makes a rectangle of n Ъу п + L Again,
n(n + 1) ^. лл / n\3
1 + 3+6+ ••• + I = 2' + 4' + ... + I 2-)
0Г
according aa n is even or odd. Hence
Ai + Ai+ — + An^i+ Лх+ ■•• + An «
Numbers simultaneously triangular and pentagonal have been treated.
124II BolL di Mat. Sc. Fis, Nat., 12P lfllO-11, No, 1, p. 16, No. 3F p. 24.
m Trane. St. Louia Aca<L Sc, 20p 1911, 37-57.
ш Matbeeie, D)p lp 19U, 74-5,
ш L'cnseignemcnt math., 14P 1912,19-30. Cf. Bftrbette^7
51i Monatehefte Math. ТЬув.у 23,1912, 267-273.
*" Sphinx-Oedipe, 8P 1913, 110F 121-2 A907-8, 173; 19llF 75).
"ЧШ., 156,172-3.
ш Ihid.t 174.
™* Revieta Soc. Mat. I^pafiola, 2,1913, 333-5.
40 MatheeiH, D)p 3, 1913, 20-22, 80-Й1. a E

36 History of the Theory or Numbers. [Chap, i
* S. Minetola**1 gave a combinatory definition of the numbers In
TartagUa's triangle.
N. Alliston and J. M. Child232 proved that no triangular number > 1
is a biquadrate.
An anonymous writer233 proved that if a number 4n ■+■ 1 is а ЕЯ, it is a
sum of two triangular numbers c(c -h l)/2 and d(d + l)/2, where d may be
negative; then с and d are of the same parity.
G. Metrod234 stated that, if p* is the «th polygonal number of q sides,
the g.c.d. of p* and p"*1 is the g,c.cL of n ■+■ 1 and q — 3 unless the latter
are evea and then is the g.c.d. of (n + l)/2 and q — 3. The g.c.d. of p*
attd P%ilQ n or n/2 according as n is even or odd.
A. Ger&rdin254* noted that 2 дх — 1 is a prime for x ^ 9. He2346 gave a
series for дя • Av = A] with the law of recurrence *«+t = 6гя — 2n_i + 2,
га *= 3, zi^20. He33^ gave a general solution of &a + д& = e2 + 2Atfj a
special case having been noted by Euler60, and noted the examples a=2s+lJ
Ь - 4e, d ** 3s, e = s -(- 1; a - 6s + 2, Ь = 4s - 1, d = 5s + 1> e = * - 1.
E. Bahier235 found sets of three m-gonal numbers in arithmetical pro-
progression : p^ -h p*n — 2рп- Multiply each p by $(m — 2) and add (m — 4J
to each product. By Diophantus' relation A), we get
Pi + Pi = 2/^ PA ^ (ffl - 2)BX - 1) + 2.
Hence, by Ch. XIV,
PA = ± (ж* - 2ry - y*)> P? = x* + y\ Py=x* + 2xy- y2-
Then X> pt v are found in terms of xf y, rn by us& of the above equation
defining PA. The conditions that X, д, у be positive integers are discussed
at length.
S. Ramanujan236" obtained expressions for the number of representations
of л as a sum of 2s triangular numbers.
Notes* from l'interm^diaire des mathematiciens,
A. Boutin,1* I, 1894, 91; 2,1895, 31, noted that the square of each term
of the series 0, 1, 6> 35, - • -, u* = 6w*_i — u^.^ - •" is a triangular number
A, and stated that the A's in this series (viz., 0, 1, 6 up to щ4) are the only
AJs whose square is a A. He gave all solutions x = 8, 800, • • • of
x* + Ax = □ and stated that j/3 =b 1 = A* only for у = lp 3, 16, 20;
i e» 0, 1, 7, 90, 126. An incorrect solution of the latter by E. Fauquem-
M Boll» di Matematica, Roma, 12P J913? 214-22*
"* Math. Quest. Educ. Times, 25, 1914, 83-5.
"* Nouv. Ann. Math., D), 14, 1914, 16-lS*
ш Sphuu-Oedipc, 9, 1914, 5,
»*« Ibi±t p, 41.
«»/«d..p.75pp. 146.
1140 /Wd.k p. 129.
ш Recherche . . . Triangles Rectanglce en Notnbree Entiera, 1916, 217-233,
=*" Trans. Cambridge PhiL Soc., 22, 1918, 269-272.
♦For a more extended account вес G6rardin.ao The present notes were obtained inde-
independently.
** Jour» dc math» 61dm., D), 4, 1895, 222. Cf. Lionnet»1^

Chap. 11 POLYGONAL, PYRAMIDAL AND FlGUHATE NUMBERS. 37
bergue, 4, 1897,159-162, was corrected later, 5,1898,257. P. F. Teilhet, 11,
1904, 11-12> verified that aside from 0, 1, 6 there is no д with fewer than
660 digits whose square is a A.
G. de Kocquigny, 2, 1895, 394, noted that every triangular number
except 1 and 6 is a sum of three, each ф 0, since537
) + Л(р), ДCр) = 2дBр) + А(р - 1),
- АBр) + АBр + 1) + А(Р).
On A's expressed as а зшп of two or three A's, see 4, 1897, 158. For solu-
solutions of (x + IK - x3 = Д„ see 4, 1897, 262-4; 5, 1898, 18, 110-1 (and
Mathesis, B), 8, 1898, 126).
E. Fauquembergue, 4, 1897, 209, noted that Ax + Av = z- is equivalent
to Bx + IJ + (?y + 1}* ~ B« + IJ -h Bг - IJ, which by Euler's for-
formula for the product of two sums of two squares has the solution
2x + I = ac + bd, 2y -\- 1 = be — ad, if &c -f ad = ac — bd + 2. Cf,
Cerardin12 of Ch. XXIV. A. Palmstrom, 210, noted that the problem is
equivalent to
(x + y)(x - у + 1) = 2B 4- у)(г - у).
ОпЛ1-ЬДу=г5 see 7, 1900, 250. Е. 13. Escott, 11, 1904, 82, noted
that A* + Av = 2s is equivalent to Bx + 1)- -h By + IJ - 2Dг^ + 1),
a necessary and sufficient condition for which is that every prime factor
of 4г5 + 1 be of the form 4?г + 1; and gave solutions for z = 1T 4, C, 9,12,16.
On Ax = у* + г2 see 3P 1896, 24S; 4, 1807, 129-132, 255.
Any number N is & sum of three pentagons (Зх? ± x)/2 since
3 = S (fa ± 1)-
3
is solvable, 4, 1897, 157. On A' + д" = Д, 4P 1897, 158; 5, 1898, 70.
Thesumn(n + l)(n + 2)/6 of the first n ATsisa д for н. = 1, 3, 8, 20, 34P
but for no further n < 316, 4, 1897, 159; 6, 1899, 176; 7, 1900, 102; 16,
1909, 236; 17, 1910, 110; and is a square for n = 1,2, 4S, but for no others
< 10", 9, 1902, 279; 10, 1903, 235. The sum n(n + l)Bn + l)/6 of the
first n squares is a A for n = lt 5, 0,85 by 6, 1899, 175; 7, 1900, 211; 9,
1902, 278.
P. Tannery, 5, 1898, 280, and C. Berdcllci, 7, 1900, 279r gave algebraic
and geometric proofs that, aside horn 6, every ?bgonul number is a sum of
p — 2 triangular numbers > 0.
A prime бтг + 1 - dp2 + q* h a sum of 3 A's >0, 4, 1897, ПО. Since a
prime $n -h 1 equals 8m* + Bp -f- 1)-, it equals A*m H- D 4- Pt w^here
P - mFm - 1} is pentagonal, 8, 1901, 183.
O. de Rocquigny, 7, 1900, G5, 195; 8, 1901, 52; 9, 1902, 116, 176, 230;
10, 1903, 5-6, 40, 122, 205-6, 285, 300-2; 11, 1904, 09, 150, 158, ИЯМ,
189, 214, 237; 15T 1908, 181, stated many theorems of the following type:
every sixth power is a sum of a square, cube and triangular (or hexagonal)
number; every number > 7 is a sum of three Д/s and three squares each
* 0. A. G6rardin, 18, 1911, 177-184, 199, 275, discussed these theorems.
™ Sameby It. W. D. Cbristie, Math. Quest. Edac. Times, 09T 1898, 48.

38 Hibtobt of the Theory of Nukbebs. [Chap, i
G. Kcou, 9, 1902, 115, noted that a* - Ъ(Ь + l)/2 « 2** for
a = 2*+l + 2Я±1, b = 2«+1 + I or 2»*1 - 2.
H. Brocard, 10, 1903, 24-6, noted the solution
a - (9-2- - 2)/7, Ь = 8B- - l)/7.
P. F. Teilhet, 10, 1903, 240-1, gave a somewhat general discussion.
That 1 + 6Л * cube Ф 0, see 10, 1903, 97, 197.
P. Jolivald, 12, 1905, 16, 152, gave an erroneous proof that unity is the
only number simultaneously a A, square and hexagon. As noted by 1VL
Rignaux, 24, 1917, 80-1, a hexagonal number rBr — 1) is triangular, &o
that we have only to solve rBr — 1) = y2, whence By2 + 1 = EJ, whose
solution is known.
A product of 3 consecutive A's may be a square, 11, 1904, 158,
IL B. Mathieu, 16, 1909, 34, gave identities showing that the square of
any number ф 1 [4, 16}, and not a multiple of 3, is a sum of a A and a
square, each not zero [three A's].
A. Arnaudeau, 18, 1911, 132, deposited with the library of the Institute
of France the manuscript of his unpublished table of triangular numbers.
A. Gerardin, 1911, 205-7, gave solutions of x* + у4 + z4 = 27*/Р, where
T and t are triangular numbers Л. He cited, 273, Fuss47 note giving ftr+5,
9x + 8 as linear forms of numbers not a sum of two A's.
To decompose (n + 1)Б - пь into three A's see 19, 1912, 37, 104r-6\
For
(* + 1У + (* + 2)' + • • • + (x + my = Al+« - Alf
see 19, 1912, 114. L. Aubry, 19, 1912, 231; 20, 1913, 108, noted that
Al- &l = z% for x, у ^ (8m4 d= 12m* - 4m1 - l)/3. A. S. Monteiro,
20, 1913, 18-20, obtained solutions from the fact that the sum of the cubes
of any number of consecutive integers equals the difference of the squares
of two A's.
R. Niewiadomski,w 20, 1913, 5-6, gave many algebraic identities be-
between polygonal numbers, also expressions for nk, n* -\- 1, л8 + (n + 1)',
etc., as polygonal numbers.
U. Alemtejano (a pseudonym), 21, 1914, 169, stated that if 4m + 1 is
a sum of two squares, m is a sum of two A's, and conversely, since
Also, 9 is the only number 4 A» + 5 which is a square of a prime and not a
sum of two squares. Again, Ai*+* + A«_i — n — ra* + (n + aJ. Proofs
by L. Aubry, 22, 1915, 69. Alemtejano, 22, 1915, 8, gave
{4(A. + A.) + 1}' = Ba + 1L2» + II + {4(A* - Ал)\\
Several, 22,1915, 167-8, proved that every square is expressible ш the form
A« — 2 A# in an infinitude of ways. On the last digits of A, see 22, 1915,
235-6.
*■ Abo in Wladomeci M&t^WaieawTl7j 1913,91-9вГ~" "

3 POLYGONAL, РЕВАНША!, AND FlGURATE NU1CBEB& 39
A. Gerardin, 21, 1914, 133-5, considered numbers expressible simul-
simultaneously ш the form (x + l){x + 2) --<* + /»)/pI for p *= 2, 3, -•-.
He, 22, 1915, 203-5, considered the representation, of numbers by & + Sf*
4.^ + 10, where w is polygonal.
The question pVfeyl1*] of tetrahednd numbers which are squares
reduces to IP — N = 6n*t which was treated incompletely by L. Aubry,
26, 1919,85^87.
A. Bout in, 26, 1919, 35, 123, proved that no number is simultaneously
triangular, hexagonal, and a square.
For minor remarks on triangular numbers, see Glaisher*8 of Ch. Ill;
Euler»ofCh.VII; Realis"andpaper8ofCh. XIII; Pepin,1» and Cunning-
ham*» (on д, = са„) of Ch. XXI; Mathieu»2 of Ch. XXII.
In VoL I of this History were quoted theorems on triangular numbers
by G. W. Leibniz, p. 59; V. Bouniakowsky, pp. 283-4; U. Lipschitz, pp.
291-2; E. Barbette, p. 373; H. Brocard, p. 425; and R Jolivald, p. 427.
"РАГЕЪЯ OI^ P0I>Y0ONAI» ОН Г1ОПКАТЕ NUMBERS NOT ДУЛПЛВЬБ FOR REPORT.
G. U. A. Vieth, U«ber fig. ZaUen, Prop, Dfflaau, 1817,
J. P. L. A. Rocbe, Dem, nouv. dee formules dee piles de boukta, Toulon, 1827.
H, Anton, Anth- Rcihen hob. Ordr u. die fig, Z.t Progr. Olsp 1850.
A, Wicgand, Trigonaltmden ш arith. Progree,, Halle, ]850.
J. Van Cleeff, Verhandeling over de polygonoal of veelfaoekige getallen} GroningeDJ 1B55F
N. NicoIaSdee, Lee Mondea, 7P 1865, 893; 8, 1865, 615, 708.
J. U A. Le Cointe, Lee Mondcsf 8,1865, 707.
Soufflet, Lee Mondea, 13, 1867, 336 past 3 papers on fig. numbers].
J. Talir, Aiitb. Reihe b6h. Ord. u. fig. Z., ftogr., Waidbofon, 1872.
O. de Rooqiugny^Adanson, Lee nombree tri&ng., Mouline» 1896.

CHAPTER IL
D10PHANTINE EQUATIONS AND CONGRUENCES.
Solution of ax + by = c-
The Hindu Aryabhatta1 (fifth century or earlier) knew a general
method of solving indeterminate equations of the first degree. The original
of his treatise (on astronomy mainly) has been lost. Such a method of
solution is given in outline by Brahmegupta without the clear details of the
later presentation by Bhdscara.
Brahmegupta2 (bom 598 A.D.) gave the following rule to find a constant
" pulverizer." From the given multiplier and divisor, remove their greatest
common divisor (found by mutual division)» The thus reduced multiplier
and divisor are mutually divided until the residue unity is obtained, and
the quotients are written ш order» Multiply the residue unity by a number
chosen so that the product less one (or plus one, if there be an odd number
of quotients) shall be exactly divisible by the divisor which produced the
residue unity. After the above listed quotients place this chosen number
and after it the quotient just obtained. To the ultimate add the product
of the penultimate by the next preceding term fete]. The number found>
or its residue after division by the reduced divisor, is the constant pulverizer.
Thus if 3 and 1096 are the reduced multiplier and divisor, the single
quotient is 365. Multiply the residue unity by the chosen number 2 and
add 1. Dividing the sum by 3, we get the quotient 1. Hence the series
is 365, 2, 1, so that the pulverizer is 1+2-365 = 73L [We have
3-731 - 1 = 21096.]
Again (§ 27, p. 336), let the reduced dividend [multiplier] and divisor
be 137 and 60, while the augment or additive quantity is 10. By reciprocal
division of 137 and 60, we get the quotients 0, 2, 3, 1, 1 and last two re-
remainders 8 and 1. Since the augment is now positive and the number of
quotients is odd and since 1 - 9 — 1 is divisible by 8, we select 9 as the chosen
number. The constant pulverizer is said to be found as before* Its product
by 10 is divided by 60 to give the desired multiplier 10; 10 -137 + 10 = 60 -23.
There occur various problems (§§ 52-60, pp. 348-360) on astronomical
time leading to a linear aquation in two or more variables, special values
being arbitrarily assigned to all but two of the variables. One equation is
fy — 13fc = 266; without detail, the constant pulverizer is said to be 2
and the multiplier 4 = c, whence the quotient gives у = 135»
Mshaviracarya3 (about 850 A.D.) gave a process essentially that due
te Brahmegupta, though not requiring that the initial division be continued
until the remainder unity is reached. To find x such that 31a: — 3 is
1 Algebra, with arithmetic and meaauiation, from the Sanscrit of Brahmegupta and Bhieeaia,
translated by H. T. Colebrooke, London, 1817, p. x.
* Brahme-ephut'a-flidd'hfiQta, Cb. 13 (Cuttaca -algebra}, ftll-l*- Colebrooke,1 pp. 330-1 r
* Ganita-Sara-Sangraba- described by P. V. S. Aiyar, Jour, bdian Matii. Club, 2,1910,2l68
41

42 History of the Theory or Numbers. [Chap, п
divisible by 73, employ
31 = o-73 + 31, 73 = 2-31 + II, 31 = 211 + 9,
11 «1-9 + 2, 9 = 4'2 + 1.
The least remainder of odd rank is 1. Choose a number a — 5 such that
a*l — 3 is divisible by the last divisor 2, the quotient being 1» By use of
5, 1 and the quotients 2T 2, 1, 4 after the first, we derive
2 2 1 4 5 1
172-2-73+26, 73 = 226+21, 26 = 1-21+5, 21=4-5+1.
A smaller answer than 172 is given by 172 — 2-73 = 26.
In the second example, 63z + 7 is to be made a multiple of 23. Here
63 *= 2'23 + 17, 23 = 117 + 6, 17 =2-6 + 5,
6 = 1-5 + 1, 5 = 4-1 + 1,
the division being carried an extra step so as to yield the Last remainder of
odd rank. Here a = 1 makes a-1 + 7 divisible by the last divisor L
Discarding the first quotient, we have 1, 2} 1, 4, 1, 8 and then get 51, 33
13, 12. Since 51=2 -23 + 5, an answer is 5.
Bha'scara Achrirya4 (born, 1114) gave detailed methods of finding &
pulverizing multiplier (Cuttaca) such that if a given dividend be multiplied
by it and the product added to a given additive quantity, the sum will be
exactly divisible by a given divisor»
First (§§248-252), we reduce the dividend, divisor and additive by
their g.Crd, If a common divisor of the dividend and divisor does not
divide also the additive, the problem is impossible.
Next (§§ 249^251), divide mutually the reduced dividend and divisor
until the remainder unity is obtained» Write the quotients in order,
after them write the additive, and after it zero» To the last term add the
product of the penult by the next preceding number» Reject the last term
and repeat the operation until only two numbers are left» The first of these
is abraded by the reduced dividend, and the remainder is the desired quo-
quotient. The second of the two, abraded by the reduced divisor, is the
desired multiplier.
Example (§ 253): Dividend 17, Divisor 15, Additive 5. The quotients
are 1, 7r so that the series is 1, 7, 5, 0. Since 0 + 7*5 — 35, the new series
is 1, 35, 5. The final series is 40, 35» Abrading them by multiples of 17
and 15 respectively, we get 6 and 5 as the desired quotient and multiplier
£17-5 + 5 ^ 15-6].
♦Lflivati (Arithmetic), Ch. 12, §$24&-26в, Colcbrooke1, pp, 112-122. [It is nearly word
for word the same as Ch. II of Bhaac&r&'a Vija-ganit» (Algebra), §§ 53-74, ColebrooVe*
pp, 156-169; Bija G&nita or the Algebra of the Hindus, tr&nel. into V-ngibth by E.
Strachey of the Persian traasV of 1631 by Ata Alia Haeheedee of ВЪ&асага Acharya,
London, 1813, Ch. / of Introductioni pp. 29-36. LUawati Or a Tre&tise on Aritn. &
Geom. by ВЬаясага A^harya, tranal. from the original Sanskrit by Jobn Taylor, Bom-
Bombay, 1$16, Part III, Sect. I, p. Ill; the PeraUn tianal. in 1587 by Fyri omitted tbe
chapters on indeterminate problems- Lilawati was the name of Bh&ecai&'g daughter.^

, iij Solution op ax-\-by=c. 43
In case (§ 252) the number of quotients is odd, the numbers found by
the above rule must be subtracted from their respective abraders to give
the true quotient and multiplier. Thus (§ 255) for Dividend 10, Divisor 63,
Additive 9, the successive series are 0, 6, 3, 9, 0 [and 0, 6P 27, 9 and 0, 171,
27, and 27 = 2-10 + 7, 171 = 2*63 + 45], so that 10 - 7 = 3 is the
quotient and 63 — 45 = 18 is the multiplier [eheck: 10-18 + 9 = 3-63}
Concerning a "constant pulverizer" {§ 263, pp. 119-120), we may
golve the first example above by first treating the related problem: Dividend
17, Divisor 15, Additive 1, then multiply the deduced multiplier 7 and
quotient 8 by the former additive 5, abrade and get 6 and 5 as the quotient
ftnd multiplier when the additive is 5.
As to a "conjunct pulverizer" (§§265-6, p, 122), if there be a fixed
divisor and several multiph"ersT make the sum of the latter the dividend,
the sum of the remainders the subtractive quantity, and proceed as before»
Thus, to find a number whose products by 5 and 10 give the respective
remainders 7 and 14 when divided by 63, take Dividend 5 + 10, Divisor 63,
Subtractive 7 + 14; reduced Dividend> Divisor and Subtractive are 5,
21, 7; the desired multiplier is 14.
Bna^scara5 gave a rule for solving linear equations in two or more un-
Itnowns. In case there are к equations, eliminate h — 1 of the unknowns
and proceed with the single resulting equation as follows. Assign arbi-
arbitrarily special values to all but two of the unknowns. In the resulting
equation in two unknowns, solve for one in terms of the other and render
it integral by use of the pulverizer.
For example, of two equally rich men, one has 5 rubies, 3 sapphires,
7 pearls and 90 species; the other has 7, 9, 6 and 62 species; find the prices
(yt c, n) of the respective gems in species. Thus
% + 8с + 7и + 90 = 7у + 9с + 6т* + 62, y = c+j+8^
Take n = 1, and use the method of ft pulverizer to find с so that
у = (- с + 29)/2 shall be integral. We get
с = 1 + 2pf y = U-p,
where p is arbitrary. For p = 0, 1, we get (y, c, n) = A4, 1, 1), A3, 3, 1).
Again (§ 161, pp. 237-8), what three numbers being multiplied by 5, 7, 9
respectively, and the products divided by 20, have remainders in arith-
arithmetical progression with the common difference 1, and quotients equal to
remainders? Call the numbers c, n, p; the remainders yt у + 1, у + 2.
Thus
5c^2Oy «y, y = 5cf2l;
In - 20(t/ + 1) - у + 1, у = Gл - 21}/21;
9р - Щу + 2) = у + 2, у = (9р - 42)/2L
By the first two values of у, с = {In — 21)/5. By the last two,
_ n = (Op - 21)/7,
1 Vij*«ft£ita (Algebra), $$ 153^6; Coid^ooke/ pp. 227-^32.

44 History of the Theory or Numbers. CChap. II
which by use of the pulveriser gives n = 91 + 6, p = 7 J + 7. Then
с = F3* + 21)/5, which by the pulverizer gives с - 63fc + 42,2 = 5Л + 3.
Hence л = 45Л + 33, у = 35Л + 28, t/ « 15Л + 10. Since the quotient
equals the remainder, which cannot exceed the divisor, we must take A = 0.
What two numbers, except 6 and 8, being divided by 5 and 6 have the
respective remainders 1 and 2; while their difference divided by 3 has the
remainder2; their sum divided by 9 has the remainder 5; and their product
divided by 7 leaves 6 (§ 163, p. 239)? The conditions other than the last
give 45p + 6 and 54p + 8 as the numbers. As the product is quadratic,
take p = 1 [provisionally~]- Abrading the product by multiples of 7,
we get 3p + 2, which must equal 7/ + G. By the pulverizer, p ^ Ih + 6,
and the second number is 378ft + 332, The additive D5p) of the first
number multiplied by 7h is its present additive, so that the first number is
315ft + 51.
What number multiplied by 9 and 7 and the products divided by 30
yields remainders whose sum increased by the sum of the quotients is 26
(§ 164, p. 240)? Answer, 27.
What number multiplied by 23 and the product divided by 60 and 80
has 100 as the sum of the remainders (§§ 166-7, p. 241)? Taking 40 and 60
as the remainders, we get the number 2401 + 20. Taking 30 and 70,
we get 240J + 90; etc.
Bachet de Meziriac6 stated that if A and В are any relatively prime
integers, we can find a least integral multiple of Л which exceeds an integral
multiple of В by a given integer J \j. e., solve Ax = By 4- J^\> Proof was
given in the 1624 edition, pp. 18-24. It suffices to solve AX = BY -f 1.
Bachet employed notations for 18 quantities, making it difficult to hold in
mind the relations between them and so obtain a true insight into bis correct
process. Hcuce wre shall here carry out in clearer form his process for his
example A ^ 67, В = 60. Subtract the smaller number В as many times
as possible from the larger number A, to give a positive remainder C.
If С = 1, A itself is the desired multiple of A which exceeds a multiple of В
by unity. Next, let С > 1 and subtract С from В as many times as possible,
continuing until the remainder 1 is reached:
A) 67= 1-60+7, 60 = 8-7 + 4, 7=1-4 + 3, 4 = 1-3 + 1.
From the last equation we deduce
B) 3*3 = 2-4+ 1,
by the rule that if a = mb + 1 then ab + 1 — a is the least multiple of Ъ
which exceeds by unity a multiple of a. Multiply the third equation A)
by first coefficient 3 in B) and eliminate the term 3-3 by use of B); we get
C) 3-7 = 5-4 + 1.
•Clavde Gaapar Bju»hetp Problemes PLaiaane ct Delectablee, Qui ее font par lee Nombrea,
ed. 1, LyoDp 1612, Prob. 5; ed. 2, Lyonp 1624; cd. 3, Paris, 1874, 227-233; ed. 4, 1S79;
cd. 5, 1884; abridged ed., 1905. See Lagnuige."

. П] Solution of ах+Ъу = с. 45
ltipty the second equation A) by the coefficient 5 in (Z), and eliminate
the term 5-4 by use of C); we get
D) 43-7- 5-60 + 1.
Finally, multiply the first equation A) by the coefficient 43 in D) and
eliminate the term 43 - 7 by use of D); we get
E) 43'67 = 48-60 + 1,
go that the least X is 43 and the corresponding Y is 48»
Bachet's first step, leading to A), is Euclid's algorithm for finding the
greatest common divisor of A and B. His next steps are the elimination
from equations A) of the terms in 3, 4, 7, respectively, in a special way
go that negative quantities are not introduced.
John Kersey7 treated Problems 18 and 21 of Bachet,* but ''without
following Bachet's very tedious and obscure method of solution/* To
solve 9a + 6 = 7b, start with б and by successive additions of 9 form the
series 15, 24, 33, 42, - • -; next, form similarly the multiples 7, 14, 21, 28,
35, 42y — - of 7; the common number 42 yields a = 4, b = 6. Another
method is used for 49a -f 6 =* 13b; find the multiple F5) of 13 which just
exceeds 49 + 6; divide 49 by 13; since in 55 = 65 - 10, 49 = 39 + 10,
we have remainders differing only m sign, we add and get 104; then
b — 104/13, a — 2. If one remainder had been merely a divisor of the
other remainder, we first multiply one of the equations. Neither of these
cases arises for 121a + 5 = 93b. Then 126 = 186 - 64), 121 = 93 + 28,
and we seek с and d such that 93c -f 60 = 28d. After the latter is solved
by the former process, we deduce a and b as in the preceding case. In a
new type of problem, the constant term occurs in the member with the
smaller coefficient, as in 71a + 3 = 1736. Take 2>71, which increased by 3,
gives a sum < 173. Since 145 = 173 - 28, solve 173Л + 1 = 71B as
above to obtain A = 16, В = 39. Multiply the latter equation by 28
and subtract the former. Thus 173A6-28 + 1) = 71 -39-28 + 145, whence
b - 16-28 + 1 - 449, a = 1094.
Michel Rolle* A652-1719) gave a rule to find integral solutions; he
applied it as follows. For 12z - 221Л + 512, divide the larger coefficient
221 by the smaller 12; the largest integer in the quotient is 18. Set
z = 18Л + p; we get 12p = bh 4- 512» By the same method ^dividing
12 by 5} h ™ 2p + st 2p = 5* -f 512. By the same method, p - 2s + m.
Then 2m = $ -\- 512, and we have now reached a coefficient which is unity»
Eliminating s and p from
s = 2m — 512, p = 2s -f m, h = 2p -f s, z = 18Л 4- p,
we get the desired solution
__ г = 221m - 47104, k = 12m - 2560.
7 The Elements of AJgebra, London, Ip 1673, 30b
■Trait6 d'Algebre; ou Principce generaux pour resoudre lee questions de m&them&tique,
Paris, 1690, Bk. l. Ch, 7 t "i ? л"* «л ™

46 History of the Theory of Numbers» [с*ар. ii
But for Ills — SOly - 222, it is simpler to begin with 301 = 3411 - 32,
rather than with 301 = 2 111 + 79.
Thomas Fantet de Lagny* gave examples of a "new method" of solving
indeterminate equations. To make m — (\9n — 3)/28 an integer, douhle
28n — A9n — 3) and subtract the result 18n 4- 6 from 19n — 3; thus
n — 9 is to be divisible by 28» Hence n = 9 4- 28/, where/is any integer»
Later, he10 gave (pp. 587-595) the following rule for solving у = (ax 4- q)lp,
where a and p are relatively prime, and (as may be assumed) a < p> q < p:
Take a from p as many times as possible and call the remainder r; take r
from a as many times as possible and call the remainder tf; etc», until the
remainder 1 is reached. Then make the same divisions for q and p as were
made for a and p, having regard to the signs. According as the last re-
remainder is - s or -f st we have x — s or p — $»
L» Euler11 gave a process to find an integer m such that {ma + v)/b is
mtegral, where v > 0. Set a = ab -f c» Then А = {тс -f v)fb must be
an integer» Thus m = (Ab — y)/c» First, if v is divisible by ct we get a
solution by taking A = 0. Second, if v is not divisible by ct set b = /fc -f d.
Then m will be integral if (Ad — v)/c is integral Thus we set с = yd 4- e,
etc» Euler remarked that the process is therefore that of finding the
greatest common divisor of a, &, continued until we reach a remainder which
divides v. His formula for a solution of ma + v = nb is equivalent to
1 1,1 1 \ - /1 1 1 \
ab be cd de /' \bc cd de /'
in which the series are continued until we reach a remainder dividing v.
For the case a, b relatively prime, these results have been given by C.
Moriconi»1*
N» Saunderson13 (blind from infancy) gave a method to solve ax-by = cf
where с is the g»c.d. of a, fr. Let a = 270, Ъ = 112, whence с = 2. He
employed the equations and successive quotients
la-0b
0a - 16
a- 2b
2a -5b
Divide the term 270 of the first equation by the absolute value 112 of the
term of the second, to obtain the quotient 2» Multiply the second equation
by 2 and add to the first; we get the third equation» The division of 112
■IN'ouveaux Eiemens d'Arithmetique ct d'Algebra, ou Introduction &ux Mathematiqiiee,
Paris, 1697, 426^35.
11 Analyse generate; ом m^thodes nouvelies рошг r&oudre ]cs ргоЫётея detous lee Genres A
de toua led Degra к Tinfini, Paris, 1733, 612 pp. Same in M6m» Acad. Roy, d»
Soiencea, И, 166в^1в&9 [1733], annee 1720, p. 178,
» Conim. Acad, Pctrop<5 7t 1734г-5, 4fr-66; Corom. Arith. CoU., I, 11-20.
11 Periodico di Mat., 2, 18§7y 33-40. а. С Spelts, GiomaU di Mat,, 33» 1895, 125.
"The Elements of Algebm, Cambridge» lt 1740,275-288» The solution of the first problem
was reproduced by de la Botticrc, Mem» de Math, et Fhys., presentee . . . dive»
eavanfl, 4,1763, 33-41, Cf- Lagmnge,M
270,
- 112,
46,
- 20,
2;
2;
2;
6a
17a
- 12b =
- 41b =
- 94* =
6,
2,
3;
2;

щ Solution of ах-\-Ьу—с. 47
by 46 gives the quotient 2; the product of the third equation by 2 when
jdded to the second gives the fourth equation; ete. But on dividing в by 2
we use 2 and not the exact quotient 3, шлее the latter would lead to an
equation 56a — 1356 = 0 with constant terra zero.
Our sixth and seventh equations each solve the problem. Other solu-
solutions follow by adding to either equation 56a — 1356 one or more times.
The process must succeed since the formation of the constant terms is
identical with Euclid's process to find the g.c.d. of at b.
The determinant of the coefficients in any two successive equations of
ihe above set is d= 1. From the pairs of coefficients form the fractions*
0 1 2 5 12 41 94
1' 0' V ' 2* 5 ' 17' 39'
They are alternately less than and greater than ajb and converge to it;
if/ and F are two successive fractions of the set, ajb lies between them and
differs less from F than from/. Also ajb is nearer to F than to any fraction
whose denominator is less than that of F. This method of approximating
to fractions is attributed to Cotes and is said to be simpler than the methods
of Wallis and Huygens.17
L. Euler" proved thatifnanddarerelativelyprimeja + HCfc = 0,1,...,
И — 1) give n distinct remainders when divided by n, во that the remainders
are 0,1, • • -, n — 1 in some order. Since one remainder is fcero, a-\- xd = yn
is solvable in integers.
W. Ешегеод" used the first method of de Lagny* to solve ax = by + с
Let d and/ be the remainders obtained by dividing Ь and с by a. Subtract
some multiple of (dy + / )/ft from the nearest multiple of y. The resulting
"abridged" fraction or some multiple of it is to be subtracted from the
nearest multiple of y, etc., until the coefficient of у is unity. Thus
x ^ (Ыу — 11)/19 is subtracted from p; the product of the difference by 4
is subtracted from y; we get (y 4- 6)/19, an integer p, whence у = 19p — 6.
The same rule and same example was given by John Bonnycastle."
J. L. Lagrange,17 to find integers pt and q± satisfying pqt — qpx = ± 1,
where pt q are relatively prime, reduced p/q to a continued fraction E 29>
p' 423). As noted by Chr- Huygens, De scriptio automati planetarii,
1703, we get a series of fractions converging towards p/q, alternately less
than and greater than pfq. Hence take pi equal to the numerator and дг
equal to the denominator of the convergent immediately preceding p/g.
Then pqY — qpi = -f 1 or - 1 according as pifat < or > p{q. To apply
A8) to py ~ qx ~ г, where p, q may be assumed relatively prime, multiply
the former equation by ± r and subtract. Thus
x = mp ± гръ у ^ mq rt Tqt.
* The last и replaced by a[b if the final quotient bid been taken as 3-
u Novi Comm. AokL Ptetrop., 8, 1760-1, 74; Сотш. Arith. Coll., Ip 275.
" A Treatise of Algebra, London, 1764, p. 215; same paging m 1806 ed.
11 Introduction to Algebra, ed. 6,1808, London, 133.
17 МёЪ. Aead, Bcriin, 23, annee 1767, 17Й9, $ 7; Оеитая, 2,1868, 38o-8.

48 History op the Theory of Numbers» [Chap. II
Lagrange18 proved as had Euler" that, if Ъ and с are relatively prime,
there exist integers у and z such that by — cz = a. Next, if у = p, z = q
is one set of solutions, every set of solutions is given by у = p + me,
* = Я + mk If a = a'd, с =* c'd, where af and c' are relatively prime, then
p is divisible by d> say p = p'd. As in the proof of the initial theorem, we
can find m such that 7/ + me' is divisible by a\ Hence we can always
find a value of у which is a multiple or of a; then z is a multiple a's of af,
and frr — c's *= 1. From a set of solutions r, « of the latter, we get
у = та ~f ?nc, 2 = sa' + rafe*
Lagrange1* noted that his17 method is " essentially the same as Bachet's/
as are also all methods proposed by other mathematicians." To solve
39x — 5% = 11, employ
56 = 1-3G+17, 39 = 2-17+5, 17-3-5+2, 5-2-2+1, 2*2-1.
By means of the quotients 1, 2, 3, 2, 2, we get the convergents
1 3 10 23 56
1' 2' 7 J 16J 39'
Thus x = 23-11 + 56m, у = 16 11 + 39m,
L. Euler30 employed the method of always dividing by the smaller
coefficient, thus following Rolle8 in essence. For bx - 7y + 3,
, 2y + 3
x = у + —-• •- *
о
The numerator must be a multiple of 5, Thus 2y + 3 = 5z,
у = 2* + -^-, «-3 - 2u,
whence у - 5г/ + б; ^ = 7м + 9. He showed that the process is equivalent
to that for finding the greatest common divisor of 5 and 7:
7 = Ь5 + 2, x = 1-y +z,
5 = 2^2 + 1, у - 2-г + щ
2 = 2a + 0, г - 2-u + 3.
Jean Bernoulli21 applied LagrangeJs19 method to find the least integer
и giving an integral solution of A — Bt — Сщ when В, С are relatively
prime, in the special cases A = JC, §C + 1, ?(C ± 1). For example, if С
is even and A ~ C/2, then u = (B - l)/2, i - C/2. If С is odd and
4 = KC+ l),thentt = h{B + s - 1), t = §(C + r), where £r - C* « 1,
r/s being the convergent just preceding C/I? in the continued fraction for
the latter»
M Mem. Ac»d. ВсШп, 24, апибе 17GS, 1770, 184-7; Oeuvtea, 11^ 659.
»• /Ш., 220-3; Oeuvrcs, II, 69ft-9. Additiona by Lagnmge to Vol. 2 of the trand. by Jean
Ш ВсгРоиШ of Euler's Algcbm, I.yon, 1774,517-523 (Edcr's Opera Omnia, (I), 1,1911,
^74-7; Oeuvnes dc Lagrange, VII, 89-95).
*А1«еЬга, 2, 177D, K^23; French trcnsL, Lyon, 2, 1774, pp. 5-29; Opera Огиша, A), I,
326-339.
n Nouv» M&n. Acad, Roy. Berlin, anu£c 1772y 1774, 2S3-5,

р. Ш
Solution of ax+by = c. 49
J. L. Lagrange*2 used the method of Saunderson13 and noted that the
process is equivalent to the usual one of converting bfa into a continued
fraction. He21 gave a more popular account [results as in Lagrange17}
C. F. GaussM employed the notations
Apply the g*c*d. process to a and b which are relatively prime and positive,
with a ^b) let a ^ cib + c, b — $c + dy с = yd + ^, - - -, те = /m -|- 1,
so that
a ^ [V ti, —, t, ft «1 Ь « [n, /i, • ■ -, 7j 0}
Take x = [>, " •, Y, Я, у = Су, ■ ■ -, y, A «J Then ax = by + (- 1)*
if fc is the number of the terms a} 0, - -, ^ n» Cf. Euler.3*
Pilatte35 solved o^ -f axY = b, where aY and a are relatively prime,
a > flu by applying the greatest common divisor process:
a = a^i И- иг, «i = «2^2 + a3j - -4, aR_i = д„.
Replacing a by its value, we get x = x% — qxxXt where x7 = (Ъ — а^хл))ц1
must be integral. Thus a2si + n^ — 6- Proceeding similarly with the
latter equation, we get a&2 + a&i = bt • • •, ^„_x ^- ал_1Хл = b. Eliminat-
Eliminating s*-i> xn-2, • ■ *, we get x = zt <xb T аагя, where a is an integer deter-
determined by the process.
P. Nicholson* gave a method best explained by his example
500 - Их ЛА Их - г 1Л
*-—86 14-'-Зб-' Г = 1°-
Divide 35х Ьу Их — г to get the remainder 2x ^- 3r. Then divide Их — г
by 2x + 3r to get the remainder я — 16т, in which the coefficient of x is
unity» The remainder 20 from the division of 16r = 160 by 35 ia the least
positive x. But in the example
200 - 5x 5x~r _
y li ** и"' r = 2'
we reach the remainder x + 2r in which the sign is plus; thus 11 — 2r = 7
is the least x.
G. Iibri27 gave as the least positive integral solution x of ax + b = cy,
where a and с are relatively prime,
с — I I y?
. иат
sin —
с
n Jour, de Ip6oo]e polyt., cah. 5, 1798, 93-114; Oeuvres, VIIP 291-313.
nlbid<t cahfl. 7, 8P 1812, 174-9, 208^9; Keprint of Leconsflem. sur mat
normaJee, 1794-5; Oeuvres, Vlly 184-9T 216-9.
^Disq. Arith., i80l, 5 27; Werkc, I, 18G3, 20; German tranal., Maeer, 1889, 12-13.
M Abnales de Math, (ed., Gergonne), 2, 1811-12,230-7. Cf- E. Catalan, Nouv. Ado. Math.,
3, 1844,07-101.
* The Gentleman'* Math. Companion, Londony 4, No. 22, 1819, 849-60.
27 M£m. prfoentee рахй divers savants a 1'acad. toy. »c. dc Tlnstitut de France, 5, 1833, 32-7
(read 1825); extr. in Ann&les de Math,, ed4 Gergonne, 16^ 1825-6, 297-307; Jour* fQr
Math., 9, 1832, 172, Cf. A. Genocchi, Nouv. Согтеяр. Math., 4, 1878, 319-323»

50 History of the Theory of Numbers» [Скат, и
The number of integral solutions x, 0 ^ x < c, is
1 « W 2u{ax + Ъ)т
A. L. Crelle," after proving the existence of solutions of алх1 =* а&г + kt
where Ox and fli are relatively prime [Euler14], solved it by setting
also by the modified equations in which the left membera are all <ti, or xlt
There are given three more such methods» The sixth method uses a prime
factor «i of a! = aifiu and a primitive root щ of «i. There exists an integer
n such that ацг? = *i«i =b L Multiply the proposed equation by т\\
Thus ZiXi *= /ЗцгГач + xs, where Хз - (Air!1 =f Xi)M is an integer. The
latter gives хг = =F (<*iz3 — far*)* Here x> must satisfy aaOfc - ^ PiZt + *ifc,
which is treated as was the initial equation.
Crelle2' considered ay — bx + 1, where а, Ь are relatively prime and > L
If Zq, ^o give the least positive solution, the general solution is х„ - № Л- *о,
У« = Ф + 9o (m =* 0, ± 1, db 2} .. ■)• If Уо < Ь/2, the numerators of
Ш V^ Vx У^г Ух
xe} x_i' ^i7 x^f x2*
increase alternately by Ъ — 2y0 and 2y0, and the denominators alternately by
a — 2xq and 2z&, and no one of these fractions differs more from a/b than the
next fraction. There are similar theorems on series of fractions involving
only positive or only negative subscripts. If yJxH — b/a = к > 0,
v/u - b/a = \ > 0, where j » | < | y^x j, j и \ < | лм+1 j , then X > h
If X < 0, he found the number of fractions vju for which к > X, ц being
given»
J» P. M. Binet* treated ax — Ay = 1, Л > a, by a process for finding
the g,c.d. of a and A in which A is always the dividend» On dividing A by
a, a>i> <h, - •, let p, p^ p?J — * be the quotients and — ci, — ait — aZl
the remainders* Then
F) appf.pf.,, = ai + Л{1 +pi_x + p»--ip,--s + ■•• + Pi-i---p2Pil-
Let an be the divisor when the remainder is zero» Since а„ divides A,
it is the g.c.d. of A and a if it divides a. But if an is not a divisor of a,
proceed as above with a and a* and call the remainders — bi, — h, • • ■,
— brf, the last corresponding to the remainder zero. Then аЛ} bn-t c»"> - * -
form a rapidly decreasing series and one of them will be ±1» If ал = ± 1,
F) for t = n gives a relation of the form a? = ± 1 + AP^
E. Midy" used Euler's14 result to solve by — сг ^ a by trial.
" АЬЬ. AJtiuL Wi«*. Berlin (Math-), 1836, 1-Й' ^^ "
w/Wfp 1840, 1-57.
«Cbmptea Rendue Paris, 13,1841, 349-353; Jour, de Math., 6,1841, 44&-4G4.
«Nouv.Ann.Matii^4,1845IU6; С A.W. Berkhan, Lehrbucb der Unbeet Analytik, HaUe, 1,
1&55, 144; A. D. Wheeler, Math. Monthly (ed., Runkle), 2, 1860, 23, 55, 402-6; L. H.
Bie, Nyl Tidaakrift for Mat.y Kjobenh&vn, D), 2, 1878, 164; J. R Gram, ibid., 3, B,
1Ш, 57,73; Б. W» Grebe, Arduv Math. Рпун., 14, 1850, 333-5.

Chaf.H] Solution of ax+by — с 51
J. A. Grunert** solved bx — ay = 1 by a process for finding the greatest
common divisor of b7 a in which the divisor is always at while the dividend
is the sum of b and the preceding remainder, a process due to Poinsot,
Jour, de Math., 10, 1845, 48.
V. Roimiakowsky*3 would solve ax + Ъу = к by adjoining a'x + b7y =* h\
whose coefficients are arbitrary. Set D = ab' - ba\ p = b'fD, q = h'jD,
r = a'jD. Then x = kp — bq, у = aq — krt subject to ap — br - L
A. L. Crelle34 gave over 4000 pairs of positive integral solutions xt < alt
X2 < о,ь of axXi = a*ri + 1, for aY Ш 120, 0 < a2 < al} with a2 prime to
аъ and indicated methods used to simplify the calculation of the table.
V. Bouniakowsky35 integrated by parts
■ b')»-ldx
to obtain an identity giving a solution of ЬЩХ — br*Y = 17 where x =* a\
у — a is a particular solution of bx — h'y = L For m = n = 2, the
identity is
H. J. S. Smith36 reported on a recent method to solve ax = 1 -f Py
fno reference]. Join the origin to the point (a, P). No lattice point
(i. e., with integral coordinates) lies on this segment; but on each side
of it there is a point lying nearer to it than any other. Let ($lt 71) and
(fr, Vi) be these two points and let fj/^i < fs/^. Then the £Ts and Vs are
the least positive solutions of атц — Pfc = lt at\t — Pfc = — L
G« L* Dirichlet*7 solved ax — by = I by continued fractions, using the
algorithm due to Euler.3*
C-G.Keuschle39 found the general solution of ax + by = с by combining
it with ax + fly = m, where m is an arbitrary integer, while a and 0 are
integers determined so that a/3 — ba = db 1 [cf. Bouniakowsky**}
J. J» Sylvester40 noted that the number of positive integers < pq which
are neither multiples of p or q nor can be made up by adding together mul-
multiples of p and q is §(p — l)(q - 1) if p and q are relatively prime»
H. Brocard41 solved ax + by = 1 by a process of reduction» It suffices
to find the residue of a modulo a — b to obtain an equation x + у = f
consistent with the given one» Thus, if b = 563030, a = b + 7, then
a = b = 5 (mod 7), 3-5 = 1, and the given equation may be combined
with x + У = 3 to get integral solutions x, y> A table gives the successive
« Arcbiv Math. Phye., 7, 1846, 162.
» Bull. Acad, Sc, St. PtSterebourp, 6y 1848, 199.
"Bericht Abd, Wisa. Berlin, 1850, 141-5; Jouj. fur Math,, 42, 1851,299-313.
*Bull. CLPbya.-Matb Acad, Sc. St. Petersburg, 11, 1&53, Ь5>
» Report British АваосЛог 1859, 228-267, §8; CoU. Math. Papera, I, 43.
"Zfthlentbeorte.tt 23-24, 1863; ed. 2,187l; ed, 3y 1879; ed. 4, 1894.
"Comm. Acad. Petrop., 7, 1734-5, 46 (Еи1егж). Novi Comm. Acad. Petrop>, 11, 1705, 2S;
se^Euler,«Cb. XII. Cf. Gauss."
"Zeltschrift Math. Phya., 19, 1874, 272. S&me by J. Slavik, Casopia, Prag, I4t 1885, 137;
V. Schawen, Zeitechrift Math, Naturw. Unterricht, 9, 1878, 107 [194, 367],
« Math. Quest. Educ, Timea, 41, 1884, 21.
a. Acad. Sc. Lettres Montpellkr, Sec. Sc., 11^ 1885-6, 139-234. See p. 153.

52 History of the Theory of Ntjmbers. [Chap, II
values of x + у when a — Ь = 1, 2, - - ■, 100. The paper ends with a
twenty page bibliography and history of linear dtophantine equations.
C. A. Laisant42 constructed the points having as abscissas 1, 27 • -■, p
and as ordinates the corresponding residues < p modulo p of r, 2rt - — , pr
(r prime to p). The lattice defined by these points leads to an immediate
solution of rx — pz = a since every point of the lattice has the coordinates
x, У = ™ - Pz-
W. F. Schiller4* gave a collection of 374 problems on linear Diophantine
equations and an extract from Bachet* with a German translation.
E. W. Davis" used points with integral coordinates to solve ay — Ьх = k.
P. Bachmarm4* gave an extended account of Euclid's g.c.d. algorithm,
continued fractions, and related questions»
A. Pleskot" treated 13s + 23y = с somewhat as had Kolle':
с = Щх + 2y) - Zy = 3D* + 7y) + x + 2yt
*c + ly = t> x + 2y = с - Ш, x = - 7c + 2St, у = 4c - Ш.
J» Kraus47 solved ax — ofy = c, where ar — л = к exceeds <x and c,
by use of otrx - rx+1 = kaXi 0 < rA < k, 0 ^ ал < a, X = 1, 2, ■. -, thus
representing rjk as a number with the digits ak, ax+lt • — to the base a.
P. A. MacMahon4* proved that, if the continued fraction for X/^ has a
reciprocal series ah a2j • *», a*, di of partial quotients, 2i - 1 in number,
then the fundamental (ground) solutions of \x = py + z are (xj, y,, i/^+j-i),
j - 1, — -, <r, if X > (i, where <r - 1 + ax + aa 4- a5 4- ■ • • + a* H- <*i + <*i;
but are (x,, yh xm_,) and (^, y,,, 0), ; = 1, - •., <r - 1, if X < д, where
<r = 1 + Os + 04 + " ' +614 + 02+lj not including я,- twice. When the
partial quotients are even in number, the fundamental solutions depend
upon both the ascending and descending sets of intermediate convergents
to X/ji. He484 had proved that the fundamental solutions are always
fa, Vj? *j)> J' ^ !> " •, <r, where the yy/s, are the ascending intermediate con-
convergents to X//i.
A. Aubry*g plotted the points with integral coordinates О Ш x < nt
О ^ у < n7 as well as the lines у = xt у = ах, у = Ъх, — >, where 1, a, b,
* > - are the integers < n and prime to n. Thus we can read off the integer
s yjx (mod n) and hence solve ax — nz — g.
N.P. Bertelsen4Ja solved Ьх -су==Ьг, 1^1/<Ь70^а;<с,1 ^г<6,
by use of the convergents Ъг(сг to the continued fraction (do, alt ■ • ■, ал)
for b/c. Then у is a linear function with positive integral coefficients of
br + kbr+i (k = 1, 2, • ■ *, a^ — 1), and x is the same function of the
Cr + кСг-ц.
*Аиюс. fran^j. av. вс., 1бр IIP 1887, 218-235. ~~
«LehrbuchderunbeetimmtenGL I Grades, Stuttgart, l, 1891,176 pp. CKleyers Encykl.},
« Amer. Jour. Math., 15,1893, 84,
41 Niedere ZahleDtheorie, 1, 1902, 99-153.
« Zeit. Math, Natyrw. Unterricht, 36P 1Ш5У 403 [33, 1902, 47].
47 Archiv Math. Phys,, 9, 1905, 204.
41 Quar. Jour. Math., 3BP 1905, 80-93.
«■ Trau*. Cambridge Phil. Soc, 19, 1901,1.
*» LTenaeignenient math., 13, 1911, 137-203. Cf. G. Arnoux, Arith. Graphique, 1894, 1906.
•• Nyt Ticfagkrift fgr Mat,, Br 24, 1913, 33-53.

Solttiton of ах+Ъу=с> 53
дЬЬ^ Bossut, Соигв de Math,y II, 1773; ed. 3, Г, I78l, 418 (g.c.d.).
Y. Paob\ Element) d'algebrb, Pisay 1794, Ip 15Я (Rolle'a1 method),
j, C. L. Hellwig, AnfojagBgrunde unbeat. Analytik, Braunschweig, 1803, 1-80,
g' F. Lacroix, Complement des EKmcns d'AIgebre, ed. 3, 1804, 273-^9; ed. 4, 1817, 287-292
(g.c\d. and continued fractions).
f\ Pczzi, Memorie di Mat. e Fis. Soc. Ital. Sc, Modeua, 11, 1804, 410-25 (cont. fr.).
j. G. Garnicr, Analyse Aig^brique, рагшу ed. 2P 1314, 58-65 (cont. fr.).
L. Castcrman, Annales Acad. Lcodien&ifi, Iifcge, 1819-20 (cont* fr)s
p N. С Egcn, Handbuchder Allgemcinen Aritb.p BerHny 1819-20- ed, 2t 1833-4; ed. 3, II,
1849, 431.
M, W. Grebd, Ueber die unbesi. Gl. 1 Gr.y Progr. Glogau, 1827 (cont. fr.).
A. J. CheviHard, Nouv. Ann. Math,T 2P 1843, 471-3 (oontr fr.),
F, Heimc, Arith* Untereuchuogen, Progr, Berlin, 1850 (Eulcr14),
T. Dieti, Nouv. Ana. Math., 9, 1850, 67 (g.c.d.).
1L Schefflcr, Uobeetim. An^lytik, ITanover, 1854 (cont. fr.).
T. Thaarupy Nyt Tidaakrift for Mat,, Ay I, 1 fe.c.d,).
V. A. Lebe^gue, Exerciccs d*analyee n\im^riqucy Pane, 1859, 48-54.
Lebeegue, Introduction & la thtforic dcs nombros, 18G2, 39—47 (g.c.d.).
J, J. NeJ*<ilJP Euler'e Aufloaungs-Metbode unbeat. Gl. 1 Gr., Progr* I^aiba^h, 1863,
J^ A* Temmey Bemcrkungcn . , . unbelt, Gl., Progr* Mimster, 18fi5 (Eulcr1').
b. L Cbse^, Ann, de l'ecole norm, вир., 4У 1807, 347 (g.c.d,)*
O. Porcelli, Giornalc di Mat,, 10, Ш2, 37^6 (t^ont. fr.),
J, Knirr, AuflCeung der unbest. Gl., Propr. Wien, 1873 (EuleriC)»
C. de Comberouase, Algcbro аирбпсиге, IT 1887, 161-173 (g<c.d.y cont. fr.).
L, MatthJcBscn, Kommentar zur Satnmluiig . . . Aufgaben . . . E. Hcib, ed.4,1902, 98-9; 1897,
221,
H. Schubert, Niedere Analysis, I, J902, 116-126 (cont. fr.); ed. 2, 1908.
G. Calvitti, Suppl. iJ, Periodico di Mat., 1905 (Еи1сг1С).
M. Morale, ibid., 1909; Periodico di Mat,, 2o, 19Ш, 182-3 (cont, fr.).
A. Bindoni, II Boll, di Matemataa, 11, I9l2y 151^3 (Eulcr31*).
E. Caben, Theorie doa nombres, I, 1914, 90-108 (al»o graphic)»
A. Sartori, 11 Boll, di matcmaticiie e ec, fi«., 18, 19l6, 2-19.
Papers not available yor rbiport.
Ikrtrand, Analyse md^teririin^e du premier dcgi6>
L. Casterman, Petitur ut acquationcs indet. 1 Gr., Grandy 1823.
C. L. A. Kunze, Einfache u. leicbte Methodc die unbewt, Gl. dee 1 Grade mit 2 unbekaunten Z»
aufzutoeen, Pnogr. Weimar, Eisenach, 1851»
W. Korn (Dorii?), Die Lchre von d«n unbest. Gl. dea 1 u, 2 Grades mit 2 verSnderlichen GroeecD,
Progr. lnoebruck, 185G.
J. Ыегтев, Gl. 1 u. 2 Grades Bchcirtatisch aufgeluet, Leipzig, 1862,
G, FJoweon, От indet- Eqvationer af 1 Gradcn, Profcr* Lulea, 1865.
K. Weihrauch, Beitrag zur Lehre unbest, GL 1 Gr., Arcnsberg, 1800.
H, Dembachick, Unbest. Gb 1 u>2 Grades, Progr. Straubipg, 1876*
Ferrent, Jour de math, deb., B), 3, 1884, 121, J55, 169, 193, 217, 241.
E. Sanczer, New method for in<tetei\ firfit degree (Polish), Craoow, 1887,
G. M. Teeti, Sulla ri^erca di una soluzione di Una equaiione di primo grndo a due incognite,
Livorno, 1902, 4 pp.
E> Ducci, Le mie tezioni di analisi indeterminata di primo grado . <. P, Bologna, 1903.
8. Soechmo, Suppl, al Periodico di Mat., 12, 1908-9, 20-22.
ft, Guatteri, ibwi., 52-3; I3y 1909-10, 76-9 (both on Euler!0).
G. Bernard^, Kuovo metodo di rieoluzione dell'equazione ax + Ъу ^ с in numeri iuteri e positivi
. . ., Bologna, 1913y27 pp.
L. Carlini, II Boll, di Matcinatica, 12, 1913, 128-Ш-
F. Palatini, ibid., 284.
L- Struiste, Die iinearen diophant. Gl,, Progr., lanebruck, I9l3.

54 History of the Theory or Numbers.
Solution of ax = b (mod m) without Fekmat's Theorem.
C. F. Gauss50 noted that ax = b (mod m) is solvable if and only if Ь is
divisible by the g.c.d. d of a = de and m = df. Let b = dk. Then x is a
root of the proposed congruence if and only if ex = к (mod/), while the
latter has a unique root modulo/. For a compoiste modnlus win, a second
method is often preferable. First, employ the modulus m as above and
let x = v (mod mjd), where d is the g.c.d. of m and a. Then x = v + я'то/d
is a root of ax = Ь (mod wm) if and only if x'a/d = (b — <w)/m (mod n).
P. L, TchebycheP1 proved that, if the g.c.d. rf of a and p divides 6,
ax = b (mod p) has the d roots a, a 4- pfd, • •«, a + p(rf — l)/<f, where
ла/d = h/d (mod p/d).
C, Sardi" considered the congruence a^x = 6 (mod p) in which p is a
prime not dividing au and & < p. Dividing p by Oif let a2 be the remainder
and (jj/aj the quotient, where [nj is the greatest integer Ш n. Multiply
our congruence by \j>ja{]\ we get
a2x = — Ъ Гр/aQ (mod p).
Let Оз be the remainder when p is divided by a2. Let the decreasing series
аъ аг, а*у • • • end with a, = 1. Then
C* LaddH showed that if a is prime to Af = Af t ■ • «ЛГ* and if г, is deter-
determined by azi -h 1 ^ 0 (mod Af,-), the root of ax -f- Ь = 0 (mod Af) is
x « b{2zi -f в2г^- -f а*2г&рк +♦■■}•
L. Kronecker" reduced the solution of ax sz 6 (mod ти), where a is
prime tom = Пр^ to the case in which rn is a power pr of a prime. Then
a root can be expressed to the base p ш the form
where each £< is an integer chosen from 0, I, < -«,p — 1. First find the root
of a£0 = b (mod p). Then seek ^ from c(f0 4- fip) = Ь (mod p2), whence
fl£i ^ F — c^o)/p (mod p), etc. Again, if N is the denominator of the
next to the last convergent in the continued fraction for ajm, then x = ±bN
(mod m)_
M. Lerch*6 showed that, if p is a prime,
where ft] is the greatest integer = /, and hence solved ax — py — I. If
h., 1801, Arts, 29; 30; Wcrke, Ir 1863, 20-3; Maser*s German tr&neL, 13-15.
11 Theorie der CongruenzeD, in RjqssUd, 1849; in German, 18S9, § 16, pp. 58-63,
"Giornaledi Mat., 7, 1369^ 115-6.
« Math. Quest. Educ. Times, 30, 1879, 4l-2.
u Vorleeungen iiber Zablentheorie, 1, 1901, 108-120,
« Math, Aimdcn, GO, 1905, 483,

Solution of ax^b (mod m). 55
ia any odd number relatively prime to* ф(т), then
where the summation extends over all positive integers b which, are < m
and prime to m7 while Р(тп) = A — p)(l — p') • • -, if p, pf, ■ - - are the
distinct prime factors of m.
15. Buscbe54 obtained graphically the number of solutions of az = b
(mod та), including solutions called improper of transfinite,*7 introduced
whe£ a and m have a common factor > 1* As the ordinary (proper) solu-
solutions may be restricted to the integers 0, 1, * ■ ♦, m — 1, we are at liberty
to designate the improper solutions by numbers её т. The simplest case
is one like 3z = b (mod 15), in which 3 and 15/3 are relatively prime; then
there is defined an improper solution designated by 15 if b = 0, 15 + J if
Ъ = j (j = 1, - - >, 4), 15 if b — 5, 15 + j if b = 5 + j (j — 1, * -, 4), etc.
Solution of(k = 6 (mod m) by Febmat*s or Wilson's Theorem,
J. P. M. Binet" noted that, if a is a prime not dividing b, bx — ay *= 1
has the solution x — b^2, the corresponding у being integral; while, if
p} p', * ■ ■ are the equal or distinct prime factors of a,
bx *= 1 - A - 6^0A - Ъ>'~1) * • *
gives an integer or, leading to an integer y, such that xt у satisfy the same
equation. The same method was found independently by G- Iibri.w
A, Cauchy*1 expressed Binet's method in the following form: let
- = I -kK.
Then for k prime to n, 1 — kK is divisible by n, so that kx = k (mod n)
has the solution x = hK (mod n).
V» Bouniakowsky* proved that, if at b are relatively prime positive
integers, ax =F fey = с has the integral solutions*
x = со**»-1, у - ^(e*® - 1).
G, de Paoli82 gave the last solution, with ф(Ь) replaced by Ф(Ь)/2 when 6
is divisible by 4* To solve ax — by — cz = e, where a, by с, е have no com-
common divisor, let a = dAt Ъ ~ dB, where d is the g.c»d. of a, fc; then e -h cz
* By Ф(ш) is meant the number of integers < m which are prime to m,
» Mitt. Math. GeaeU, ШшЬш«, 4, 19OS, 355-380.
" Jmaginaiy by Gauss, Di$q> Arith., Art» 3i; G. Arnoux, Arithm^tique graphique, 2, 1906,
20. Both excluded euch solutioDs.
«Jour, de Г6со]е polyt4 cah. 20, 1831, 292 [read 1827]- communicated to the Sod6t6
Philomatique before 1827.
" M^moirea de Matb. et de Phye^ Florence, 1829, 65-7» Cf. LibriJ« of Ch. XXHL
« Exerdcee de Math., 1829, 231- ; Oeuvres, B), IX, 296.
* Мёш. Acad, So. St. P6tereboiirg tMath. Phys.), F), 1, 1831, 143-4 [read Apr. 1, 1829].
a Opuacoli Mat. e Fie. di Divert Autori, MiLaaot 1,1832, 269. He stated that the paper veas
written ш 1Б30 without knowledge of that by Bluet.

56 HlSTOEY OF THE ThEOHY OF NUMBERS. [Chap, 11
must be a multiple du of d; the equations Ax — By = u} du — сг = е are
each solved by Fermat's theorem; similarly for n variables (pp. 327-338).
A. L, CreIleM noted that ax = \ (mod m) has the solution о*(жИ.
A. Cauchy*4 obtained independently the result of Bouniakowsky.*1
J. P. M. Binet** employed Wilson's theorem to solve ax =* 1 4- pyt
when p is a prime. We may take 0 < a < p* Then x = — (p — 1Oа*
Whether p is prime or composite, we may also proceed as follows. Divide
p by a and call the quotient q and remainder й\,' divide p by ai and call the
quotient qi and the remainder a*; etc., until the remainder an = 1 is
reached. Then
V. Bouniakowsky* employed (p, n) ^ p(p — 1) • - (p — n + 1). Then,
ifb<p,
(P + br P) = (P> P) + (j) <P, P " l)№r 1) + B) (P, V - 2)Ff 2)
Divide by (pt p) and write a = p + &. We get aE = 1 4- P^, where ^
and X are integers* if p is a prime» Hence we have solved ax *= 1 -|- py
in integers if a > p and p is a prime. To solve
Mx - Ny = 1, N = pYr^ ■ -,
where p, q7 rt --» are distinct primes, determine ль ^^ • ♦ ■ so that
Max - PPt = 1, Л*а* - ^2 = 1, Маъ - rft = 1,
as above» Raise Max — 1, Afaj — 1, • * * to the powers X, p, ■ • ** Then
where е1? с*, * * •, and A below are integers- By multiplication,
MA + (- i)»+"+"+- = NB, В = AXA" • •..
According as X -f /i -h p -h * - is odd or even, у = В or — В.
L. Poinsot67 noted that Lx — Afy = X has the solution x — Z.1" if
Lm як 1 (mod Af), e. g., if tw = Ф(Л/)* Не also expressed the method in
terms of regular polygons» Thus, for 12я — 7y = 1, take 7 points Pu * •»,
P7» Take the first, the fifth after the first, etc. E being 12 — 7); we get
PiP9PiPtPjPtPz> Since Pt is now the third point after Pu we have x ~ 3.
We get у from the equation or by use of 12 points.
^« АЬЬи Akad, Wiea. Berlin (Math.), 1836, 52.
*Comptea Rendus Paris 12, 1841, 813; Oeuvt^e, (l)t VI, 113. Exetcices d'Aaalyse et dc
Physique Math., 2, 1841, 1; Oeuvres, B), Xll. Sec Vol. 1, p. 187, of this History. CU
report by J. A. Gnjncrt, Archiv Math. Phys.r 3. 1843, 203.
ш Comptee Rcndu» Pane, 13, 1841, 210-3.
м Мёл, Acad. Sc. St. Pfitersbourg (Math^ Phy*), F), 3, 1844, 2S7.
*E - (p + b - 1I ^- !p!M} ie an integer by Catalan,51 p, 266 of Vo]. I of thia History»
♦7 Jour, de Math., A), lO, 1845, 55-59.

лрЛ1] Chinese Problem or Remainders. 57
J. G. Zehfuss6* gave the formula of Cauchy60 and noted that, if
=: amfl* * ■ >, and if A is not divisible by the prime ay В not by 0, - ■ ■,
then
H- • ■ • =1 (mod /i).
por A — В = • — =* a, let the left member become k. Then ax = b
(mod /u) has the root kbfa. It also has the root A — AB • * -)J>/°/ where
i.
= 0 (mod л"*),
В=11+a* —I =0 (mod 0я), • ■-,
where <za is the least positive residue of a modulo ay since, by Wilson's
theorem, <za + (or — 1)! a is divisible by the prime a.
M» F, Daniels** noted that, if pt* • >pn =. ± I (mod к) by Wilson's
generalized theorem, then p,x = 1 (mod k) has the root ± pi-- p»_i
P»+i* "Рв' Further, if к = pV" ■ * s^^ if aci ^ I (mod p), acj ^ 1 (mod
q)} < • >, then ux = 1 (mod A;) has the root
x — - [1 — A — aci)"(l — (iCzY-' '\-
J» Perott70 noted that U. a and и are relatively prime and if a belongs to
the exponent t modulo u, ax = 1 (mod u) has the unique solution x = a'
(mod u). He admitted he was anticipated by Caucby.
Chinese Pkoblem or Remainders»
Sun-Tsu," in a Chinese work Suan-ching (arithmetic), about the first
century A.D., gave in the form of an obscure verse a rule called t'ai-yen
(great generalisation) to determine a number having the remainders 2,
3, 2, when dtvided by 3, 5, 7, respectively. He determined the auxiliary
numbers 70, 21, 15, multiples of 5-7, 3-7, 3-5 and having the remainder 1
when divided by 3, 5, 7, respectively. The sum 2 • 70 + 3 • 21 + 2 -15 «= 233
as one answer» Casting out a multiple of 3<5>7 we obtain the least answer
23» The rule became known in Europe through an article, "Jottings on
the science of Chinese arithmetic," by Alexander Wylie,*2 a part of which
was translated into German by K. L. Biernatzki.73 A faulty rendition by
"Dies. {Heidelberg), DarmHtadt, 1857; Archiv Math. Phys,, 32, 1859, 422.
•• Uneaire Congruenties, Disa., Amsterdam, 1800, 114, 90.
7» Bull, dee Sc- Math,, B), 17, I, 1893, 73^4,
* Y. Mikamip Abh. Geschichte Math. Wibb., 30, 1912p 32.
"North China Herald, 1852; Shanghai Almanac for 1S53. Cf. remark by G. Vacca,
Bibliotheca Math., C), 2 1901, 143; H. Cordier, Jour. Asiatic Бос., B), 19, 1887, 358.
» Jour, fQr Math., 62, 1856, 5&ЧМ* French transl. by O. Terqliem, Nouv. Ann. Math,, B)r
1, 1862 (Bull. BibL Hist.), 35-44; 2, 1803, 529-540; and by J. Bertrand, Journal dea
Savsnta, 1869. Cf, Matthieseeo P

58 History op the Theoey of Numbers. [Chap, it
the latter caused M. Cantor74 to criticize the validity of the rule. The rule
was defended by L, Matthiessen," who pointed out its identity with the
following statement by C, F. Gauss»14 If m = тхтгтг- • -, where Шц т*}
m*, — • are relatively prime in pairs, and if
cti = 0 (mod m/m,-), a< = 1 (mod m>) (i = 1, 2, 3, • • 0,
then x = ctiri + a2fi + - - is a solution of
x = rx (mod wii), x = r% (mod mt),
This method is very convenient when one has to treat several problems
with fixed mh mtl • • >, but varying ri9 r2l
Nicomachus77 (about 100 A J),) gave the same71 problem and solution 23.
Brahmegupta*8 (born, 598 AJX) gave a rule which becomes clearer
when applied to an example: find a number having the remainder 29
when divided by 30 and the remainder 3 when divided by 4. Dividing
30 by 4, we get the residue 2. Dividing 4 by 2, we get the residue zero
and quotient 2. Dividing the difference 3 — 29 by the residue 2, we get
— 13» Multiply the quotient 2 by any assumed multiplier 7 and add the
product to — 13; we get 1. Then 1 -30 + 29 = 59 is the desired number.
This problem forms the second stage of the solution of the "popular"
problem (§7, p. 326): find a number having the remainders 5, 4, 3, 2
when divided by 6, 5, 4, 3, respectively» The answer is stated correctly
to be 59.
The priest Yih-hing" (t 717 A.D.) in his book t'ai-yen-lei-scbu gave a
generalization to the case in which the moduli m{ are not relatively prime.
Express the l.cm. of m^ m^ • • >, mk as a product m — juuw * -д* of rela-
relatively prime factors, including unity, such that m divides m^ Then, if
a,- ss 0 (mod nifta), a< == 1 (mod д.) (г = 1,- ••, fc),
x = a^ + cttn + " * is a solution. Other solutions are obtained by sub-
subtracting multiples of m. Yih-hing proposed to find the number of com-
completed units of work, the same number x of units to be performed by each
of four sets of 2,3, 6,12 workmen, such that after certain whole days' work,
there remain 1, 2, 5, 5 units not completed by the respective sets. The
l.c.m, of mi = 2, •. -, TO4 = 12 is m — 12. Taking ^ = /*i = 1, мз ~ 3,
/к = 4, we get <*i = or* = 12, a5 = 4, <xA = 9,
x= M2 + 2-12+ 5*4+ 5-9 = iqi - 17 (mod 12).
u Zeitechrift Math, Phye,, 3, 1858, 33G; Dot repeated io hie Gcscbichte der Math., ed. 2, I,
643. H.Hankel.Oeschichted. Math.in Alterthumu-Mittelalter, 1874, erred in identify-
identifying the Chinese rule with the Indian cuttaca.4
n Zeitechrift Math, Phys., 19,1874, 270-1' Zeitschnft Math. Naturw. Unterricht, 7, 1876, 80.
"Dieq. Arith., art. 36; Werke, I, 26. Cf. Eulcr."
n Pythagorci in trod, arith. libri duo, rec. K. Hochey Leipzig, 1866, Supplement, proh» V»
71 Br&hme<ephut'a-eidd'habta, Ch. 18 (Cuttaca = algebra), §§ 3-6, Colebrooke,1 pp. 325-fl,
" L, Matthieesen, Comptes Rendu» Pari9, 92, 1881,291; Jour, fur Matb., 91, 1881,254-261;
Zeitachr. Math. Phye», 20, 1881, Hiat.-lit. Abt.r 33-37 (correctioo of Biematzki"),

. ii] Chinese Phoblem of Remainders. 59
For x = 17, the completed part is 8-2 + 5-3 + 2*6 +1-12 = 55. We
0iay equally well take /ii = 1, /i2 = 3, & = 1, /x* = 4 and get ax = 12,
tf2 = 4^ oj = 12, «4 = 9, 2*^ = 125 = 17 (mod 12).
A condition on the solvability of the problem is that r* — r, be divisible
by the g.c.d. of т<, т^
Ibn al-Haitam80 (about 1000) gave two methods to find a number,
divisible by 7, which has the remainder 1 when divided by 2, 3, 4, 5 or 6.
The first method gives the one solution 1 + 2-3-4-5-6 = 721. The second
xaethod gives a series of solutions 301, et<5.; in effect JF + 2n-7J0 + 1,
where n is an integer such that 6 + 2л-7 is a multiple of 4.
Bhascara81 (bom, 1114 A.DO treated the problem to find the number
having the remainders 5, 4, 3, 2 when divided by 6, 5, 4, 3 respectively.
By the first two conditions, tlie number is 6c + 5 = bn + 4. By use of
the " pulverizer," the integral value of с — {Ъп — l)/6 is с = 5p + 4. The
number 6c -f 5 = 30p + 29 must equal ±1 + 3. Hence p = D1 — 26)/30,
which is converted by the pulverizer into 2h + 1. Thua
30p + 29 = 60Л 4- 59
is the answer.
Again (§ 162, p. 238), what number being divided by 2, 3, 5 has the
respective remainders 1, 2, 3, while the quotients divided by 2, 3, 5 respec-
respectively have the remainders 1, 2, 3? Call the quotients 2c + 1, Ъп + 2,
5f + 3, Then the number is4c + 3 = 9n-f-8 = 25J-flS. Applying the
pulverizer to the first equality, we get с = Яр + в. The resulting number
36p H- 35 must equal 252 + 18, whence p = 25h + 3 and the answer is
900А-КШ.
Leonardo Pisano*2 treated (p. 281) the problem to find a number Ny
divisible by 7, which gives the remainder 1 when divided by 2, 3, 4, 5 or 6»
By the latter condition, N exceeds 1 by a multiple of 60; but 60 has the
remainder 4 when, divided by 7, while we need the remainder 6; thus we
multiply 60 by 2, 3, • • • until we reach 60 X 5 with the remainder 6.
Thus N = 301, to which we may add a multiple of 420 = 60-7. Simi-
Similarly, 25201 is the multiple of 11 having the remainder 1 when divided by
2,-.,10.
To find (p. 282) a multiple of 7 having the remainders 1, 2, 3, 4, 5 when
divided by 2, 3, 4, 5, 6, we take 1 from a multiple of 60 such that the dif-
difference is divisible by 7; the result is 2-60 — 1 = 119. Similarly, to find
a multiple of 11 having the remainders 1, 2, - -, 9 when divided by 2, 3,
-", 10, we subtract 1 from the least common multiple 2520 of 2, ■• >, 10
and get 2519, which being a multiple of 11 is the answer.
He employed88 (p. 304) in effect the rule t'ai-yen71 to tell what number
not exceeding 105 a person has in mind if the latter gives the remainders
"Arabic MS. in Indian Office, London. Of. E. Wiedem&on, Sitsungpber. Pbys. Medic.
Soo. Erl&Dgen, 24, 1892, 83.
n Vljarganita (algebra), § 160, Colebrooke,1 pp. 23*-7.
* liber Abbaci A202, revised 1228), ptib. by B. Boacompagni, Rome, I, 1B57.
*" M. Curtee,Zeitechrift Math, Phys., 41,1Й9вр Hist. lit. АЬЦ81-2, remarked that if Leonardo
bod found the rule independently, he would haveeo stated and would h&ve given a proof.

60 History or the Theory of Numbers. [Chap, 11
(say 2, 3, 4) obtained by dividing it by 3, 5, 7:
2-70 + 3-21 H- 4-15 *= 263, 263 - 2-105 = 53 - am.
Similarly for the number not exceeding 315, given the remainders upon
division by 5, 7, 9: the remainders are to be multiplied by 126, 225, 280,
and from the sum of the products is to be subtracted a multiple of 315.
Ch'in Chiu-shaow gave a method applicable to the problem to find a
number z having the remainders ru * • -, rn when divided by mu • - -, mn,
which are relatively prime in paics. Let M be any one of the quotients
Mk = m>\- * -ninlmk, and seek p so that Mp = 1 (mod m — mk). We may re-
replace M by its residue R modulo m. On dividing m by R, let the quotient be
Qi and the positive remainder be ?ч ^ R. Divide R by л to get the quotient
Q2 and positive remainder r2 ^ n; divide n by r2 "to get the quotient Q8
and remainder ^ r2; proceed until we reach an j\ — L Let Ai = Qu
A, = A,Q2 + 1, Аг = ^*Q3 + Alt ^4 = AiQ* + Ab • ■ •. Then p - ^,-f
and X = ГхМ^ + пМгрг + - - - + гпМярл-
A German MS.W of the fifteenth century proved a general rule corre-
corresponding to the Cliinese t'ai-yen rule.
Regiomontanus85 A436-1476) proposed in a letter the problem to find
a number with the remainders 3, 11, 15 when divided by 10, 13, 17» It is
possible" that he got acquainted in Italy with the work of L. Pisano.
Elia Misrachi*8 A455-1626) reproduced L. Pisano*2 (pp. 281-2) and
gave answers to similar problems.
Michael StifeP9 gave the correct result that if x has the remainders
r and s when divided by a and a + 1, respectively, then x has a remainder
(a + l)r + d4 when divided by a(a + 1).
Pin Kue90 treated in 1593 the problem given by Sun-Tsu.71
The problem to find a multiple of 7 having the remainder 1 when
divided by 2, 3, 4, 5 or 6 was treated also by Casper Ens91 and Daniel
Schwenter.62
Frans van Schooten93 treated the problem to find a multiple of 7 having
the remainder 1 when divided by 2, 3 or 5, He used 30fc + 1, where
Jc = 3 is chosen so that the number is divisible by 7. He gave what is
really the t*ai-yen rule, but attributed it to Nicolaus Iluberti; it leads
here to the multipliers 3-5-7 - 105, 2-5-7 ^ 70, 3B-3-7) - 126, each
with the remainder 1 when divided by 2, 3, 5, respectively.
M Nine Sections of Ма1Ъ, (about 1247), Cf. Mikami/i pp. 65-9.
» M. Curtz&, Abh. Geschichte der Math., 7, 1S95P 65-7.
nC. T. de Mnrr, Memorabilia Bib], publ. Narimbergensium et Uoiveraitatfe Altdorfiuae,
Pare I, 1786, p. 99.
tJ Cantor, Geschiehte der МлШ^ o*L 1, II3 263,
*G, Wertheim, Die Arithm«iik des E. МЬгасЫр 1893, cd, % 1896,60-61.
w Arithmetic» ititegra, 1544, Book I, foL 38v. Die Соаз Christoffo Rudolffa, Die Sch<3nen
E^mpeln der Coea Durcb Michael Stifel Gebeasert, KanigH^rg, 1553, I57t,
^Swan fa tong tsong, Ch. 5r p. 29, MS, in Bibl. Nat, Paris; abetract by Es Biot, Jour»
AwaUque, C)? 7, 1839, 193-21$.
n Thaamaturgud Math.y Munich, 1636, 70-71 ►
n Deliciae Physico-Math. odcr Math.-u. Phil. lirquick^tundeD, Niimberg, 1, 1636, 41.
M ExercitatioDiixu math. Ubri quinqne, Lugd. Batuv., 1657, 407-410.

СяарЛ1] Chinese Problem of Remainders. Gl
W. BeveridgeM treated the problem to find the least number P which
has given remainders К and L when divided by A and B, when the latter
are relatively prime. Let D be the least multiple of В which has the
remainder 1 when divided by A; let С be the least multiple of A which
divided by В leaves 1. Then P = DK + CL, as shown by a two page
proof*
To find the least number P which has the given remainders K, L, Z
when divided by the relatively prime numbers M, Bt A, first find the least
multiple F of ABy least multiple N of AM, least multiple Q of BMt which
have the remainder 1 when divided by M, B, A, respectively» Then
P ^ KF + LN + ZQ.
This is precisely the rule as given later by Euler96 and Gauss,76
*J. Wallis93 gave an empirical solution of the problem of the Julian
period.
T. F. de Lagny9 treated the problem to find the year x of the Julian
period when the solar cycle is 13, the lunar cycle is 10 and the "mdictiorT'
is 7; thus if x is divided by 28,19,15, the remainder is 13, 10, 7, respectively.
From x = 28m -f 13 = 19rt + 10, he found9 that n = 9 + 2S/, whereV
is an integer. Thus x = l$n -f 10 - 181 + 532/. Since x — 7 ts to be
divisible by 15, the least/ is 3.
L. Euler9* treated the problem to find an integer z which has the re-
remainders p and q when divided by a and hf respectively, where a > b.
Thus z — ma + V = nb + q. He solved the second equation by use of
the process for the greatest common divisor for a7 b, continued until one
of the remainders c, d, e, — • is reached which divides ^ = p — q. He thus
deduced the result
in which the series is continued until we reach a remainder dividing v.
At the end of the paper, Euler gave a rule generally attributed to Gauss.7*
To find a number which has the respective remainders pr qy rT ,5, t when
divided by a, 6, c, d, e, which am relatively prime in pairs. An answer i$
Ap + Bq + Cr H- Ds + Et + Mabcde, where
^4=0 (mod bede), A = 1 (mod a); В = 0 (mod aede),
В = 1 (mod b); • - В = 0 (mod abed), E = 1 (mod e).
C- von Clausbergn found a multiple of 7 having the remainder 10 when
divided hy 15.
N. Saimderson98 treated the problem to and a number which has the
remainders d and e. when divided by a and b4 a > h. Let I be the g.c.d.
и Inatitutjomim Chronolograriim libri II. IJnA гщй toKibtii Arithmeticc* Chronologicae
libelfo. TerCluilielm BevercRium LoDilini 16G0, lib. II, pp. 2S3-G,
M Opera, 2, 1693, 451-5. a. HutUm.
^Comra. Acad, P^trop,, 7r 1734-5, 4.G-GG; Comm. Ariib. Coll., I, 11-20.
91 Dcraonetrativc Keohnonkun^t, l7'S2, § 1360, § 1493,
«The Elements «f Algebra, Cambridge, 1, 1740, 316-329, Rpprxxlurrd by <lc Ь Botticro,
Mdm. dc raa.th. phy^rJ prceent^a . . . divers savans, 4p 1763, 41-65.

62 History of the Theoby of Ncvbkvs.
of a and 6. Evidently I must divide d — e. I^t this condition be satis-
satisfied and determine A and В so that Aa - Bb = — L Multiply the last
equation by (d — e)jL Then
is an answer. Other answers follow by adding any multiple of the Lc.hu
M of a, b. Next, let there be three divisors a, b, с sad corresponding
remainders d, e, /„ By the first problem find a number g having the re-
remainders d and e when divided by a and b, and then a number h having the
remainders ^and/when divided by M and с From the answer h we obtain
others by adding any multiple of the l.c.m. of a, b% c.
♦A. G. Kastner," • Ludicke1" and С Huttonm treated problems on
the Julian period. To find the year x of Christ in which the solar and
lunar cycles are 18 and 8, and Roman indiction is 10, Hutton noted that
the year before the Christian era was the ninth of the solar cycle, first of
the lunar and third of the indiction. Hence the remainders on dividing
x H- 9, x + 1, x + 3 by 28, 19, 15 respectively fthe periods of the solar,
lunar and indiction cycles) must be 18, 8, 10. Thus x = 7980p H~ 1717.
To apply102 the rule in J. KeilTs Astronomy Lectures, p. 380, divide
18 4845 -f 8 4200 + 10-6916 by 7980; the remainder 6430 is the year of
the Julian period; subtract 4713, the Julian year at the birth of Christ.
A. Thacker108 proved the last rule, starting as had Hutton.101
The least number"* with the remainders 1, 2, 3, 4, 5 when divided by
2, 3, 4, 5, 6 is 60 - 1 [L. Pisano82].
K. Robinson1*5 found a number x which has the remainders 19, 18,
--,1 when divided by 20, 19, '-',2. Since
x = 2a + 1 = 36 + 2 = - - = 20Л + 19, b = 2m - 1, a = dm - 1;
then use x = 4c + 3, ete. Hence x = 232792560S - l, the least being
given by В = 1.
J. L. Lagrange1* determined n so that it shall have given remainders
ЛГ, Ni, Nif • • -, when divided by M, Mb M2, * * • respectively. Let P be
the Lcm. of M, Mu M2, • • •; Q that of M, M2, Mlt ■■■ (omitting Af0;
Qi that of Af, Afi, Af3, • * • (omitting Af*); etc» Then seek (Lagrange17)
integers p, vf /ii, vu * — such that
** Augew&ndte Math, in dei Chronologie.
i» Archiv der Math, (ed., ffiDdenbuiR), 2, 1745, 206.
101 The ЕНлпял Repository^ or Math, Register» by a Society of Mathematicians, London,
1774, 306; The ГНапап Miscellany, extracted from Ladiee' Diary, London, 2, 1775,
33-4; LeybounTft Math. Queat. piopoeed in ЬмКса* Diary, 1, 1817, 232-3.
» Ladies7 Diary, l735r 33-4, Quest, 175.
'" A Miscellany of Math, Problems, Birmingham, 1,1743, H57-&
^Ladiee7 Diary, I749r 2lT Quest. 29»; Diarian Repository . . . by a Society of Mathe-
Mathematicians, London, 1774, 501-2; C. Hatton>s Diarian Miscellany, 2, 1775,264-5; Ley-
bourn's Math. Quest, b. T>.t 2, 1817, 2.
** The Gentleman's Diaryr or Nfath. Repository, 1748; A- Davtfed., Тюодъп, 1,1814,154-5.
'«■ Меда, Acad. Boy. Sc. Ber«np 23, аопбс 1767 A769); Oeuvns, U9 519-20.

Сна*. П] Chinese Problem op Rbmainpkrs. 63
Then n = \P + N-\-ftQ-\- /i^ + p,Q, + > >>, where X is any integer.
The first of the above set of equations has an infinitude of solutions if Q
and Mi are relatively prime, but no solution in the contrary case unless
Nx — N be divisible by the g.cA. of Q, Mi.
Lagrange107 noted that the problem is to make Mi + N, MiU + Nt,
M%x *\- Nij *% • equal. The general value of t making the first two equal
is t = Ar + Mi?n, where A = Nx — Ny r is fixed and m is arbitrary» The
next step is to solve
M(Ar + JM» + N - Af,x + N2
for mt x; etc»
K. F. Hindenburg108 gave a method of "cyclic periods" to find, for
example, a number x having the remainders 1 and 2 when divided by <x = 2
and 0 — 3» The numbers 1,2, ■ • •, a are written in a column and repeated
0 times; similarly 1, 2, - * -, 0 are written in a second column and repeated
a times* The given remainders appear in the 5th row; hence x = 5.
1 1
2 2
1 3
2 1
1 2
2 3
C. F. Gauss,109 to find 2 with the remainders a and Ъ when divided by
A and Bt solved z = Ax + й^6 (mod B), obtaining x ^ v (mod 2?/S),
if 5 is the g.c.d. of A, B. Hence z = Av + a (mod M) is the complete
solution of the problem, where M = ABjb is the Lc.m. of Ay B. If we add
the condition that z = с (mod C), we get the complete solution
z = Mw + Av -\- a (mod M*),
where AT = ABCjbt is the 1.слп. of .4, B, C7 while с is the g.c.d. of My C.
We may replace z = a (mod A) by z = a (mod J/)> z = a (mod J/'),
* • -, where AfAff- • ■ = Л and Л/, A", • • * are powers of distinct primes.
Similarly, let В = BrB" • • ■. In case Br = pTf A' — p*} г Ш $, the problem
is impossible unless b = a (mod Af)y while if this is satisfied the condition
2 = a (mod A') may be dropped» In this way we can derive an equivalent
set of congruences in which the moduli are relatively prime in pairs and
proceed as above or as in Gauss76 [_due to Euler*].
A» D. Wheeler110 noted that the least integer к which has the given
remainders r7 r't • • • when divided by the given numbers dy <£', * * * is found
by reducing (x — r)/d, (x — r')fd\ • • • to equivalent fractions with a
1B> M6m. A<*d> Ш>у> Sc. Berlin, 24, annee 1768 A770), 222; Oeuvreap II, 698.
lW Leipziger Mag&zin reine q. angewandte Math.t 1786,281-324; eictr. by Lorent^, Lchrbegriff
der Matb., ed, 2, I, 406-442, and by C. A. W. ВегкЬлп, Lehrbnch der UDbeetimmten
AnaJytik, Halle, 1, 1855, 124-144.
1<ж Disq. Arith., 1801, arte. 32-5; Werke, I, 1863, 23-6; МазегЪ Germao transl, 15-13,
'"The Math» Mcmthly (ed.T Runkle), New York, 2, l860r 410.

64 History of the Tiieoey of Numbers, [Chap, 11
common denominator and taking a linear combination x — к of the new-
numerators such that the coefficient of x is unity»
L. Mattbiessen111 discussed the Chinese rules in modern form.
M. F. Daniels69 noted that if a, b} ■ ■ • are relatively prime integers,
x ^ A (mod a), x = В (mod b), • - • have the solution
T. J- Stieltjes112 noted that the congruences x^a (mod A)t •■-,
я ^ X (mod L) have a common solution if and only if <* — 0> a — yt & — yt
■ • > are divisible by (Л, Л), (Л, C), (#T C)> ■ ", respectively, where (A, B)
denotes the g.r.d. of A} B* The case in which Ay < ■ ■, L are not relatively
prime in pairs can be reduced [Yih-hmg79] to the contrary case by writing
the I.cm. of the moduli in the form M — A'B'- • -U, where A\ • • •, V are
relatively prime in pairs and divide A, ♦■•, L respectively» Then any
solution of the initial congruences satisfies also x^a (mod A'), •••,
x s X (mod L'), whence x ^ a (mod M). Conversely, the last x satisfies
the initial congruences if they are solvable.
H- J. Woodall11* found numbers with given remainders when divided
by3, 5, 7, И, 13.
J. Cullen114 gave a graphical method to solve x ^ a (mod P), • - *,
x ^ X (mod L), useful when P, •■•,!, are very large.
G. Arnoux116 gave implicitly the theorem that, if щъ • - -, тпя are rela-
relatively prime in pairs, M — m^ • -m^ ^ = M/miy and if au • ■ -, am are
integers such that ацц = r (mod m^ for г ^ 1, < • •, nT then a^i + • ■ ■
+ ««ми = ^ (mod Af). Proofs were given by C. A. Laisant1" and T.
Hayashi.11&
A oh the рновькы of
gl, Math. WorterbucbT 3, 18OSP 792-800.
J. C. Schtfer, Die Wunder der НтсЬбЫсшшк, Wdmar, 1831p 1842, Prob. 60^
И, Kaiser^ Arthiv Math. Phya^ 25, 1855, 76.
G. Doetor, iMdv 63, 1879, 224.
V. A. Lebesgu^ Exercices dTanalyse num^rique, Paris, 1859, 54-8.
Sseaic, Van der Kongraenz der Zahlen. Progr. S<5brimmT Ш73.
A. Domingucs, Lea Moodce (Revue Hebdom, dee Scieoccs ^t Art»), Paris, 55, 1881 62.
G, de Rocqui&by, ibid., 54, 1881, 304,
D, Marchand, ibid., 54, 1881, 437.
NtrMBER o) of Positive Integral Solutions of ax -\- by = n, Where
a and b Are Positive and Relatively Prime»
P. Paoli117 noted that if ax + by — n has integral solutions, any common
factor of a and 6 must divide n and hence can be removed from every term.
i" Zeitechr. Math, N&turw. Unterricht, 10, 1879, 106^110; 13, 1882, 187-190,
™ Armaks Fac. Kc. Toulouse, 4, 189OP final paper, pp. 31-32,
*l Math. Quest» Educ. Times, 73p l900r 67,
114Proc- London Math. Soc, 34, 390Ь2р 323^34; B), 2, J905, 138-141.
^Arith. Graphiquc, Paiie, 1^06, 2^-31.
IU L'cnedKneinent math., 10p 19OSP 220-5; 12, 1910, I4l-2r
^Opuscula analytic^ Liburni, 1780, 114. Ь one place in the text and in his example, be
erroneously took p between — 6/2 and b/2, instead of positive.

Chap, 11] NUMBER OF SOLUTIONS OF ОХ-\-Ъу = П> 65
Let henceforth a and b be relatively prime and positive» Let 0 denote
the least positive integer such that n — afi is divisible by b. Then every
solution is given by
л , , n — a0
x = 0 + urn, # = — _ am.
b
The values of m making x and у positive are 0, 1, -■ ■, E, where E is the
largest integer less than (n — a$)!{ab). Thus there are w = E + 1 sets
of positive integral solutions ;r, #.
P. Barlow11* employed positive integers pf q such that aq — bp = + 1»
Then all solutions of ex + by — ti are given by
x = ny — 7ttb; i/ = ma — яр.
Let [0 denote the greatest integer ^ L Then
-И-И
or one less according as nq/b is not or is an integer. In fact, m must be
less than nq/b and > np/a to make x and у positive.
Libri27 expressed w as a sum of trigonometric functions.
C. Hermite11' employed the integers
-—И
Then every positive integral solution of ax -\- by = n is ^ven by
и -[?]+m--<-
where f, ^ take the cj -\-1 sets of integral values ^ 0 which satisfy f -h t\ = w.
Here w is such that rc<° = wab. Thus if т is the greatest integer ^ n/(ab),
and n = та£ -h u, then w = т or r + 1, according as ая -|- &y = v has
positive integral solutions or not.
M. A. Stern120 gave Barlow's118 result.
A» D. Wheeler121 noted that if ax+by = c has the least positive solution
x = ^ it has the solutions x = » + &, etc., and hence n positive solutions
if с > ?шЬ. The least and greatest values of с for n positive solutions are
(n — l)ab -\- a -\- b and (n + l)ab> If с = nab there are exactly n — 1
solutions. If с = nab + as' + by% there are n + 1 solutions»
i» Theory of Numbera, Lond^T^lX 324.
lll>Quar. Jour- Mfttb., I, 1855-7, 37(b3; Nouv. Ann. Math., 17, 1858, 127-Ю.
I, 440. Cf. СгоссЫ»1»
u* Jour, fur Matb.f 65, 1858, 210.
Jn The Math- Monthly («L, Runkle), New York, % I860, 56,183-4.

66 History of the Theory of Numbers. [Сваг.н
J, J. Sylvester*2 stated two theorems on the number (n; a, b) of positive
integral solutions of ax + by = r for the values r = 0, 1, • • -, л:
(; , H- a + Ь + 2n' - 1) + («'i <*>
if к and n/ are positive integers for which n + 1 = frab 4- «'»
where a\ br are positive integers such that abf — baf = 1, a' < а, Ъ < Ъ.
E. Catalan120 made use of the known fact that the solutions of
ax -\- by =* n
are x = a ~ W, i/ = /3 + a*, if a, ff is one set of positive integral solutions.
Let а, Ь, n be positive. Then the positive solutions have В < a/b, 0 > we,
which are equivalent to $ < лв/(а6), в > (аи - n)/(ob). Hence ы = [«MJ
or [я/аЬ]+1. Writing ««abg+n', 0^л'<аЬ, he proved that аг+ЬууП
has 3+1 or q positive solutions according as axf + by' = «' has a positive
solution or none,
С de Polignac1*4 remarked that ax ^ by ^ n may be solved graphically
by means of a lattice whose initial rectangle has the base a and altitude b.
He concluded that, if т * [nlab^ * = т if the remainder obtained by
dividing « by ab is < bft where ^ is the least positive y, while бз ^ т + 1
in the contrary case»
E. Catalan125 stated and E. Cesaro proved that, if we count the integral
solutions ^ 0 of each of the equations x + 2y = n — I, 2z + Zy = л — 3,
Зг + 4^ = и - 5, -., the total number of solutions equals the excess of
n+2 over the number of divisors of ti+2. For, px-\-(p-\-l)y = n—Bp-l)
has
solutions ^ 0, where € - 1 or 0 according as p + 1 is or is not a divisor
of n 4- 2.
E. Cesaro stated and X Gillet1* proved that if we oount the integral
solutions ^ 0 of each of the equations x + 4i/ = 3^ - 1,4a? + 9y = 5n - 4,
9s + 16t/ = 7n — 9, ■ • •, the total number of solutions is n.
E. Catalan stated and R Cesaro and H. Schoentjcs127 proved that if
we count the integral solutions ^0 of each of the n + 1 equations
m7>>mptes Rendus Paxisr 50, 186Or 367: Col). Math. Papers, IIT 176,
»" Melanges Math., 18G8, 21-23; Мбш. Soc. Sc. Liege, B), 12, 1885, 23 (Melanges Math, lh
Mathcsid. 30r 1890, 22О-2.
l" Bull Math. Soc. Fnuree, 6, lS77-8? 158, E. M. Laqui^re, Oft, 7, 187МГ 8,
PoligDac's work. A resume of both is given by S. Gunther, ZeJtechr, Matb-
Untcrricht, 13, l882r 98-101.
ltt Nouv. Ann^ Math., C), 1, 1882, 528; C>, 2, 1883, 3S0-2*
>* Mathcsis, 2r 1882, 208: 5, l885r 59-60.
i"/l«U, 2, 1S82, 158; 3, 1883, 87-91.

Сйар. И] Number of Solutions of ах-\-Ъу=п> 67
x + 2y = n, 2х + Zy = « - 1, •. -, (и + l)s + (n + 2)y = 0, the total
number of solutions is л 4- 1.
Cesaro1** proved the last theorem with n replaced by n ~ 1, by showing
that p(x + ^ + 1) + У ~ n bas exactly Nv =* [n/p] - [nf(p + 1)] in-
integral solutions ^ 0. Also,
while JV^ + JV3 + № + ■ • ■ equals the difference between the number of
odd divisors and the number of even divisors of 1, 2, - -, n. The number
of integral solutions ^0ofz + 2y = 2(n ~ 1), 2x + 3y = 2(* - 2), • ■-,
ttx + (n + l)i/ я= 0 is the number of non-divisors of 2n + L As a general-
generalization, px + (p 4- l)V = k{n - p), for p = 1, - • -, n, have
Л/ = M l + • • ■ -h Л/ „
integral solutions ^ 0, where Mp = \_knlp] - [_(kn + ^ - l)/(p -h 1)] is the
number of solutions of the equation written; for к — 3, Л/ equals the sum
of the numbers of divisors of 3n -h 1 and Зтг + 2. [The preceding results
are special cases of a formula given by Lerch in 1888; cf. Gegenbauer,23
p. 227 of VoL I of tbis History.] As a generalization of Catalan's127
theorem, the total number of integral solutions ^ 0 of
A + jk)x +A + 7+1 k)y = k{n -j-1) (j - 0, 1, . • 0
is n. Given a set X = — <*> У = & of integral solutions of ax + by = n,
the number of integral solutions ^ 0 is [£/a] - [(or - l)/b].
Consider a set wi, и*, щ *' °^ positive integers each prime to the term
following it. Let vu Ъ, - - be integers and determine a series of w's by
WP = byUp+i - A + Vp+X)up.
If wr is the first negative term, the total number of integral solutions S 0 of
^ Cyi/«J — C"r/«r], since the equation written has [vjup"]
solutions ^ 0» The case t;p = nt up = p2, gives the result of CesAro.125
He quoted (p 273) from a letter from Hermite the result that
member being the number ^ of sets of positive integers for which
ax H" ^J/ — n> Henceforth, let a and 5 be relatively prime. Then the
number of integral solutions ^0 of ax + by = n is known to be
Nn = [njab~] + r, where r = 0 or L Cesaro noted (p. 278) that r = 1 if
the remainder R obtained by dividing n by oh is of the form pa + vb,
where p7 a are integers =£ 0, and r = 0 in the contrary case [Catalan123^
Thi theorem, which may be expressed in the form AU — Nn = [njdb^ is

0g History of the Theory of Numbers. [Сялг. ii
proved in two ways, one by use of a geometric process communicated to
him by Lucas: Given one point on the line ax + by = n with integral
coordinates ^ 0, it is easy to find all such points. If M is the point with the
maximum abscissa, we get a second point Mf by subtracting b from the
abscissa of M and adding a to the ordinate of M. From AT we obtain
similarly a new point, etc»
CesAro stated and N. Goffart12* proved that the total number of integral
solutions ^ 0 of
x + 4y - 3(» - 1), 4x + 9y = 5(n -2), 9x + Щ = 7(n - 3),
is n.
J. GilletIJ0 stated that the sum of the numbers of solutions of
pmx + (P + 1)"У = \(P + 1)" - Pm(™ - Z>" (P ~ !» '">*)
is n, a generalization of the theorems by Cesaro126 and Catalan.141
E. Lucas1*1 proved Catalan's123 result and added the remark that there
are 1(a — l)(b ~~ 1) v^ues of his nf for which ax -\- by = n' has no solutions
> 0. In the continued fraction for afh} let &}$ be the convergent of rank
rt — 1 immediately preceding a/d. Then, writing r for n\ we have the
solution Jo ^ (" 1)nr^ 2/o ^ — С— l)"ra of ах + Ъу -r. The sum of
the squares of the values x = x<> + Ы, у = yo — Qtf giving the general
solution, is a minimum for t = s/(a2 + b2), where s = (- 1)П"Ч^Л + W)'-
Let P! be the least positive remainder and — pi the greatest negative
remainder when s is divided by к = a2 -\- b2* Then the gets of minimum
solutions are given by
кгл = ar - bpu kffi = br-\- aph kxt = ar + bPb ky2 = br — ap2.
In only one of the sets are the unknowns Ш 0. Hence ox + by = r is
solvable in integers ^ 0 if and only if one of ar — bp^ and br — apz is not
negative.
E. Catalan showed by an example that Lucas' last method requires
long computations» He noted (ibid., 241-3) that, if o>(n) denotes the
number of integral solutions Ш 0 of px + qy = n,
A. S. Werebrusow1^ noted that w = (n — bp — aa)!abt if ^ is the least
positive y, and « the greatest negative x.
L. Salkin151 employed the argument of Catalan123 to show that ы = q or
+ I, according as d ^ d' or d > d', where, if — I, w is one set of solutions,
ДО], t*; = m/o - Г^/а]>
3)^ £l8S4, 39d, 539-40»
di>,6.1888,32.
^thefiifl, 10, l890r 129-132; Thdorie dee nombres, 1891, 479-484; Jour, de math» я
1886,20-22.
"»Matheeifl, Ю, I860, l»7^-9.
m Spacoueki^ Bote Matb,r Ckleaea, 19Olr Noe. 298, 299,
J" Matberis, C), 2,1902дШ9

З Number of Soltjttons of ах-\-Ъу=п. 69
V. Bernardi135 would find the r>ositive integral solutions of ax + by = к
by employing the remainders r[t ri and quotients q\t q[' obtained on dividing
k — b by a and к — a by b. Thus
Similarly, as* + fya = ^> • ••,«?* + ty*. = fc~ = k^t — а -Ъ — г^ — г^,
where гл, т2 are the remainders and qmj q^ the quotients obtained on
dividing &,*_i — b by a and fc^i — a by h. In this way we find a value
и of m such that a zero remainder results from that one of the two divisions
in which the divisor is the smaller of ay h> or such that the remainder from
the other division is zero or is divisible by the smaller coefficient. Then
ku is divisible by the larger or the smaller of а, Ь in the respective cases.
The positive integral solutions with ku divisible by a are
s« ~ kja — nb, yv = na (n — 0, 1, • • >, [kufab~]}.
Then all positive integral solutions of the given equation are
у = (-
Cf. Hermit^1*
L. Crocch]136 noted that Hermite's11^ formulas do not give merely the
integral solutions. Thus, if n < a, n < 6, they give z = ±Ъ£, у = db arjt
£ -\- rj = dt n[(ab)t which lead to fractional solutions of ox + by = n*
Crocchi therefore transformed Hermite's formulas so that the resulting
formulas give merely positive integral solutions. Set
л=|-|в + г = |г|& + *, n =n—r—s, n = \ — \u-\-r\
Then
s _ r^l _^i^ Г-П Г—П ~— Г—1-Гл1-Г-1
a Lai a La J LaJ a* LaJ LaJ LaJ+^
where C*/a3f ^ ^he Quotient by excess of s by a. Similarly>
LaJ LaJ LcJ+ LaJn-'
Taking alternate signs and adding, we get, for тп even,
)» Atti eocietA italiana per П progreeeo detle sdense, 2, 190$, 317-8.
"»11 Boll, di Matematica Gior. Sc.-Didat., 7% 190$, 229-230.

70 History of the Theory of Numbbbs.
and, for 77i odd,
[Chat. II
Then z = x' + (- l)*+lb6 у = у' +.(- 1)»+^ { + Ч = nC-)/(ab).
L. Crocehi137 noted that, if in Hermite's119 process we have reached the
dividend nt* = aQ + r9 = bQ' + i>, then n1^» = n<» - rF - i>. For
example, consider 5x -h lly — 488.
Dividends
488
481
472
460
451
450
440
Residues
by 5
3
1
2
0
1
0
0
by 11
4
8
10
9
0
10
0
Quotients
by 5
97
96
94
92
90
90
88
by 11
44
43
42
41
41
40
40
Here 481 = 488 - 3 - 4, etc. Thus
x* - 97 - 96 + 94 - 92 + 90 - 90 + 88 = 91,
y'= 1 + 1 + 1+40-43, « » 91 - Пет,
у = 43 - 5n, m + n = 440/E*11) ^ 8.
To find z' more readily, use the second, fourth and sixth entries 8, 9, 10
in the third column and set
x' = 97 - /2 - U - h - 91.
Similarly, from the second column, y' = 44 — 1— 0 — 0 = 43. But if the
number of operations had been even, we would have used /i, It, /*.
L. Rassicod,128 V. A. Lebesgue,m G. Chrystal,14* L. Aubry141 and E.
Cesaro141 evaluated w by known methods. Cf. Laguerre91 of Ch. III.
m И KtAgora, Palermo, 15, l90fr-Sr 29-33.
J» Nouv. Ann. Math., 17,1S58, 126-7.
ut Exereices d'anaJyse numerique, 1S5D, 52-3,
ш AlgebmJ 2r 1889, 445-^9; ed. 2, vd. 2, 1900, 473-6.
^ L'eneeignement math., 9r 1907, 302.
114 Mem. Soc. Roj, Sc. dc liege* C)r9t 1912, No. 13.

Chap. II] One LlNEAB EQUATION IN ThREE UNKNOWNS. 71
G. B. Mathews" proved that, if ^(n) is the number of positive integral
solutions of x + у - и in which 3x ^ 4y, 2x ^ 7y, then
For the problem in n instead of two unknowns, see Ch. III.
One Linear Equation in three Unknowns»
T. F. de Lagny10 (p. 595) treated py = ax + z by giving values to z
which are the successive multiples of the g.c.d. of p and a. The methods of
de Paoli*3 and Mac Mahon48 were given above.
Several143 found the 12 sets of positive integral solutions of
IQa? + Uy + 12z = 200.
L. Euler1" treated Aa + Bfr + Cc ^ 0. For example,
49a + 59b + 75c = 0.
Divide by 49 and set a + 6 + с = d. Thus 106 4- 26c 4- 49d = 0. Di-
Divide by 10 and proceed as before» We ultimately get all integral solutions:
a = - 8e - 7/, 6 - 13e + 2/, с = 3/ - 5e,
P. Paoh44* solved 5z + 8y + 7z = 5Q hy successive substitutions:
x + у - *, 5* + Зу + 7г = 50,
у + * = t\ Zt' 4- 2^ -h 7z - 50,
^ 4- <' = Г, 2Г' -f t -h 7г = 50.
Since a coemcient is now unity, the solution is evident.
A. Cauchy145 proved that every solution ofo£ + &y+C2 = 0is given by
x = bw — cv, у = си — aw, z = av — to/,
if the gx.d. of a, fr, с is unity.
V- Bouniakowsky147 proved Cauchy's14* result by solving
ax + by + c* - 0, a'j + 6^ + c'z = Л', a'^ + b'V + c"« = h",
the two adjoined equations having arbitrary coefficients» Then
ete., where и = (a'k'f — hfa")jAt etc., A being a determinant of order
three.
** Matb. Quest, and Sdutiona, 6,
10 Tbe Gentlerawi^ Иагуг or Math. Repository, 1743; Davis' cd^ Londoo, lr 1814, 45-7.
luOpue. впл1., 2r 1785 [1Л51 91; Comm. Arilh. Coll., II, 99.
"» Efementi d'Algebra di Retro PaoK, Plaa, 1, 1794r 162.
«•Eserefoeede math., 1,1826,234. Ocavree de Caachy, B), 6, 1887, 287. Extr. by J. A.
Gnmert in Archiv Math. РЬуя., 7, 1&46, 30fr^.
вд BulL Acad. Sc St. Petcrabooi^ 6,1848,196-9.

72 History or the Theory of Numbers. [Cb**. n
V. A. Lebesgue148 noted that, if the g.c.d of a} bt с is unity, all solutions of
A) ax + by + cz ^ d
are given by
x ~ dab + cou + tfc/Д у = d06 + фи — vafD, г = dy - Dut
where и and v are arbitrary, aa 4- 60 = P, D being the g.c.d. of a, b, and
D6 + cy = 1.
H. J. S. Smith148 stated that if а, Ъ,с and a', b'9 d are two sets of solutions
of Ax 4- By + Cz =* 0, where Д, В, С have no common divisor, the com-
complete solution is
x = at + a'u, у = to + 64 z = <* + Л,
if and only if there is no common divisor of
he' - Ъ'с, ca' - ac', ab' - а'Ъ.
A. D, Wheeler1" treated (I) by taking 1, 2, • ■ • for z until we reach a
value for which axt + by^ — d — cz<a-\-h and hence is not solvable.
By simplifying this method, he found the number of solutions.
L. H. Bie1" expressed the general solution of A) in terms of the residues
of d — pc modulo Ъ.
С. de Comberousse1" employed the g.c.d. В of a, b. Let the g.c.d. of б
and с divide d. Then d — cz = 60 has an infinitude of solutions z, 0. For
each Bt xajh + ybjh = 0 has an infinitude of solutions. If <xt 0, у is one Bet
of solutions of A), every solution is given by
x = a - Ь$ + с*', у ^ 0 + a0 + c0", г = 7 - a*' - Ъ6" (в, $\ $" arbitrary),
A. Pleskot1" treated A) by continued fractions.
While in various books™ on algebra the solution of A) involves three
parameters, that by G. M. Testi165 involves only two. Let the greatest
common divisor $ of a and b be prime to с Then
%x + -y = t} U + cz = d.
о v
The second has the general solution U — сф> ?o + &Ф* if k> ^ is one solution.
All solutions of the first are given by x = x<£ — Bbjht у = yd + fla/5, if a^,
y* ifl a solution of
Г + ^1
141 Exerdcee ^analyse mim^tique, Paris, 1859, 60.
"■BritiA Левое. Report, 1860,11,6; Coll. Math. Papers, I, 365-в.
^ The Math. Monthly (ed.r RunkJe), New York, 2, I860,407-410.
ш Tidaekrift for Mat., 2,1878,168-78.
u* Alfebra eap^iieure, lt 1887, 179-183.
ш Caaopifl, Pr»et 22, 1803,71.
>« G. J. Bertnnd, Tmite №n. d'algebre, 1850; tranel. by E. Bnii, Florence, 1862, 285.
*» Periodico di M*t.t 13,1898,177.

Chap, и] One Lineab Equation in n>Z Unknowns» 73
Thus A) has the solution a *= х£>, 0 — yQto, у = zo, and also
x - л — схцф —
у ~ p - суоф +
The latter give all integral solutions of A) when ф and в take all positive
and negative integral values and яего. A like result was given by F,
Giudice.1M
♦H. Kuoss showed graphically how to find those values of x, у, z in
A) which satisfy certain restrictions, e.g., are all positive.
One Lin^ak Equation in n > 3 Unknowns.
Brahmegupta2 and Bh&scara* assigned values to all but two of the
unknowns.
T. Mosswt tabulated the 412 sets of positive integral solutions of
17» + 21s + 27y + 36* = 1000.
C. F. Gauss15* noted that, if the constant term is a multiple of the
g.c.d. g of the coefficients of the unknowns, then g is a linear function of
those coefficients, and the equation is solvable in integers»
V. Bouniakowsky147 would solve ax + by + cz + du = 0hy adjoining
three equations a'% + ■ • • = h't etc», and solving the system. The general
solution of the given equation is thus
x = dp — cq + br, z = drr — bq' 4- aqy
у = — dp' 4- cq' — ar> и = - crf + bp' - apt
where p, gt rt p', q\ r' are arbitrary. He gave a hke result for uve unknowns
and outlined the law for n unknowns. V. Schawen1M gave the same method.
B. Jaufroid1*1 assumed that there is no common divisor of a, - -, m in
A) ax + by + сг + ■ ♦> + mu + n = 0.
First, let a and b be relatively prime» Then
aA + bAi + c = 0, • -, ah + bLx + m « 0, aM + bMt 4- n = 0
are solvable for Л, ^t, ■ ■ *, so that A) is satisfied by
x *= Az + Bff + • - + L« + Jtf - K,
tf = 4i« + Biv + ■ • ■ -h Liu -h АГЖ + at.
Second, let 5 be the g.c.d. of at b and let 5 and с be relatively prime. Set
a *= ai5, 6 = &j5, and
B) fllz + biy ^ p>
« Giornale di Mat., 36, 1898, 227. ~~
w Korrwp» BJ. f. d» ЬбЬегеп SchuJen WOrttember^, Stuttgart, 9, 1912, 48l-4>
m Ladtee' Diary, I774r 35-6, Queet» 658; T. Leyboorn'e Math. Quest. L. D42,1817.374-6/
>»DifquiatkmeBArith., l8Ol,art.4O; Werke, I, 32,
"♦Zritechr. Math. Natuiw. Unterricht, 9, 1878r 111-8.
i«NAnn. Math., 11,1862, 158.

74 History of the Theory of Numbers, СС*а*- ц
Then A) becomes 3p + cs + • • • + w« f я = 0 and is satisfied by
For these values, B) becomes a& + Ъ$ — Bxv — - •• = 0. Thus by the
first case we obtain solutions xy y, z in terms of v, • ■ >, м, t, t\ A similar
method applies when the g.c.d. of а, Ь, с is prime to d, etc.
V» A» Lebesgue1** noted that if a and b are relatively prime, we may set
aa-\-bp — 1; then the general solution of A) is "
x = Qa -\-bw, у = Q& — au>, Q=—n — cz~-*— ma.
But if no two of the coefficients are relatively prime, proceed as for
Set Ai = a^ and let Atbe theg»c»d. of au - *,a<fori = 2, - -,5. Hemove
from o« the necessary factor Д*, so that now Д* - 1. Determine integers
a.-, fit such that АД + avfia^i = At+i (t = 1, • • •, 4)» Solve each of
A^tf^ и- а^х< = A,^,- (i « 2, 3, 4), where 2/1 « Xi, and A&4 +
Thus
for t ^ 2, 3, 4. Eliminating the #'s, we get xu — -, x5 in terms of the
parameters z*, • - -, г5.
E. Betti164 gave without proof a formula for the integral solutions of
+ ■ ■ ■ + аяхл = ay where а^ • - •, ал have no common divisor, and
^«„isa particular set of solutions:
si = «1 + ал -f cijfc + ■ ■ • + а^1вл-х + an$n,
x% = a2 - аЛ -f агв'ь + . ■. + e^jCi + апв'п>
where the л(л — l)/2 numbers ^1? are arbitrary, F. Giudice156 proved that
every solution is of this form and gave a method (based upon equations in
two variables) of obtaining the general solution in terms of n — 1 para-
parameters.
C. G. J. Jacobi1*3 treated in several ways the solution of
QftXt + ДО, -f - ■ • + «пГя =» fUy
where / is the g.c.d» of aL| • • -, <xn. Let [a7 b] be the g.c.d. of a7 b. One
obtains easily all solutions of
M Exercices d'aoalvae numdrique, 1S59, 58.
1W Jour, fur Math,, 69, 1868, 1-28; Werke» VI, 355-3S4 D31)»

З Они 1дК£А& Equation in n>3 Unknowns. 75
Adding, we get the given equation since /n = f. His second method con-
gists in treating these equations taken ш reverse order, after each Ы divided
by the /, in the second member. He noted that the method of Euler144
is applicable also to Aa + Bb + Cc *= u.
K. Weihrauch.1*4 denoted by E{M : N) the integral part obtained on
dividing M by N, and by R{M : N) the remainder. Thus [if Ai Ф 0]
C)
gives
X! - tfD : Л0 - ХгЕ(Аг : -Ai) *»ЯD» : ilj + tu
к ^~\ЩА :Аг) -хгЩАг:А0 - ... — х^ДСЛ» Г-^ОЬ
Treating the latter similarly, we get x*. Finally, we get a relation between
Xn^u Xn, whose solution involves a new parameter l*-i- Thus
D) Xi ~ Mi + a<A + • * ■ + a.^n-1 (г = 1, • ■ -, n),
in which i^!, < > -, i/n is a set of solutions of C), and
А&ц + Atatj 4- •■• + Ananj ^ Q tf ^ 1, ".,n — 1)-
The condition that D) shall give all solutions of C) is
-j-Uo-l * ±1 (г = 2, -,п;;*1, ...,л-1),
■я-i
where the symbol denotes an (n — l)-rowed determinant.
T» J. Stieltjes1*5 reduced а& Ч- • t ■ 4- c*h-i^f»+i — « to an equation in
one variable. If X ^ (ob a2) is the g»e.d. of a^ at, we can find relatively
prime integers а, у such that a^a. -\- a^y = Л* Taking f} — —
5 = aj/Л, we have <x6 ^ ffy — 1. Set
Then the initial equation is equivalent to
Similarly, we can replace the first two new terms by (au а?, а^)х\\ etc,, and
finally get drf"* = u, where d = (аА, * —, c»+i) is the g.c.d. of ai, • • ■, ая+i.
Giving to ^i, •' -, xl+i all sets of integral values, we get all solutions of the
initial equation if it be solvable, viz., if и be divisible by d. A system of
n independent sets of solutions is fundamental (Smith307) Я the g.c.d, of
then+1 n-rowed determinants is unity.
W. R MeyerIW solved C) by use of recurring series obtained by simplify-
simplifying and extending C. G. J. Jacobirs generalized continued fraction
algorithm.
** UnteroicbuDgeii Uber eine Gl. 1 Gr, Dies. Dorpat, 1869. Zeitechrift Math. Pbys., 19,
I874r 53.
*• Annalee Fac. Scr Toulcmse, 4, 1890, final рлрег^ pp. 3847.
**♦ Verband, dee etBten Intern. M&th.-Kotigreeees, 1897r Leipzig, 1898, 168-18L
i» Jour, for Math., 69, 186S, 29-64; Werke, VI, 385-126.

76 History of the Theory op Numbers. [Снлр.п
R. Ayza1" treated ax-\-by-^-cz-\-du^ - — — hby means of
ax + by = fci, cz + #" = fc2, •••, Ai = A — A2 — fa — • "f
where A», A3, * — are arbitrary. For m linear equations in m + n variables,
successive elimination gives one equation in m + n variables, one in
m -\- n — 1 variables, " •, one in n + 1 variables, which is solved as above.
A» P. Ochitowitscb1*9 treated Za#< = 0. If aPf aq are relatively prime,
y, ~ sa« + n£, 3/, = - zap + ту\, apy'p + ee^ 4-1=0,
where г = 2a$i for г Ф p, g. For d - рГ1" ^Г*» where pl7 • • , рл are
distinct primes, a set of solutions of 1 + fli^i + а&2 = 0 is given by
A
"V
where zu • • •, гя are to be chosen to make the indicated fractions integral*
E. B- Elliott170 recalled the fact that all seta of positive integral solutions
of a linear diophantine equation in n variables are linear combinations of
a finite number of ff simple " [fundamental] sets of solutions (*lf ••• *, an),
■•*, (wi, ••-, w«) and that a linear combination of these simple sets is
always a solution. He noted that two such combinations may give the
same solution since the simple sets are usually connected by syzygies.
For example, the three simple sets A03), B30), A11) of solutions of
3x « 2y + z are connected by the syzygy A03) + B30) « 3A11), so that
s « ft + 2fa + U, У = Ok + 3f2 + th z = 3fi + 0t2 + U
give duplicate solutions unless we restrict h to the values 0, 1,2. In this
sense, he obtained formulas giving each solution of an equation in n variables
once and but once, making use of generating functions.
G. Bonfantmiin noted that, if a, ••■> l7 к have no common factor,
ax + by + • ■ ■ + hi *= к has integral solutions if and only if a, — -, I
have no common factor»
Several1'5 found all positive integral solutions of
Ш + 211 + 29m + 37n = 300.
* P. B. Villagrasa175 treated C).
D. N. Lehmerul° proved that C) with Л = 1 is satisfied by tbe co-
factors of the elements of the last row of a certain determinant of value
unity, those elements being any integers whose g.c.d, is 1. The general
solution of C) is deduced»
"VArehivo de M*temiiticaSr Madrid, 2r 1897, 21-25r
w Text on lmear equations, Kasanr 1900»
м Quar. Jour. Matt., 3^r 1903, 348-377.
>n II BolL di Matematica, Gior. Sc. Didattico, 3, 1904, 45-47»
i* Math. Quest. Educ. Times» 7» 1905г 21-Э2.
"* Bevista de la Soried^d Mat, Espoftola» 3, 19l4r 149-156.
i7u Proc Nat. Acad. Sc.» б, Ш19, 111^; Amer, Math. Monthly. 26» 191^ 355-6.

ij System of Linear Equations, 77
Since the problem to solve n — ax -\- by + - ■>. in positive integers is
the same aa to partition n into parts a, b, * ■ ♦, reference should be made to
Ch. HI, Iй particular for theorems on the number of solutions»
System op linear equations»
Chang CnJiu-chienlW (sixth century A.D.) treated a problem equivalent
to
x + у -f z = 100, 5x + Zy -f \z =* 100,
and gave the answers D, 18, 78), (8, 11, 81), A2, 4, 84),
Mahavlracarya175 (about 850 A.D.) treated special cases of
x + y+z-\-w = ft, az + &y + cz + tfw = P'
Shodja B. Aslam17* (about 900 A.D.), an Arab known as Abu Kamil,
found positive integral solutions of x + у + * = 100, 5x + j^/20 + z ^ 100,
whence у = 4x -\- 4*/19, x = 19; x + у + г = 100 = |x H- $y + 2г,
whencez « 60 - 9yflO,y = 10mt m *= 1, •••, 6;
* + У + * + » e 100, 4jf + TVy + J« + « « 100,
whence X = /try 4- i^, with 98 sets of solutions (two of which are omitted).
When the last equation is changed to 2x + \y + \z -f- и - 100, there are
304 sets of solutions. There is no solution of
x + у + z - 100 - дх + у/20 + }г.
There are 2676 seta of positive integral solutions of
x + y + z + u + v= 100, 2x + §y + Ь + i« + » « 100.
Alhacan Alkarkhi177 (eleventh or twelfth century) treated the system
8 — x J- у -}- Zj w =
by taking 3 = 1, whence ж ^ 33, у ^ 13. He treated the problems of
Diophantus I, 24-28, as had Diopbantus, by making the indeterminate
problems determinate by assigning a value to one unknown.
Leonardo Pisano/" in 1228, treated various linear systems, the first
being that of Alkarkhim without the final condition:
m Gtmita-S&n-San^aha.1 Gf. D. E. Smith, Bibliotheca Math., C), 9,1909,106-10.
»«H,SuterrBibUothecaMath.rC), 11, 1911,110-20, gave a German tranal, of a MS» copy
of about 1211^ A.D.
1» Extwut du Fakhrt, French transL by P, Woepcke, Parie, 1853, 90, 95-100-
171 Scrittf di L. Plaano, 2,1862, 234-6. Cf. A. Onocehi, Annali di Sc. Mat- e Fie., 6, 1855,
169; O. Tenmem, itat, 7,1856,119-36; Nouv. Annr Math., Bull. Bibl. ШЦ 14,1855,
1ГС9; 15, 1856, Hi, 42-71.

78 History of the Theory op Numbers. EChaf. II
These determine x, yt z, t in terms of u. Since 71 ^ 47u, he took v = 7,
whence* = 47,x ^ 33,y =* 13,z — 1. His next indeterminate problem179 is
-f z3), f + j| = 4(x< -f- xt),
Since the problem is impossible if xx and x* are positive, change X\ to *- xi.
Now z* *= 4zi. Take xt = 4, whence Xi ^ x» = 1, x< = 4, * — 11.
For** x + y + г *= 30, |x + & + 2z ^ 30, we have у + lte = 120,
у + z < 30t z ^ 9. The case z ** 10 is impossible. For z = 11, we get
у = 10, x = 9. Tbe same problem with the constant term 30 replaced
by 29 .or 15 is treated similarly.
Finally,1*1 consider the system
Hence 2y -f- 27г + 42i - 288, у + z + t <Ж Thus г is even and < 10.
The cases z = 6, z = 8 are impossible. Thus there are only two solutions:
г « 2, t « 5> у = 12, x = 5; ar = 4, < = 4, у « 6, a: = 10.
Regiomontanus A436-1476) proposed in a letter (cf. de Murr^ p^ 144)
the problem to solve ш integers
* + ?+* = 240, 97x 4- 56y + Зг - 164L7.
J. von Speyer gave the solution 114, 87, 39 (de Мшт, р, 167).
Estienne de la Kochelte treated the solution in integers of
x + у + z = a> tftt + «У 4- P2 = Ь.
His rule ^applied to the case a = Ь — 60, w = 3, n = 2, p — J] is as follows*
Let p be the least of m, «, p. From the second equation subtract the
product of the first by p; we get
(m - p)x + (n - p)y * b - ap [|x -f fy = 30].
To avoid fractions, multiply by 2. Thus 5s + 3y = 60. Although
x = 60/5 gives an integral solution, the corresponding у is zero and Ш
excluded. The next smaller values 11 and 10 for x lead to fractions fox y,
while x = 9 gives у = 5 [whence z — 46]. For x = 1, 2, • • •, the least я
yielding an integer for у is x = 3, whence 9 = 15, г = 42. The problem
may be impossible, as shown by the case a — Ъ = 20, m = 5, n = 2, p = i,
whence 9x + 3y = 20.
l7*Scrittir llr 238-9 (De quatuorhoaUDibuaet bursa), Genoccbirl7¥ 172-4.
in the account by Terquenu
li0Scrittir llr 247-8 (De aulbus emendie). Genuccbi, 218-^2. For analogous problem*,
eee Liber AbbacirScrittip ir 1857, 1в5-б.
^Scrita, llr24ft (Item j>ae&erea)r Genocchi, 2224.
M Lariametique & Geometrier Lyoat 1520r fol. 28; 1538. Cf. L, Rodet, Bull. Math. So*
Prance, 7r I879r 171 [162].

з System of Linear Equations. 79
Luca Paciuolo18* treated the solution of
р + с + т + а = 100, \p + \c -\- тг H- 3a = 100,
«riving the single solution p — 8, с = 51, тг — 22, a — 19» Many solutions
were found by P. A. CataldL1M
Christoff Rudolf!166 stated the following problem. To find the number
of men, women and maidens in a company of 20 persons if together they
pay 20 pfennige, each man paying 3, each woman 2 and each maiden §.
1Ъе answer is given to be 1 man, 5 women, 14 maidens. [The only solu-
solution of x -\- у + z -= 20, 3x -|- 2y -\- \z — 20 in positive integers is x — 1,
У ~ 5, z ~ 14»] The solution is said to be found by the rule called Cecis
от Virginum.
C. G. Bachet de Meziriac1" solved in integers the system of equations
x + у + z = 41, Ax + 3i/ + J* - 40.
Multiplying the second by 3 and subtracting the first, he obtained
\\x -\- 8y — 79. Since у — 9J — lfx, x must have one of the values 1,
. - ~t 7» By the value of 8z in terms of x, 1 + 3x must be divisible by 8.
Hence x = 5, so that у = 3, 2 — 33. He treated Rudolff's185 and a similar
system and found 31 sets of positive integral solutions of
x ±y + z + w = 100, 3xH-i/H-^H-4w = 100.
«L W. Lauremberg187 described and illustrated by examples the rule called
Cecis [Coeci] or Virgirmmlga for solving indeterminate linear equations,
referring to the Arabs [^although known to the Indians].
Ren6 Francois de Sluse1" A622-1685) treated the problem to divide a
given number b into three parts the sum of whose products by given
numbers г, gt n shall be p. Call the first and second parts a and e. Then
/* л p — nb -\- ne — ge
ж + ge -h п(Ь - a - e) - p, a = — — •
Take p - Ь - 20, z - 4, g = §, n = i Then a - F4) - €>/15.
Johann Pratorius190 solved the following problem: Anna took to market
10 eggs, Barbara 30, Christina 64). Each sold a pait of her eggs at the
same price per egg and later sold the remainder at another price. Each
"* Summa de Arithmetic*, 1523, foL 105; £Stnn» . . ., Venice, НЗД; «мае edution by
N. TarUgJlb, General TratUto di Nvmen * * * f If 1566.
1M Kegola delb Quaritita о Сова di Сева, Bologna, 1618, 16-20.
'" KuiujUiche Recimungr 1526; NQxcbei^ 1534, f. nvij a mod b; Numberg, 1553 and Vienna,
1557, Г Rvii a and b.
i« Diopb&nMie Alex. Airlh., 1621,261-6; comment од Dkjfii_, IV, 41.
JOT Anthmetica, Sorae, Denmark 1643,132-3. Gf.lLG.Zeatnen>rutcnne<llairede8m&tJi.r
3r 1896, \S2r-3.
iU According to O, Tcrquem, Noov. Ann. MMib, 18,1859, BulL BiiL, 1-2, the term problem
of the virgin* amee from the 45 arithmetic*) Greek ерфщшм, Bacbet,1*1 pp. 349-370,
and J, С Heitbronner, Histoiia Math. Uttivereae, 1742,045. Of. Sylreeter** of Ch. Ill,
111 MS. No. 10248 du fonrfs I&tin^ Bibtiotbeque Natkmab de Pane, f. 1M, "De pioblem&tibua
arith. iftdefinitaa,'' Prob 2.
1M Abentheuerticber GhickatopT, 1669, 440. Of.

80 History of the Theory of Numbers. [Chap, и
received the same total amount of money. How many did each sell at
first and what were the two prices? The answer given is that at first A
sold 7, В 28, С 49 at 7 eggs per kreuzer; the remainder were sold for 3
kreuzer per egg» Thus they received-1 + 9, 4 + 6, 7 + 3 kreuzer each.
There111 are eleven sets of positive integral solutions of
x + у + z =- 56, 32s + 20y + 16* - 22-56.
T. F. deLagny1* (p. 583) treated the problem of Diophaotus192 II, 18,
to find three numbers such that if the first gives to the second £ of itself + 6,
the second gives to the third £ of itself + 7, the third gives to the first $
of itself + 8, the results after each has given and taken shall be equal.
To avoid fractions call the numbers 5z, 6y, 7г. Then the first gives x + 6
and receives z + 8 and becomes 4x + z -f- 2. Thus
4x + * + 2 = 5y-M-l = 6* + 5f-l.
Eliminating z and у in turn, we get
19» + 18 17x -f- 12
»-—ИГ"' »--аГ~-
Their difference Bx + 6)/26 must be an integer. Multiply it by 8 and
subtract from z\ thus x — 36 and hence x — 10 is divisible by 26. Since
2x + 6 and 2(j - 10} are divisible by 26, while tbeir difference is 26, the
problem is possible. We may take x = 10 + 26£ and get an infinity of
integral solutions. He employed the same method to treat any such
" double equalities " of the first degree, which may be reduced to
±fli±? ± Ьх ± d
The principle is to get acicby elimination.
N. Saunderson13 (pp. 337-354) and A. Thacker1* treated two equations
in xt yt z in the usual way.
L. Euler1M discussed the regula Coeci. Given
p ± q + r - 30, 3p + 2q + т - 50,
eliminate r. Thus 2p + g = 20, whence p may have any value ^ 10.
In general, for
A) * + tf + * e «, /ж + ЯУ + Лг = &, f±g^k,
Ь £/(« + У + ж) - /в, Ь 1= *(* + У + ^ = Ад,
while b must not be too near these limits /a, fca. By eliminating z, we get
ocx -b /fy = c, where a and 0 are positive» A similar pair of equations in
y, 1709-10, Queet. 8; С Hutton'e Diari*n МмоеШлу, 1, 1775, 52-^3; T»
y Math. Qu«^. L. В,, 1, I8l7r 5.
Di^phanius used 5xt fizr 7x and got at - 18/7. G. Wertbeim, in Ыя edition of Diopbantue,
1890, proceeded as hiid de Lagny»
А МиееШшу of Math. ГгоЫета, Birmingbur>, 1, 1743,161^9.
• Algebra, IIr1770r Cap. 2, Sft 24-30^ 1774, pp. 30-tt; Орелодаш^явг. 1,1,1911,358-31^

CBAPr n] System of Linear Equations. 81
four variables is treated; also
3x -f- fyf + 7* « 660, 9x + 25y + 49* =* 2920.
E. Beaout1M solved ж + у + z = 41, 24ж + 19y -f- 10г = 741 by elimin-
eliminating x and showing that the integral solutions of 5y + 142 = 243 are
z ~ 5u-3, у = 57 - 14«.
Abbe Bossutt9f solved by eliminating z
z ±y + z = 22, 24x + \2y -f- 6* = 36.
A. G. Kastner1" treated the problem of Pratoriusm and its generalisa-
generalisation: Three peasants have a, b, с eggs, respectively, where a, fct с are distinct
numbers. They sold xy y, z eggs respectively at the price m per egg and
the remainder at n. Each received the same total amount of money.
Find xt yt z, m/n. We have
mx 4- «C« л x) — тУ Ч- л№ ~ v) — ю* + w(c — г),
where xt a — xt etc», are to be positive integers. We get
m Ь — a _ с — Ъ (Ь — c)x-\- (с — a)|/
n x — у v —z v> — a)
Give successive values to x and solve the equation in yt z,
A. G. Kastner19* discussed the " Regel C6ci/' From A),
- Ь-ah- (f-h)x
y" g-h
whence
Ъ ~ ah
, ag + (/ — £)s ^ 6, so that we have limits for z.
J. D. Gergonne1" considered n equations in m > n variables,
aixxt + ■ • 4- «мЛ, ^= A£ (г = 1, * -, n),
with integral coeffieients, and stated a priori that
х, - TV + ilja + Bfi + • • -
where a, ^, * * * are parameters in number м — n at least. Substitute these
expressions for the x's into the given equations and equate the coefficients
of at of 0, etc. Some of the resulting conditions show that 7\, 7^,
is a set of solutions of the given equations. The remaining conditions
show that the A% the B's, - * - are sets of solutions of
an*l + ' • • + <*imVm = 0 (i « lf . . -f n),
i" Coum de M*tluf 2,1770,94-6. ^"^ "^
"*Guun<ieMfttfa.rH, 1773; ed. 3, I, Para, 1781 r 414.
ltT Lezpziger Мл^алш fur reine u, ODesw» Math,, l7S8r 215-227.
m Mstb. Anf«npgrtadef 1,2 (Fortsetmns der Rocbenkunflt, ed. 2,1801, 630).
w АпмЬ de Mmth. (pd., Gereonne), 3,1812-13,147-156.

82 History of the Thbort of Numbers. [cw. п
and hence are determined by the matrix (a,-,-). The same discussion was
given by J\ G* Gamier,200 who remarked that the determination of the A'a^
B% * — is facilitated by the use of determinants.
J. Struve201 reduced the solution of A) to an equation in 2 variables
V. Bouniakowsky202 discussed the solution of one от more indeterminate
equationsr chiefly of linear type.
G. Bianchi** treated three linear equations in x3 у, г, и, solving: by
determinants for x, y, z as linear functions of г*-and determining by inspec-
inspection what positive integral values, if any, may be given to и such that the
expressions for x, y7 z become integer».
C. A. W. Berkban104 noted that if A) have positive integral solutions,
the xJs are ш arithmetical progression, with the common difference g — h~
I. Heger** considered a system of homogeneous equations
B) fcuXi + — + к<я^пХт+л = 0 (г « I, ■ • •, n),
with integral coefficients. I^et Хц be the numerically least value + 0 of Xt
in all possible sets of integral solutions, and let хц, * * >, Zim+* be one such
set. Their products by £i give a set of solutions. The only possible Xj's
are multiples of Хц. 1й B) set
zi - XnSi, x,- - xxrft -f x\ (j - 2, - -, m 4- л).
Then
hi&i + * •• + ^«+Ли-, = 0 (i - 1, • •-, n).
As before, xj — x^ti where x^i is the numerically least value + 0 of xi in
all sets of integral solutions. Let Xiir * * 'j xi «+>• be such a set. Proceeding
in this manner, we get
If the determinant of the coefficients of x^+i, * ♦ -, xm+n in B) is cot zero,
those variables are definite linear functions of X\, • • *3 Xm> whence
where the x^ „,+/ may be taken integral. Giving arbitrary integral values to
Zu ' *') %mt we obtain all integral solutions of B).
For n non-homogeneous equations in m variable», л < m, let all the
determinants D of order n of the matrix of coefficients have the g.c.d. /;
"•Couth d'Analyee Algebrique, ed. 2, Fairfc, 1814r 67-79-
M EriuuteraDg einer Regd fur unbeet, Atifpaben . . ., Ahon*, 1819,
ж Bell. phys. matb. acad. ве. St. Pg^z^bouzg, 6, 1S4S, 196.
*" Memoric di Mat. e Ft». Бос. Itaiiana Sc^ М«Ыц 24, II, 1850, 2SO-9.
** ШкЬщуЬ der Unbeat. Analytik^ Ha)!^ 1,1855, 46-53.
"* Denkechriften AbsA. Wiae. Wien (Math. Nat.), 14, II, 18S8» 1-122. Extract in
her. AkeA. Win. Wien (Mathjr 21, 1856r 55(МЮ.

] Згахдам of Linelab Equations. 83
consider the determinants К in which the constant terms appear in one
column, and let F be the geA. of the /)'s and /Ts. There exist integral
solutions if and only if/ =* f; while///" is the least common denominator
of all sets of fractional solutions. Cf. Smith*7 and Frobemus.1"
V. A* Lebesgoe** would select, if possible, two equations до* — F(xif
♦- .f Хш) And a'xi — Fi(x^f * - *9 jcb) from the system of linear equations such
that л, a' are relatively prime» Determine ry s, py q so that or — a'* — 1,
ap — a'q = 0. Then Xj = rF — sFi, p^ — jf\ = 0, whence the system Ы
reduced to the former and equations in хг, - --, jn only. To solve
дд; + Ay = ее, a'j? + Ъ'у — e'l, where the g.c.d. of a, b, с is unity, we may
set Zr — Du, where D = cux + b^isthef.c.d-ofa,i* Thosx « ели + 6i?/D^
^ = c&u — av/D. Then the second equation becomes Аи + Bv = c't,
which may be treated as was the first. Given a system of m linear equa-
equations in тп -\- n unknowns in which an m-rowed minor D is not zero, we get
Dxi = fi(yir • * -, yO, г = lt - , m. It remains to solve the congruences
/» — 0 (mod /)), which can he treated by the method for linear equations,
H, J. & Smith107 proved that if the excess of the number of unknowns
above the number of linearly independent equations is m, we can assign m
sets of integral solutions (called a fundamental system of sets of solutions)
such that the determinants of the matrix formed by them admit no com-
common divisor > 1. Every set of integral solutions of the equations can be
expressed linearly and with integral multipliers in terms of the fundamen-
fundamental system. By use of this concept be proved the theorem of Heger:** A
system of linear equations is or is not solvable in integers according as the
g.c.d. of the determinants of the matrix of the coefficients is or is not equal
to the g.c.d. for the augmented matrix obtained by annexing a column com-
composed of the constant terms (cl Frobenius™). Use is made of the im-
important elementary divisors*
H. Weber8" considered the system of equations
Л, ** **!*!, -f ■ * ■ 4- m*rfi + A> (i - 1, ■ -, p)
with integral coefficients <r,> of determinant 6. IS & # 0 we obtain every
set of integers Ab * * -,hp and each $*~l times if we take all possible combina-
combinations of integers for mh > - -f mp and let X,, * * ■, X, run independently
through л complete set of residues modulo & If S ^ 0, we can apply to the
fn's sudb a substitution of determinant dz 1 that the matrix (tr^) is trans-
transformed into one with columns of zeros at the right. Then by a linear sub-
substitution on hu " -t hp of determinant db 1, we get a matrix having zeros
except in the 9-rowed minor in the upper left-hand corner.
E. d'Ovidio30* treated algebraically a system of л — r independent
linear homogeneous equations in n unknowns and the conditions that it
have the same <*>* solutions as a second such system.
y qe, Parier 1850r 66-75»
PhiL TmoL Lndoo, 151,1861,293^326; «bet/, in Proc. Hoy. Soc^ 11, l$6lr 87-0. Coll.
Math, ftpe», 1,367-400.
ш Jour, fur M*liur 74,1872, 81.
*" Atti R. AccmL &?. Tocifta, 12,1876-7,334-350.

84 History of the Theory of Хтшвекв. £Chat. ii
G, Frobeniufl21* proved the following generalization of the theorem of
Heger:** Several non-homogeneous linear equations have integral solutions
if and only if the rank I and the g.c.d. of the l-rowed determinants of the
matrix of the coefficients of the unknowns are the same as for the augmented
matrix obtained by annexing a column formed by the constant terms.
Again, sets of integral solutions of m independent linear homogeneous
equations in n unknowns (n > m) form a fundamental system if and only
if the {n — m)-rowed determinants formed from them have no common
divisor. He discussed (pp. 194-202) the equivalence under linear trans-
transformation of determinant zt I of two systems of m linear forms in n vari-
variables; on this subject, see Smith,2*" G. Eisenstein,211 and G. Frobenius.2"
Ch. Meray2" considered a system of m linear forms
C) ф< *= a# + b$ + ••• +j> (t-1,-,»)
in n > m unknowns. Multiplication of this system by the matrix
/ Xi mi • • • <«i \
is defined to be the operation of forming the system of m forms
tfi = Х1Ф1 -j- Х2ф2 + - — ■+- Хя.^, - —j ^m = wi^i -h «4Ф2 + • • * + «Ж*
If we multiply the Utter system by a second matrix, we get a system which
can be derived from C) by multiplication by the product of the two
matrices» Given a system of m forms C) with integral coefficients, the
m-rowed determinants of whose matrix of coefficients are not all zero and
have the g.c.d. d} we can assign a matrix D) of rational elements of deter-
determinant lid, and a linear substitution on n variables with integral coefficients
of determinant unity, such that after multiplication by the matrix and
transformation by the substitution, we obtain a system of forms =b xif
±.Xzt "., rt xm. Then the system <fn + hi = 0 (t = 1, • • -, m) have
integral solutions if and only if the тта-rowed determinants of the coefficients
of the 0Js have for their g.c.d. a number d dividing all the wHrowed deter-
determinants obtained from the preceding determinants by replacing the elements
of an arbitrary column by the fc's [Heger*06]- When the equations have a
set of integral solutions £, . ■ *7 ф, all sets of integral solutions are given
without duplication by
where the 0's are arbitrary integers and the coefficients of any 6j satisfy
the system ^ = 0 (i - 1, • - -, m).
A. Cayley214 stiggested that, to solve a system of linear homogeneous
M Jour, fur Math, 86» 1878/l7b3. Cf-
«l Bericbte Akad. Wias. ВсгНц, 1852» 350,
*■ Jour, fur Math.» 88, 1879» 96-116.
™AnnaJes sc. de lp6cole nonnale вир.. B), 12, 1883, 89ЧО4; Gomptes Rcndus Paris.
18S2, 1167.
«"Quw.Jour. Matb.p 19» 1883, 38-40; CoU. Math. Papera, XII,

Chap. II] SYSTEM OF LlNEAB EQUATIONS. 85
equations in the unknowns A, B, • • -t we first equate to zero аз many
unknowns (say A, >>>,#) as possible such that there exists a solution
with F 41 0; we may take F = 1 and have a solution " beginning with
F = 1." Next, set F = 0 in the initial equations and equate to zero as
many of the earlier unknowns (say A, Bt C) as possible such that there
exists a solution with D Ф 0; we may take D = 1 and have a solution be-
beginning with D — 1 and having F — 0. The third step might lead to a
solution with A — 1, D = F — 0. Then we have a system of three
standard solutions.
E. de Jonquieres*1* discussed the equations, arising in Cremona trans-
transformations,
Z Mi = 3(л - 1), 2 i*a, = n* - L
G. ChrystaP16 proved that if x\ y\ z' form a particular set of solutions of
ax + by + cz — d, a'x + bfy + c'z ^ d',
and if « is the g.c.d. of the determinants (bc')f (ca7), (afrO, while г* is an
arbitrary integer, all solutions are given by
x = x* -h (bc')u/t, у =y' -h (cay)t*/«, г =* г' + {ahf)u^
Т. J. Stieltjesm gave an exposition of the results by H. J. S. Smith397»
L. Kronecker"8 gave a simple proof by inductioa of the theorem due
to Frobenius*10 that every n-rowed square matrix with integral elements
can be reduced by elementary transformations (interchange of rows or
columns and simultaneous change of sign of one row or column, and the
addition of one row or column to another) to a matrix in which every
element outside the diagonal is zero while every element =f 0 in the diagonal
is positive and a divisor of the following element. A matrix has a single
such reduced form.
P, Bachmann*1* gave a detailed account of the theory of systems of
linear forms, equations and congruences. For a summary account, see
Encyclopedic des Sc. Math», tome I, VoL 3, 76-89.
J. H. Grace and A» Young230 gave a simple proof that any systsm of
linear homogeneous equations with integral coefficients has only a finite
number of irreducible solutions in integers ^ 0, a solution being called
irreducible if not the sum of two solutions in smaller integers.^ 0*
J. Kdnig9* treated, from the standpoint of modular systems, systems of
linear equations and congruences whose coefficients are polynomials in
assigned variables.
«Giomale di Mat., 24, 1886,1;Cooiptes Bendue Paria, 101,1886,720,857,921. 'Pam-
'Pamphlet, Mode de solution d'une question d'&nalyue indetermipee . . . th^orie dea trajM-
fonnationa de Cremona, Pww, 1885.
<" Algebra, 2,1889, 449; ed_ 2, vol. 2,1900,477-3.
*" Annalefi Fac. Sc. Toulouse, 4,1800, final paper, pp. 4&-103.
•» Jour, fur Math., 107r 1891,135-6.
*" Aritb der Quadratiacheu FormeD, 1898, 288-370.
»»Algebra of Invariants, 1908,102-7.
ш Eialeitung . . . Algebraiechen Groasen, Leipaig, 1903, 347НЮ0.

86 History op the Theoky op Numbers» CChap. ir
A. Chatelet222 gave a brief summary of results, especially Heger's.20*
E. Cahen*21 gave an extended treatment of systems of linear equations,
congruences, and linear forms.
M. d'Ocagne2» solved x + y + * + t = n, 5x + 2y + z + $t = n to
find the number of ways to pay a sum of n francs with 5, 2, 1, \ franc coins,
n in alL For similar problems, see Schubert1" and d'Ocagne178 of Ch. III.
One lineak congruence in two or mobe unknowns,
Т1ь Schonemann22* considered the number Q of sets of solutions of
<h*i + • •• + а~Ы = 0 (mod p)t
with {j, --, £щ distinct and with the understanding that the solutions
obtained by permuting equal elements a count as a single solution, and
p is prime. Let д of the a's be equal, v further ay$ be equal, etc» If
«i + • • • + e» + 0 (mod p) and m ^ p,
But if Oi + - >> -|- Ощ ^ 0 (mod p)} while the sum of fewer a*s is not divisible
V; A. Lebesgue,2» by specialization of his" result in Ch. VIII of Vol. I
of this History, found that, if p is a primitive root of the prime p,
0, pe + Ai + •" + (>*** => 0 (mod p)
each have p*^1 sets ^i, ••',** of solutions ^ 0, but have
£fo - i)((p - l)-1 - (- l)*-M, ^ {(p - i)fc - C- l)fc)
sets of solutions > 0, respectively.
AL A. Stern*" proved that, if p is вп odd prime, any integer can be
expressed modulo p in exactly P = B* — l)/p ways as one or the sum
of several distinct numbers chosen from the set 1, 2, - ■ -, p — 1. For
example, 3s=l-j-2 = l+3 + 4 (mod 5). Restricting ourselves to an
even number of summands, we find that zero can be expressed in
b(P + p- 2) ways, while 1, 2T • • -, or p - 1 can be expressed in J(J> - 1)
ways. We shall report in the chapter on quadratic residues on his results
when the aet is 1, 2, ■»., (p - l)/2.
фваг la tfeorie dee DOmbtra, 1913, 5ЬЯ.
Tbtetit (fee nombrfflr l, 19H, 110-85, 2M-62r 278, 299-315, 383^T, 405-6.
L'eoiK%Dement m»tb.718P 1916, 45-7. Cf. Amer. Math. Monthly, 26,1919, 2]5-8.
• Jour, fiir 11*41^ 19, J839r 292.
•Jour, de Math., C2), 4, ]859P 366.
* Jpur. ftr Math., 61f 1863, 66.

Ю One Lineak Congruence in Two ok More Unknowns. 87
E. Lucas258 noted that, if a is prime to n, the points (x, y)t where x = 0,
1, • ■ ■, n and у is the residue modulo n of ax, lie on a lattice (composed of
equal parallelograms), and are said to form a satin na. These satins lead
graphically to all solutions of tnx -j- ny = 0 (mod p).
L. Gegenbauer2*' gave a direct proof of Lebesgue's225 results. Let the
number of sets of solutions each ф 0 of dXi -[-■"+ a*z* + Ъ = 0 (mod
p), where each a is not divisible by the prime p, be S'k or St according as
b is or is not divisible by p. Let N be the number of all sets of solutions.
Since a*xk -j- b ranges with x* over a complete set of residues modulo p,
N is the sum of the numbers of sets of solutions of the p congruences
«l^i + ■ ■ ■ + at^iX^i + с = 0 (mod p), с = 0, 1, - • -, p — 1; while the
number of those of these sets of solutions whose elements are prime to p
equals the sum of the numbers of the sets of solutions of like property of the
p — 1 congruences <*& + • ■ ■ + a*-iX*_! + c' s 0, d = 0, •. -, Ь — l,
Ь + 1, '-,p-l. Hence
tf - p*-1, Й = (p - 1)S^U Sk = tfw + (p - 2)S*_X.
K. Zsigmondy2* proved that, according as a is not or is divisible by the
prime p, kn+'ki-l- '•• + £,_! = a (mod p) has ^0? — 1) or ф(р — 1) — 1
sets of solutions in which each hi is prime to p, where ф(п) is the number of
congruences of degree n with no integral root modulo p. The system of
congruences
h + • - +'*^i з0, Ai + 2tf + • • • + (p - 1)^г з « (mod p)
has $(p — 2) or ^(p — 2) + p — 1 sets of solutions prime to p according
as а ф 0 or « = 0.
R. D. von SterneckM1 found the number (л), of additive compositions
of n modulo M formed of г summands which are incongruent modulo Mt
L e», the number of solutions of
л = xi + xt + • • • + Xi (mod M), 0 Ш Xi < x, < • • • < Xt < M.
Let (n)J denote the corresponding number when each summand is not
divisible by Mt so that 0 <xx < - - < x4< Af. Define /(я, О to be zero
if any prime occurs in d with an exponent which exceeds by at least 2 its
exponent in n; but when the primes pb —,Pi occur in d with exponents
which exceed by unity their exponents in n, and the remaining prime
factors of d occur in n at least to the same power as in d, let
Application de Parith. к la construction de I'&rmuro dee eatins r^guliera, Pane, 1868.
Prmcipii fondamentAli della geometri* dei tewuit, I'lugegnere Civile, Turin, 1880; French
tranaL in Assoc. /rang. &v. ВС., 40,1911,72-^87. See S. Gunther, Zeilsehi. Math. Naturw.
Untemcbt, 13,1882,93-110; A. Aubiy, I'enfieicnecient math.r 13,191IP187-203; L
CChVI
gr Akad.Wiae. Wien (Math.), »&, Па, 1890,793-4.
Monatehrfte Math, Phy»., gr 1897,40~l.
Sitnm«pber. Akad. Wiea. Wicn (Math.), Ill, IIat 1902, 1507-1601. By simpler methods,
and removal of the restriction on the modulus M, ibid.t 113, Ila, 1904t 32G-340.

88 History of the Theory of Numbers. tcaAP. Ii
where ф is Euler's function; finally, l&t/(», d) — 0(d) if no prime occurs in
d to a higher power than in n, so that n is divisible by d. Then
summed for all the divisors d of ilf, where (J) is & binomial coefficient
and is zero if j is not an integer. By the sedond formula, /(те, М) equals
the difference between the numbers of representations of n by an odd and
by an even number of summands not divisible' by the modulus M.
Von Stemeck232 proved that the number [n~\* of representations of n as
the residue modulo M of a sum oft elements chosen fromO, 1, • • -, M — 1,
repetitions allowed, is
summed for all the divisors d of 3f. If the elements are chosen from the
numbers ei, * • •, ep mcongruent modulo Mt then
Von Sterneck2" determined (n)i and CnH^ ^or a prime power modulus^
О. Е. Glenn234 found the number of sets of solutions ofX-j-M + v^O
(mod p — 1) and of X-h/t-h^ + f^O (mod p — 1)T the order of X, /i,
— being disregarded, and p being prime*
D. N. Lehmer286 proved that aYxx -|- * - + a*v* ± a*+x = Q (mod 7»)
has m1^1^ solutions or no solution according as the g.c.d. 6 of au * * -, ал> тп
does or does not divide ал+ь
L» Aubry1* noted that if Д is prime to N and if Л/ ^N is not integral,
Д^ =2 ^ (mod jV) is solvable in integers ф О numerically < VSf-
System of ltneab congruences»
A. M. Legendre^* treated the problem to find integers x such that, if
a and b, af and &', * — are relatively prime,
ax — с a'x — c'
are all integers, The first condition gives z = m + bz* Then the second
condition requires that a'bz + a'm — c' be divisible by б', which is im-
d/WjBe. Wien (Math.), 114, Ш, 1906, 7l 1-730.
Л, 118, Па, 1909, U9-132.
г. Math. Monthly, 13,1906, 59-60,112-4.
ш1Ш.,70,1913,155-6.
V Mfttbra, D), 3t 1913, 33-5,
шТЪбопе dee nombroe, 1798, 33; ed, 2, 1805, 25; ed. 3,1830,1, 29; Maaer, It p. 29

;] System of Likeab Congruences. 80
possible if the g.c.d. в of b and V is not a divisor of a'm — c'; but, if в be
such a divisor, the general solution is of the form z — n + z'6'/0. Thus
x = ?w' + #'z, where 2?' is the l.C-m. of Ь, Ъ\ Similarly, also the third
fraction will be integral if x = M + Вг, where В is the Lena, of 6, h\ btr.
* M. Fekete215 treated the general system of linear congruences in one
unknown.
C. F. Gauss2*9 discussed at length the solution of n linear congruences
Ш n unknowns» His second (more typical) example is
dx + 5y -h * = 4, 2z -h 3y + 2z = 7, 6x + у + Зг = 6 (mod 12).
We first seek integers240 £, £', f" without a common factor such that the sum
of their products by the coefficients of у (and by those of 2) is congruent
to zero:
5£ + 3{' + $" e 0, f + 2f + ЗГ = 0 (mod 12).
Thus £ — 1, Г ^ — 2, $" = b Multiplying the congruences by these,
and adding, we get 4x = — 4 (mod 12). Similarly, the multipliers 1,
1,-1 give 7y = 5, while the multipliers - 13, 22,-1 give 28z = 96.
Thus x = 2 -^ 3f, у — II (or 11 -f 12r), г — 3u. The proposed congru-
congruences now give three equivalent to
19 + 3* + и ^ 0, 10 + 2t + 2u = 0, 5 + 5^ + Zu = 0 (mod 4),
which are all satisfied if and only if и = I + 1 (mod 4)» Thus
(s, у, г) - B, 11, 3), E, 11, 6), (8, 11, 9), A1, 11, 0) (mod 12).
H, J. S. Smith noted that the theory was left imperfect by Gauss» In
A) AaZi + • • • + Ainxn = Ain+i (mod M) (г - 1, • • -, n),
denote the determinant | An | by Z>. If D is prime to Mt there is one
and but one set of solutions. Next, let D be not prime to M = p^pT * ■ -,
where the р'з Яте distinct primes» A necessary condition for solvability is
that there be solutions for each modulus p?\ Conversely, if there be Pi
sets of solutions for modulus p7\ there are PtPt • * * sets of solutions modulo
M* Hence consider A) for the modulus pm, and let Ir be the exponent of
the highest power of p dividing all the r-rowed minors of D. Then, if
/„ — /n_i ^ m, the congruences, if solvable, have p1* sets of solutions»
But if I* — /*_! > m, we can assign a value of r such that
/r+i — It > m ш It — L-i
and then the number (if any) of sets of solutions is pkt where
к « L + (n - r)m.
w Math. & РЬув» Lapok, Budapeit, 17, 1908, 328-19, ~
*■* Disq. Arith., Art. 37; Werkc, I, 27-30.
«•R J. StudniStB, Sitsungpbericlite, Abd. Wla?-P Pragr 1875, 114, noted that they are pro-
portionai to the signed minore of the coefficients of the first column ш the determinant
of the coefficienta.
•» Report Britiah Aaoc. for ]S50, 228-67; CoU. Math. Papers, I, 43^-5.

90 History or the Theobt of
Smith107 wrote Vn for the determinant f Ац] of <1)T V^i for the g
of its first minors, ■ ■», Vi for the g.c.d. of the elements Aiit and set Vo = 1-
Let DKt D«_i, •-, Д bethe correspondmg g.c.d,'s for the augmented
matrix. Let 8< and di denote respectively the gx.d. of M, V,/v\_i, ***<*
Af, DijD^ Set d = rfr -<*,, 5 = Sr > 1Щ. ТЪеп the system of «ш-
gruences A) is solvable if and only if d — 9; when this condition is satisfied
the number of incongruent sets of solutions is d. There are similar theorems
(pp. 402-4) when the number of unknowns is either less or greater than
the number of congruences.
Smith142 employed a prime factor p of Af, and the exponents pf а„ л,
of the highest powers of p dividing M, A, V., respectively» He proved
that his preceding theorems can be replaced by the following: For the
modulus p% the congruences A) are solvable if and only if a* = a*, where
aa — a,_i is the first term of д* — а*_ь a^i — a*-*, - * * which is < p;
when this condition is satisfied the number of incongruent sets of solutions
is pk, where к = a, + (n — <т)м-
G. Frobenius2№ (pp. 185-194) proved that the congruences (I) have
Af * incongruent sets of solutions if the 1-rowed determinants of A have
with Af no common divisor and if tfee (I -h l)-rowed determinants of the
augmented matrix of all coefficients are divisible by Af, where I is the rank
of the matrix A of the coefficients of the unknowns. If the rank of the
augmented matrix is I -h 1 and the g.e.d. of the {I + l)-rowed determinants
is d', while the rank of A is I and the g.c.d- of the Crowed determinants is
d, the congruences A) have no solutions if the modulus M is not a divisor
of dfjd. The number of incongruent $ete of solutions of the homogeneous
congruences Ai&i + - • • + AiMxn ^ 0 (mod M) equals s^- - -*„, where sk
is the g.c.d. of M and the Xth elementary divisor of the matrix (-A,y).
Frobenius212 proved that a system of linear homogeneous congruences
modulo M in n unknowns has a fundamental system of n — * sets of solu-
solutions, but none of fewer than n — s, if the determinants of order s + 1 have
with M a common divisor bub the determinants of order & do not He in-
investigated the rank and equivalence of linear forms modulo M.
F. Jorcke143 treated systems of linear congruences without novelty.
D. de Gyergyoszentiniklos144 considered the congruences
* u, (mod m) (p = 1, ^., n).
Let D = \ dpj | f and V* be the determinant derived from D by putting the
u's in the fcth column. Let 6 be the g.c.d- of m and D. If ацу F* is
not divisible by 5, there is no solution. Next, let each Vk be divisible by Ь
Then Dxt = Vk (mod m) uniquely determines xk = at modub mfB. Set
Xk = ^ 4- tkfnfb in the initial congruences. Thus
(a,A + • • • + a^mlh ^ u, - a^a^ - ... - et^o^ (modm),
* Proc. London Math. Soe., 4,1S71-3, 341-0; OoIL Math. Pbpei* U, 71-SO.
w Uebei- ZahleDkongroensen and euuge AnWttdtmgen dexseben, Pro^r Frwiet*dt, 1878.
M Compt^ Rendua Phxvt 88, 1879,1311.

OF LnVEAH CotfCRtJENCES. 91
the latter system, the modulus * divides the determinant D. Hence
if some minor of order n — via not divisible by 5, while all minors of higher
order are divisible by 6, the solution involves exactly v arbitrary parameters
and there are S" sets of solutions^
L. Kronecker3* deduced from his theory of modular systems the theorem
that, for p a prime, the general solution of
Z V*Xk = 0 (mod p) (» = 1,.. -, I)
involves т — r independent parameters if the matrix of the tr numbers Vik
is of rank r modulo p.
K. Hensel24* considered a system of m linear homogeneous congruences
in n unknowns in which the coefficients and the modulus P are either
integers or rational integral functions of one variable» We may replace
the system by an equivalent system whose modulus divides P and hence
finally obtain moduli» unity.
E, Busche247 proved that the number of solutions of a system of n linear
homogeneous congruences in n unknowns equals the modulus if the latter
divides the determinant of the system. This theorem is equivalent to the
following. Write а ~» Ы£ а-— Ь is an integer» If the a(i are integers of
determinant ф 0, the number of non-equivalent solutions xlp • - -, xn of
a»i*i + - • + ai*?* ~Q (i = 1» ~", n) ш the absolute value of the deter-
determinant | Qij [ „
G» B. Mathews^" noted that a system of n linear congruences in which
the motiatt aremIr »',.*» respectrvety may be reduced to a system with
the same modulus m (the Lc^nu of mu. • -,тл), by multiplication by mfmt,
■ -, mfm* respectively- For the case erf a common modulus m the method
is to derive an eqarralectt system of congruences involving respectively
я, я — 1, - - -, 1 unknowBs. Details are g^ven only for the example
ax + by + as = d, a'z + Vy + t?z = ^, a"x + V'y + d'z = d" (mod m).
Let ^ be the g,c.<t of a, u?t a" and let В = pa + qa* -j- та". Multiplying
the congruences by pf qr r respectively and adding, we get a congruence
to• + ^ + 7^ — &- If,, for ехяшр^е, р m- prime to m, we get an equivalent
system by taking: the latter in place of the first congruence of the system*
The» eliminate x from the second and third by means of $x -j-
I G shewed that the system of linear congruences
f[ = 0 (mod p) (P = 0, • •., V ~ 2)
Has as many Кпеаггу independent sets of solutions as
1,167, a. i*pera 24-26, p. 22бгаЫ 43, p. 232
?
" Jwa. Rlr Mbitb^ 107, ISM, 241.
w Mitt. Math. GweJL ШшАиг^ 3r I»1P 3-7.
219 Tbcory of Nambem, 1888; 13-14.
*"МЫЬ Further report on p. 229 of Vol. I ofthia History,

92 History of the Theory of Numbers. QCbap. II
has distinct roots not divisible by p- Such a system of linear congruences
has been discussed by W. Burnside.*0
E. Steinitz**1 stated that all theorems on linear congruences follow
easily from one: Given k linear congruences in n variables modulo то, the
k sets of coefficients form the basis of a Dedekind Modul. If ei, > > -, en
are the invariants of this Modul (the last n — r e's are zero if the rank
r is < n) and if \e^ vy\ is the g.c.d- of e* and m, then the totality of sets of
solutions of the к congruences represent a Modul with the invariants
m m
[em> тУ *[еи тУ
Expositions in the texts by Bachmann,*1* J. Kdnig,**1 and Cahen1" have
been cited. Zsigmondy230 found the number of solutions of & system of two
special congruences;
EL Weber1*2 made a direct examination of the conditions under which
B) aitfi + а*#, + • • • + a,jVp s 0 (mod p")
shall require that each y< be divisible by p9t where p is a prime. It is as-
assumed that not every a^ is divisible by p (otherwise a solution is obtained
by taking each y,- to be any multiple of p*~l). We may assume that
Д * |а#|«-,^1.„..г is not divisible by p, while every (r + l)-rowed deter-
determinant of the matrix (a,;) is divisible by p. Denote the signed minors of
A by Aju and set
h
Thua Dk, ж Д if к = a; 2>fa = Oife^T,s+&; while, if e > r, Dk* is а т-
rowed determinant of (a»-;). Applying Cramer's rule to the first r con-
congruences B), we get
C) Ayj + 2>/tf+jf,+, + • • • + AA ^ 0 (mod Pw) U = l.-i r)-
Hence
AfairVi + - • + <!„#,) = Лт+| r|/^x + - - - + A^V, (mod p')>
where
equals a {? 4- l)-rowed determinant of (ou) and hence is divisible by p.
Tims, if r < p, B) are satisfied when yr+1, * **, j/ are divisible by p''1-
In order that B) shall require that each y< be divisible by p* it is therefore
necessary that т = p* This condition ts ftbo sufficient, since C) then
reduce to Ay2 = 0, - • •, Aj/, ^ 0, whence yi, • • •, yw are divisible by p',
F- Rieaz^ stated that, if the «^ and ft are reai, the congruences
£лм*ь s ^ (mod 1) (i ^ 1, • * ♦, m)
r
d. Detttecb» M^HL-Vcma., S, )Я9Б [190Ц 87
>Lehrbuchder AJeeboi,2, хдабу«7-8; Ы.2,1806,94. CLft
* Gottptes ^cvfaxi F^n, 139, 1504, 439-4Й2.

Chap.iij Real Linear Forms, Approximation. 93
are solvable in integers when the 0's are arbitrary, with a desired approxima-
approximation, if and only if 2aikxk ^ 0 (mod 1) are not solvable exactly in integers
not all zero.
U. Scarpis2* proved that a system of n linear homogeneous congrueDces
in n unknowns has solutions not all divisible by the modulus M if and only
if the determinant A of the coefficients is not prime to Л/. The problem
is reduced as usual to the case M = p", where p is a prime. Then let
some /crowed minor of Л be prime to p, but all Arrowed minors (k ^ p -j- 1)
be divisible by p. Let p* be the highest power of p dividing A and all its
£-rowed minors (fe s p + 1). Then p of the congruences are linearly
independent. We may assume that | a,-,- j , where i} У — 1, --,/>, is prime
to p. Then the last n — p of the congruences can be replaced by congru-
congruences in x^u • • -,zn in which each coefficient is divisible by -p\ If m = 1,
no more than p of the initial congruences are linearly independent; the
valuesofxi, - -,x, are uniquely determined in terms of x^l% •• -, xn which
are arbitrary, so that there are pw~^ sets of solutions.
Linear forms with real oobffzcents; approximation.
J. L. Lagrange** noted that, if a is a given positive real number, we
can find relatively prime positive integers p, q such that p — aq shall be
numerically smaller than r — as for г < pt s < q, by taking p/q as any
principal convergent to the continued fraction for a with all terms positive.
Lagrange*2 determined a fraction m/a, with given numerator or denomi-
denominator, which shall approximate as closely as possible to the given fraction
B/A < 1, where At В are relatively prime» For example, let m be given.
Take as a the quotient found on dividing Am by B. If ^ С is the remainder
numerically < §£, then Ba — Am = ±C, В/А = т/а ± C/(Aa). Start-
Starting with С/A, determine similarly n/Ъ, with n given, by using the quotient
Ь and remainder =F D when An is divided by C, whence CfA =n/b±Df(Ab).
Similarly, DjA = р/с ± E/(Ac). It follows that m ^ a, n < 6, p ^ c,
■ • • and that A,B,C, D, - — form a decreasing scries terminating with zero:
In case the denominators a, bf c, • • • were given and all equal, we have
expressed B/A to the base a. Finally, suppose that neither m nor a is
given, but are to be found such that m < £, n < A, and such that m/a is
as close an approximation to B/A as possible- Hence must С = ± 1»
Then frt and a are found by Euclid's g.c.d. process. Saunderson13 had
already treated the approximation to a fraction and cited earlier writers.
С G. J, Jacobi** proved that integral values not all zero can be assigned
to zt yf z such that ax -j- a'y + arfz and bx + b'y + b"z are less than any
assigned quantity» Cf. Sylvester108 of Ch. III.
*■ Periodico Л Mat., 23r 1908,49-бГ " """ ^"^ ^^
» Additioiu to Euler'e AJ^jbra, 2, 1774, 445; Oeuvrea, VH, 45-57.
*■ Jour. Шг Math,, 13,1835, 55; W^rke, II, 29-31.

94 Histohy of the Твдюкт of Numbers. [Свар, и
G. L, Dirichlet*' stated that it has been long known from the theory of
continued fractions that, if a is irrational, there exists an infinitude of pairs
of integers x, у for which x — ay is numerically < 1/j/, He proved the
following generalization: If a^ - • •, «* are such that
vanishes for no set of integral values of Xq, - - -, xMf not all zero, there exists
an infinitude of sets of integers Xo, -*-, s», with X\, ^ , хш not all zero,
such that / is numerically < l/em, where s is the greatest of xh • • -f s*.
Similarly for several forms/. For example, if a = cr^i 4- - • + «^ and
0 - /З^ + - • - 4- ft»** vanish simultaneously only when xt9 -•-, x«, are
all zero, there exists an infinitude of sets of integers хь '.-л not all zero
for which | a | < A}^ \P\ < Bis™-*-*, in which A and Б are constants
depending on the at, ft, while a is any constant between 0 and m — 2.
Ch, Hermite2" remarked that, if A and В are given irrational numbers,
we can readily find the linear relations Aa + Б6 + с = 0 (if existent),
where а, Ъ, с are integers. In fact, a = mA - mf and /5 = m^ — m" can
be made as small as one pleases [by choice of integers m, m\ m"]. Since
aa + b& — — am' — &m" — cm is an integer^ it cannot be made < I witfi-
out reducing to zero. Thus to find m, m\ mff, we have only to convert
pfa into a continued fraction to obtain the desired relation,
Hermite2" proved by means of the тштИтпплш of a binary quadratic form
that, if а, Д are real, there exist integers m, n such that
(m - апУ + П2/Д2 < | |
whence |m — an J < 1/(«V3), Let m', »' be the integers corresponding
to Д' =^ Д + 5, where 5 is an infinitesimal. Then mnf — nm' — =fc 1.
P. L, Tchebychef1* proved that, if a is Irrational and b is given, there
exists an infinitude of sets of integers xf у such that у — ax — Ь is numer-
numerically < 2/| x |.
Hermite2 proved that, in Tchebychefs result, we may replace 2/j x \
by 1/12 [ г | } and in fact by ^/27/|л|.
L, Kronecker*2 treated the problem to find integers u>, w' such that
aw + ftV takes a vnlue as near as possible to £, where a, a\ £ are given
real numbers. In general, consider a system of p equations
+ h а*ю, - ii (i = 1, • • -, p),
with real coefficients. Let г be the number of these equations whose left
pr. Ak*L Wim. Balm, 1842, 96; Wttte, I» 635-8.
«Jour. ШтЬС^Ъ, 40, 1850,261; Oeuvree, Ir 101.
•■ /Ы., 41,1851,1вб^7; OeavnsA I, W-171.
"*Zttpbki Ac»d, плок St. Paerabonri, 10, I866r Sappl. No, 4, p. 50; Oeuvree, 1,1899, 679.
~ Jour, fur Math., 8&J879.10-15; Ouevies, ГП, 513-9-
^Momiteber. Akad. WW, Berlin, 1884, 1179-93, 1271-«9; Weite, Ш1т47-109. «.t&id4
1071-80; Comptea Bendafl Pane, 96, 1383, 93-8, 14S-53, 21U-21; 99, 1884, 765-71,
Werke, Шъ 1-44, fur mpplfc&tott t ib it

Chai\II] Real Linear Forms, Approximation. 95
members are linearly independent, so that r is the ordinary (absolute) rank
of the rectangular matrix
(aik) (i = I, -. ,p; к = 1, -,g).
This matrix is said to be ol relative rank (or rank of rationality) R if R is
the least number such that, by means of a linear substitution on the rows
with arbitrary coefficients, the matrix can be transformed into a matrix
all but т of whose rows contain only zero elements and all but R rows
contain only integral elements. Necessary and sufficient conditions for
the approximate solution in integers of our equations are expressed in dif-
different forms: R of the £'s can be given arbitrary values, while the choice
of r — R of the £'s is limited only by certain conditions of rationality,
while the remaining p — r £7s are uniquely determined in terms of the earlier
ra
A. Hurwitz proved that if » is irrational there exists an infinitude
of pairs of integer» xt у for which j y/x — ы \ < l/(V5x2). Likewise,
| yfx — v I < l/{ <J§x*i if o> is not equivalent to A + V5)/2.
EL Mmkowslri*4 found by use of lattice points and other geometrical
concepts the fundamental theorem that, if/b - - •,/* are linear homogeneous
functions of Xty - - -, x* with any real coefficients whose determinant Д is
not zero, we can assign integral values not all zero to хг, - • •, xn such that
\fi [ ^ ЩМ for i = lf - - -, n. If at, - - -, a«^i are real, we can find
integers Xi, <« <, xn without a common factor and with xn > 0 such that
For» > 1, consider n linear forms /i, • * ->/» in Xi, * - -t xn with a determinant
Л Ф 0, such that r of the forms have real coefficients ands = (n — r)/2 pairs
have conjugate imaginary coefficients, and let p be any real number = 1.
Then integral vnlues not all zero can be assigned to Xi, - , х* such
±
except ior p = 1, s = 0, n = 2, when the members may be equal; here Г
denotes the ordinary gamma function. He obtained (p. 161) Lagrange's2**
result on the minimum of x — ay,
A. Hurwita*5 gave an elegant analytic proof of Minkowski's*1 theorem,
and the fact that the inequality sign may be taken in n — 1 of the n relations.
Ch. Hermite*67 remarked that Euclid's gx-d. process leads to approxi-
m Math, AnnaJen, 39, JS9I, 279. TM4 and papers с«Ш1 on p. 158 of Vol. I of this History
give approximations by use of Farey series.
»* Geometric <fer Zahleii, 1896,104-123. Extracts in Math. Papera СЬк»£о Coogreae, 1896,
201-7; French tr&fisT, Nouv. Ann, Math., C)] 15, 1896, 393-KB.
99 Also in Compt«B Rendue Pan», 112, 1891, 209; Wcrke, I, 261^3.
"•GottingeD Nachnchtcn, 1897, 139, French tnuisl,, Nonvr Ann» Math., C)f \jt 1898F
64-74. Cf. P. ВасЬтлпп, AUgemeine Arith, d. Zahlenltdrpcr, 1905F 335-41; G. Humbert,
Annaleade Ы Fac. Sc. Toulooee, C)y 3r 1911, 8-12,
*7 Le Matematiche pure ed applies^, Citta di CasteDo, Ir 1901,1-2; Werke, IV, 552-3.

96 History of the Theohy or Nxxmbers. [Chaf- ц
mations to a fraction by means of a seiies of fractions m/nf the error being
< А/те2, Не gave a slight modification of Diriehlet's**7 method.
K. Minkowski2*8 proved that if £ = ax + &y and 17 =* yx + fy have
any real coefficients of determinant a$ — /37 =5= 1 and if £0, ijo are any given
real numbers, there exist integers x, у for which | (£ — &)(ij — »&>) J ^ }.
In particular, if a is irrational and b is not an integer, there are integers xt
у for which I (y — ax — h)(x — c) | < J; the case с = 0 gives a better
approximation than Hermite's,*1 since 1/4 < V2/27-
E. Cahen*' discussed the approximate solution in integers of a system
of linear equations.
E, Borel270 proved that if а, Ь, M are any given real numbers, integers
x, y, zf numerically < M, can be assigned such that
в being independent of а, Ъ, M (but not found)* Again, intervals (Anj Вл)
can be found such that A* and B* increase indefinitely with n and such
that, if a is any irrational number between 0 and 1» integers p», tfn exist
for which
_ 1
At least one of three successive convergente to a satisfies the first inequality
[cf, Hurwitz**}
Minkowski271 proved that if о is real we can choose integers xf у such
that I xjy — a I < lfy* and deduced the existence of solutions of si — ry = 1
if 5, т are relatively prime integers. He gave a new proof, suggested by
D. Hilbert,*71* of his*4 theorem on n real linear forms. He discussed (pp.
47-53) the maximum value of the minimum of | f \p + [ 17 |p, where £ and
rj are real linear forms. He treated (pp. 68-82) the equivalence and mini-
minimum of three linear forms £, 17, f, and gave theorems on the values of their
sum or product,
B, Levi27* proved Minkowski's** thoerem, and for the limit case in
which no integers, not all zero, make each |/* I < lf proved his result
that then at least one of the/» has integral coefficients.
"■Matb- Annalen, 54,1901 r 9Ы24, все ррЛ08,116 (Gee. Abhandl,r Ir320); French tr&nd.,
Ana. <fe I'ecole поппд]е sup., C), 13, 1896, 45. For an account of Mmkowakire investi-
investigations, see Yerb&Ddl. dee diitten intern. Math. Congra&ee Heidelbergr 1905, 164>
Proof by J.Uepensfcjj, Appbcatione of continuoiiH parajnetere in the theory of numbere,
8t- Peteraburg, 1910; cf. Jahrb. Fortechritte Math,, 1910, 252.
**Bull. Soc, Math. France, 30, 1902, 234-212. He also made additions to the subject in
Ыл article in the Encyclopedia dee Sc. Math., 1906, tome I, vol. Ill, 89-07.
17i Jour, de Math., <5), 9, 1903, 329-375; Cotoptee Rendue Рагвдг 1вЗ, 1916, 59в-8. Lecona
eur la thebrie de la croiseance, 1910, 143-154, Cf. A. D<mjoy, Bull, Soc. Math, de
France, 39, 1911, 175-222.
*" Diophfijitische Approxim&tionen, Leipzig, 1907, 1-19, 28r
*7le Cf. J. Somraer, Vorleeangen uber Zahlentheorie 1907, 66^72; French transl. by A. Levy.
I9llr
XTi Eendiconii Circolo Mat. Palermo, 3lr 1911, 31&-340.

п] Real Linear Forms, Approximation. 97
S. Kakeya** proved the theorem (Minkowski,*4 p. 108) that if «i, - -',a*
are real there exist integers Xi, - - -, xn, z such that xi — ахг, <—,хл — о»*
are as small numerically as we please. He proved that these forms approach
indefinitely any real numbers. He274 gave a generalization to any linear
functions.
R. Remak27*1 proved arithmetically Minkowslri's*88 first theorem.
H, Weber and J. WelisteinWi* gave a new arithmetical proof of Min~
kowski's*4 initial theorem for both real and imaginary linear forms,
H. F, Blichfeldt27* proved a result which in Minkowski's notations
becomes
For small values of n this limit is higher than Minkowski's2" (p. 122), but
for large n*s it is smaller. Given the positive numbers <x1} ■. • , a*^ and
any positive number & < J, we can find integers Xu ■ • •, Хл-Ь Z such
that the n — 1 differences | XijZ — m | are ^ 2b and
7 (n - l)Z-*H*-v
2>/(«-i) г
1
Except for n - 2, this approximation is closer than that obtained by
Hermite,*" Kronecker, and Miakowski.*4
G. H. Hardy and J. E. Littiewood*74 proved tiiat if tft, - - -, Bm are ir-
irrational and connected by no linear relation with integral coefficients not
all zero, and if an, • * >, a*» are any numbers such that 0 ^ ац < 1, there
exists a sequence of positive integers nif n±, • • - such that the fractional
part of nl$p approaches ctiv as r increases, for I = 1, • - -, k; p ~ 1, - - »f m
(the case к = I being due to Kronecker*2), Given X, there is therefore a
function Ф of к, m, X and the 6's and aJs such that the difference between
the fractional part (n'0p) of nl99 and aip is numerically < 1/X for some n < Ф.
When the 0's are given, а Ф can be found independent of the a's. When
all the a's are zero, а Ф can be found independent of the 0's. An upper
bound for Ф, in this last case, was later ^ven by H, T» J, Norton,277 IL
Weylm<l went further by showing that the numbers (n'0p) are " uniformly
distributed '* throughout the unit cube 0 s* xiv ^ I in space of km di-
dimensions [i.e., if we associate with n the point whose km coordinates are
x[p = (nlBp) and denote by n¥ the number of the first n points which lie
inside an assigned part of the cubs, of volume V, Шеи п9 *~ nV when
П -* со}
™8<лепое Reporte ТЫюки Univeraity, 2r I913r 33^4.
J» TMioku Mfttb, JourM 4F 19l3-ir 120-131.
«* Jour, ffir Math., 142, 1913, 278^82.
»*M«th. Annalen, 73, 1913,275-85r
m TnnBs Amer. Math. Soc., IS, 1914, 227-235.

98 History of the Theory of Numbers. [Chap. 11
Hardy and Littlewood™ also considered the same problem when nl
is replaced by an arbitrary increasing sequence >« with infinite limit and
obtained the same resultr but with the exception of a set of values of 0 of
measure zero, R. H. Fowler*77* established uniform distribution, with an
upper bound for the errorr when X* increases as rapidly as an exponential
e*1 E > 0), WeyF7* extended the theorem of uniform distribution to all
cases in which \n increases with tolerable regularity and as fast as (log nJ+s
E > 0). These questions are intimately connected with the problem of
the behavior of the series 2* exp. BтггХп) when N -> °°, which has been
considered in detail by Hardy and Little wood ,277C and WeyL277*
G, Giraud278 proved that there exist integral values of the x's and t/'s
for which
[ я\ ~ ^\У\ ~ '' • — ац>У? — A,-1 < € (г - 1, • ■ •, rt),
whatever e be, if and only if all the forms fnxXi + ■ ■ ■ + m*Xn, which take
integral values when Xi, ■ ♦ ■, Xn are replaced in turn by the p sets of
values a»-, ' ■ •, anj- (i = 1, ••-, p), take also integral values when we replace
■Xij " ■, Jin by Ai, ■'', A«.
S. L. van Oss27' proved that n real linear functions of xh - - •, xn of deter-
determinant unity have the minimum value unity for integral z's if at least one
of the forms has integral coefficients without a common divisor. This
had been proved by Minkowski for n — 3.
W. E. H. Berwick380 gave a method to find which pair of integers
x, у @ ^ у < N) gives the least value for / = ax + by + c, where a, by с
are real and not zero. Thus he finds the point with integral coordinates
nearest to the line/ = 0 and within the strip between у - 0 and у = N.
A. Brown281 noted thaty to find the fraction whose denominator does
not exceed a given integer and which approximates most closely to a given
number, Lagrange's theory gives the fraction nearest in defect and the
fraction nearest in excess, but does not decide which of them is nearest in
absolute value to the given number. A simple method is here given for
deciding between the two fractions,
A» J. Kempner28* noted that any straight Line with an irrational slope
has on either side of it an infinitude of points with integral coordinates
lying closer to it than any assigned distance»
G. Humbert283 developed Hennite's*5* method of approximating to an
irrational number u, showed that it differs very little from the method of
w Acta МдШ,г 37y 19H, 155-191; Proc. Fifth Internal CongrewMath., lr 1912, 223-9.
"* Proc. London Math. Soc.r B)r t6r 1917, 294-300,
«"• Gottingen Nachrichten, 1914r 234-244; Math. Annalcn, 77, 1916, 313-352.
"* Pioc. London Math. Soc>r B), 14,1915,189-206.
™* Acta Math.r 37, 1914, 155-238; Proc. Nat, Acad. Sc,T 2, 19l6r 583-6; 3, 1917, 84-8.
*" Soc. Math. France, Comptes Rendus dee S6ancesy Ш4, 29-32,
m HandeLLngen XVde Nederlandech Natuur- en Gene«skundig Congre8y 1915, 192-3.
*" Meaeenger Matb.r 45,1916,154-I60r
m Trans, Phil. Soc. South Africar б, Ш6, 653-7,
ш АдтОв of Math-r 19,19l7y 127,
^CompteB Hendu» Parier I61r I915r 717-21; 162, I9l6r 67; Jour, de Math,, G)r 2r 19l6r
79^103.

Chap. II] RfiAb LINEAR FORMS, AppHOXIMATION, 99
continued fractions, and found necessary and sufficient conditions that a
given fraction be in Hermite's series of fractions tending towards w. The
main condition was generalized by E, Cahen.*M
Humbert186 gave simple proofs of the theorems by Hurwitz.28*
M, Fujiwara** supplemented Hurwitz's2" second theorem аз Borel27*
had the first,
J, H, Grace2" proved that if f ^ A: ^ 2 and if xjy and x'fy' are two
consecutive rational approximations to an irrational number 0 such that
then xy* - x'y = ± 1 [Hennite2" for к - V3, Minkowski»8 for к =* 2j,
He188 proved that Minkowski's2»8 last result is final, i.e\, if к < I, it is
possible to choose a and b such that there is not an infinitude of integers
x for which | у — ax - Ь \ < k{\ x I
»*Compt«»ReadiiS Paris. 162У Шбр 779-782r ~~
** Jour, de Matb.r G), 2r 19l6r 155-167.
«»TAhoku Math. Jowr., 11,1917, 239-242. Cff 14,1918,109-115.
"' Ptoc. London Math. Soc., B), 17, 1919r 247-^58.
***1Ш.У 36

CHAPTER III
PARTITIONS
G. W, Leibniz1 asked Bernoulli if he had investigated the number of
ways a given nujnber can be separated into two, three or many part?,
and remarked that the problem seemed difficult but important. Leibniz*
used the term number of divulsions for the number of ways a given integer
can be expressed as a sum of smaller integers, as 3, 2 + 1, 1 + 1 + 1>
and noted the connection with the number of symmetric functions of a
given degree, as 2a1, Xa*b, Zabc,
L. Euler3 found relations between A - Xa, В = 2a2, С = 2o*, - -., and
a =* 2a, /3 = Xab, у *» Zabc, > - -,
<E = a* + a*b + ab3 + b4 + afc + aic + - • •,
Ф — й4 + a*6 + - • - + abed + • • •,
We have
P^S
1 -az
Й s ПA + og) = 1 + az + j
2сШ
Hence
A = а, аЛ - В = 2ft 0A - aB + С =
Next, expanding A + az), we get
Now take а ■= n, Ь = я1, с = n*f ■ ■ -. Then
A-n/(l^n), B-nV(I
Hence
' MAth, Scbiifteu (ed., GeAardt)r 3, П, 1856, 601; letter to Job. Berooulli, 1669.
* MS. dflied Sept. 2,1674. a. D. Muhnke, BlblioUifcs Mith,, C), 13,1912-3, 37.
' "ОЬэепг. аид!, de combumtioiubufl/' Comm. ActuL Petrop., I3f &d аппшп 1741-3, 1751,
64-93,
101

102 History of the Theory op Numbers, CChaf. in
p, y7 • • * being the sum of the products of n, n*f • • ■ two, three, — - at a time,
whence
The coefficient of n* in 0, % • * • is the number of ways a is a sum of two,
three, • - - distinct parts» This solves the problem {proposed to Enler by
Ph, Naude) to find the number of ways a number is a sum of a given number
of distinct parts.
By the above relations between a, fi, ■ - -, A, B, • •>, we get
To give a proof of these results found by induction, write nz for г in R.
We get A +п2г)A +п3г)-- = 1 + ост + p«V + ■-•. Ite product by
1 + яг gives R = 1 + ca + - - -. Hence we get the preceding values of
«j P, 7> " • - bet i»<w be the number of ways m is a sum of д distinct integral
parts, where the affix г (signifying inaequalus) is omitted if the parts need
not be distinct. This mpy is the coefficient of n" in
the sum of the /ith series a, 0, 7, ■ ••. Keplacing the numerator by
n»<9-i>/i^ we gg^ ^e series whose general term is m^n1, or, if we prefer,
(m + ^ij^n1". Subtract the former fraction from the latter; we get
A - n).-(l - n*-1)'
the general term of the series for which is т^^пт. Hence, transposing^
A) (m + p)^ - m<» + mi^1\
which serves ад a recursion formula. Since in the series for 1/{A ~ л)
• ■ • A — nM) \ the coefficient of n* is the number of ways s is a sum of parts
1, • • •, /Jt when the numbei of parts is not prescribed and the parts may be
equal, ml/° also gives the number of ways m — p{p + l)/2 can be obtained
by addition from 1, • - •, /x.
The second problem proposed by Naude was to find the number
of ways m is a sum of ft equal or distinct parts. To treat it, set
<r--xi-^)-"l=1 + lb + w + --
Writing nz in place of г, we get
(I ~ 7й)A + Hz + 93z* + • -) e 1 +
,r _ _^_ л = _*» *
Д 1 -n1 1 ^n* A -n)(l^n2

Chap. ПЦ PARTITIONS, 103
Hence a = Я, 0 = «93, у = n3G, S = n*$), - - -, where 1, 3, 6, - - - are the
successive triangular numbers. From the above series for «, ft - • -, we
see that
«o>
Hence «<"> is also the number of ways m — ц can be obtained by addition
from 1, " -, (i. The former recursion formula for m№ gives
B) *»<*> = (tn - /»)W + (m -
He stated, as a fact he could not prove,
C) p(*)=j|(i-**)-i-*
and that the reciprocal of the product isl+:r + 2r2 + 3-r* + 5zi + • • •,
the coefficient of x* being the number of ways s can be partitioned into
equal or distinct parts. As to C), see Euler'-*, Ch. X, Vol. L
Euler,4 in a letter to N, Bernoulli, Nov. 10, 1742, stated the preceding
facts on partitions. The answer to the second problem he stated in the
following equivalent form: mF> is the coefficient of n« in the expansion of
( ( (
) )
Euler6 gave C) and p(x) = 1 - Л + Л - P, + . • • [see Euler8].
F- K. Boscovich6 gave a method of findmg all the partitions of a given
number n into integral parts > 0, Write down n units in a liner Replace
the last two units by 2, then replace two units by 2, etc. Nextr write
n - 3 units and 3; replace two units by 2, etc» Then write n - 6 units
and two 3Ts; replace two units by 2, etc. Thus the partitions of 5 are
11111, 1112, 122, 113, 23, 14, 5.
He applied partitions to find any power of a series in xy also in a paper,
ibid., 1748» In his third paper, ibid., 1748, he showed how to list the parti-
partitions of n into parts ^ m, by stopping his above process just before a part
m+1 would be introduced» He applied the rule also to the case when the
parts are any assigned numbers. He treated the problem to find all the
ways in which a given integer n can be decomposed in an assigned number
m of parts, equal or distinct; but the solution by Hindenburg1* is much
more simple and direct. Boscovieh attempted in vain to find a formula
for the number of partitions. He gave elsewhere7 his rule»
K, F, Hindenburg8 would obtain the partitions of 8 by annexing unity
to those of 7, and supplement them with
2222, 224, 233, 26, 35, 44, 8.
* Opera poeluma, lr 1862; Corresp. Math. Phys, («d., Fusa)r 2, 1843, 691-700,
* Letter to d'Akmbcrt, Dec. 30, 1747; Bull, Bibl. Storia Sc. M*t.f 19r 1886, 143.
1 Gioraale der LeUerati, Romer 1747. Extract by Trudi," pp. 8-10.
7 Archiv der Math. (ed.r Hindenburg), 4, l747r 402.
*Ibid. 302r and Erete Samml. CombiQatoriach-Analyt. Abhand,, 1796, Ш- Quoted from
G, S, KlugeTsMath. Worterbuch, 1, l8O3p 456-60 E0&-11, for references).

History of the Thkoby or Numbers. [Снл* щ
L. Euler* noted that the coefficient of £V" in the expansion of
is the number of different ways n is a sum of m different terms of the eet
«i ft y> * • •• Tne coefficient of х'Ч* in the series giving the expansion of
is the number of different ways n is a sum of m terms of or, ft ■ ■ *, repetitions
allowed. In particular, the coefficient of z* in
ft A - &Yl = l+s + 2z2 + 3x8 + 5*< + 7*i4
is the number of partitions of n. If the product extends only to j = m» the
coefficient of x* is the number of partitions of n into parts ^ ?n. In
з = П a +«%) -
replace z by жг, so that Z becomes Z/(l + xz). Hence
A + xz){l + Рда +
By comparison of coefficients, we get
Hence the number of partitions of n into parts ^ m equals the number of
wave of expressing n + wi(m + l)/2 as a sum of m distinct parts. Applying
the same process to ПA — х'г)^1, we obtain the series
" A х){1 ж")A ж3)
-x1)
Hence the number of partitions of n into parts Ш m equals the number of
ways of expressing n + masa sum of m parts, not necessarily distinct.
If (n, m) is the number of partitions of n into parts ^ m, then
(n, m) = (щ m —* 1) + (n — m, m).
By иве of this recursion formula, Euler computed a table of the values of
(n, m) for n Ш 69, m ^ 11. The product of
is ПA — x*>), all of whose factors occur in P. Hence Q>roof by L.
Kronecker10 for [ x | < 1, to insure absolute convergency],
(i) a-?® i
• Introductio in Analyain mfimtflmm, LausaiHie, ], 1748, Cap. l6r 253-275. German traneL
by J. A. C. Mkbeben, BeriinT 1788-90. French tranal. by J. B, L*bey, Рапя, 1, 1835,
234-256.
ZAblentbeorie, 1, 1901,50-50.

, нц Partitions. 105
9o that the number of partitions of n into distinct integers equals the number
of partitions of n into odd parts not necessarily distinct»
Replace x by z3 in C) . Since ДA - z") = PQ,
Q = (l~*~* + x» + *« )I,
i = l-b*-b2*2-Mza + 5z*+'--.
Hence, by multiplication,
Thus the coefficient of x* in this series gives the number of partitions of 8
into distinct parts. Since
every integer can be obtained by adding different terms of the progression
1, 2, 4, 8, 16, • • • or of zb 1, ± 3, ± 32, * -. The latter facts were known
by Leonardo Pisano,11 Michael Sttfel,11" and Frans van Schooten,52 who gave
a table expressing each number ^ 127 in terms of 1, 2, 4, * • -, and every
number ^ 121 in terms of ±1, ±3, ±9, ■-,
Euler18 reproduced essentially his preceding treatment. He concluded
(§ 41, p. 91) that, if P(n) or n{t9) denotes the number of all partitions of n,
P(n) = P(n - 1) + P(n - 2) - P{n - 5) - P(n - 7) + P(n - 12) + ...,
the numbers subtracted from n being the exponents in C). His table of
the number п(ш> of partitions of n into parts Ш m here extends to n ^ 59,
m ш 20 and includes тп — w. He proved again that every integer equals
a sum of different terms of 1, 2, 4, 8,
Euler14 noted that the number (iV, n, m) of partitions of N into л parts
each ^ ?n is the coefficient of Xs in the expansion of (x + х* + • - + хщ)п.
Set
E) A +*+ --. +*«^)» « 1+A& + B&+ ••-,
bring to a common denominator the derivatives of the logarithms of each
member and equate the coefficients of like powers of x in the expansions of
the numerators. The resulting linear relations determine An, Bn, - ■ • in
turn, whence
X(n + X, n, я») = (fi + X - l)(n + X - 1, n, m)
— (inn + m — X)(n + X — m, n, w)
-n-J-m+l- X)(n + X — m — 1, n, m).
uScxitti L. Pisano, I, liber abbaci, 1202 (revised about 1228), Rome, 1857, 297.
m Die Сода Chrietoffe Rudolffe . . . durch Michael Stifel gcbeesert . . ., 1553.
" Exercitatiouum Math., 1657, 410^0.
14 Novi Сохши» Acad. Petrop,, 3f ad unaum 1750 et 1751, 1753, 125 (summary, pp. 15-18);
Comm. Arith. CoIL, I, 73-101.
u Ntm Comm» Acad. Pctrop., 14,1,17Ю, 168; Comm. Aritb. Coll., If 391-WO.

106 History or the Theory of Numbers. [Chap, hi
Again, by comparing E) with the corresponding relation with n replaced
by n + 1, it is found that
(N + 1, n + 1, m) = {N> n + 1, m) + (N, щ m) - {N - m, n, m).
Finally, by expanding A — хш)п and A — x)~n by the binomial theorem,
A, «, m) - (^ x ){x){ ь„т
+ \-2m-l\ _ fn\fn + \ -3m - l
\-2m ) V3/V \-3m
Euler's proofs were made for m = 0 and, except for the third formula,
involve incomplete inductions. By evaluating the coefficient of x" in the
expansion of
Euler found the number of partitions of N Ш 26 into three parts, the
first part Ш 6j the second ^ 8, the third ^ 12.
As to the problem known as the rule of the Virgins [cf. Sylvester,*4 and
note 188 of Ch. НЗт the number of sets of integral solutions p, q, • * -9
each ^ 0, of the pah* of equations
ap + bq + ■ ■ * = n, otp + Pq + ■ - = v,
is the coefficient [^not determined] of х"у* in the expansion of
A -a-yr'a-aV)-1-'--
K. F» Hindenburg16 gave a method, different from Boscovich's, for
listing all partitions of n. For n = 5, the method lists them in the order
5, 14, 23, 113, 122, 1112, 11111.
Hiudeaburg16 gave a method of listing all partitions of n into w parts.
The initial partition contains то — 1 units and the element n — m + 1.
To obtain a new partition from a given one, pays over the elements of the
latter from right to left, stopping at the first element / which is less, by
at least two units, than the final element £f — 2 in 1234]. Without
altering any element at the left of /, write / + 1 in place of / and every
element to the right of / with the exception of the final element, in whose
place is written the number which when added to ail the other new elements
gives the sum n. The process to obtain partitions stops when we reach
one in which no part is less than the final part by at least two units.
Case n = 10, m = 4:
1117 12 3 4
112 6 13 3 3
1135 2224
114 4 2 2 3 3
1225
•* Methodue nova et f&cilie eerierum infimtarum exhibetidi digmtaLee, LeipeaeT 177$.
tinomii dignitatum historiat leges, ac formulae, GotLing&e, 177&, 73-91 A66> tables of
partitions). A less interesting method is given in a Progr., 1795*
"Exposition by С, Кгатр, Е16шепя d'Arith. UniverBelle, 1S08,1339. Quoted by Trudi^

Сна?. Ш] Partitions. 107
p. Paoliir noted (p. 38) that n can be separated into m positive integral
parts in (£l!) ways, if different permutations are counted separately. The
number (p. 42) of partitions (different permutations not counted) of n
into m parts > 0 is фA) -f ф(т -f 1) -f фBт + 1) + - ■ •, where фЦ) is
the number of partitions of n — j into m — 1 parts» The number (p. 53)
of ways n can be divided into m distinct parts is \{m) + \Bm) -\- \Cm) +
• ■ *j tf МЛ is the number of ways n — j can be divided into m — 1 distinct
parts» There are (p. 63) as many divisions of n into m distinct parts as of
n — m(m — l)/2 into то parts equal or distinct» Let ф, ф} w be the number
of ways 2n -h 1, 2n, 2n -\- 1 can be divided into 2m — 1, 2m, 2m + 1 odd
parts, respectively; let ^[rj, 4>[r\ "Ы be, the corresponding numbers
when n is replaced by n — r\ Then
to = ф + ф[2т + 1] + Ф[±т + 2] + ^[Gm + 3] +
If we impose also the condition that the odd parts be distinct, we have
ф - fBm) + *Dw) H , w = 0Bw) + ф(±т + 1) + 0Fm + 2) +
The number (p. 76) of ways 2n is a sum of m even parts is фA) + ф(т -\- 1)
+ фBт -\- I) + * - -, if ф(г) ш the number of ways 2(n — r) is a sum of
m — 1 even parts. The number (p. 79) of ways n is a sum of m parts is
the number of ways 2n is a sum of m even parts» The number (p. 80} of
ways n is a sum of m distinct parts is the number of ways pn is a sum of m
distinct parts multiples of p. The number P(nt m) of partitions of n into
parts ^ m is 2P(n — j, ?w — 1), summed for j — 0, m, 2mf • - ■. The
number (p» 85) of partitions oi n into m parts equals P(n — m, m)> If
^j w denote the number of ways (m —* l)a -\- rb and ma -\- rb can be formed
additively from m and m — 1 terms of the progression a, a -f- fr, a + 26,
■ • *, then a> * 0 + </>(wi) + 0Bw) + ■ ■ •> where 0(,?) is derived from ф by
replacing r by r - j. Similarly (p* 92) when only distinct terms of the
progression are used. If (p. 98) Ф is the number of ways n is a sum of
numbers chosen from a, a -\- b> • - -, a + (m — l)b, and u that for a, • • -,
a + mby then
Finally (p» 103) the number of ways n is a sum of terms of any given series
is discussed» He gave a more extended treatment in his next paper»
Paoli18 treated linear difference equations with variable coefficients:
Z(Vix) = AJ(y - l,x) + BJ(y - 2,x) + ---±XJ(y-x,x)+ ■-.
+ AlZti,z- 1) +№& -l,z - 1) + >.- +XlZte - z,x -1) + ••-,
where Az, * -» are given functions of x, and у is a function of x. Let the
integral be
where ?w, a, n, bt ■ • • are constants» The condition that а^лх sheU be an
11 Opuscula aoaJytica, Libumi» 1780, Opusc. 11 (MeditAtioncs AritbJ, § 1.
18 Memorie di mat. e fie. soctetu ItyJiana, 2, 1784, 787-845.

108 History of the Theory of Numbers. ССнлр.щ
integral is
A* + B'jx-i + — + Х'лг* -b ■ ■ ■
* 1 - A^1 - Bjr1 - > -
Hence we get v«*; let its expansion be
Vet* = A + A'a + ilV + > ■>.
Differentiate its logarithm, regarding x as constant and a variable* Thus
A' + 2A"<rl + - ■
where r<m) is the excess of the sum of the {m + l)th powers of the roots of
the denominator over the sum of the (m + l)*h powers of the roots of the
numerator in the fractional function of tf^1 giving Va^ Hence
A' = Ar, 2A" = A'r + Ат>, Ы'" = А"т + A'r' + Ar",
which give A'/A, A"jAf — • as functions of г, т*, • •». Hence, evidently,
Z(y, x) - Аф(у) +А'Ф(у - 1) + А"ф(у - 2) + • •.,
Consider {pp^ 817-21) the number (y, x) of ways у is a sum of z equal
or distinct positive integers» Those in which 1 is a part furnish the (y — 1,
x — 1) ways у — 1 is a sum of ж — 1 parts; while those in which each
part exceeds 1 give, upon subtracting 1 from every part, the (y — x, x)
ways у — x is a sum of x parts. Hence
(V> z) = (y-z,z) + (y~l,x- 1).
It has the integral (у, х) = а^а* И а* = аг*а* + ar\ whence
The sum of the mth powers of the roots of a — 1, a1 ** 1, • ■ >, a* = 1 is
the sum B(m) of those of the numbers m, m/2, m/3, • ■ >, m/m which are
integral and = x. Hence
r<-> = 5(m + l), A' = 5{1), ^f'
- ф(у -х)+ А'ф(у - х - 1) + А"ф(у - х - 2) + - > -.
г« 1. Then А' = А"= ■■ = 1ЛУ,1) = *(»-1) +ф(у-2) +
Replace у Ъу у - 1. Thus (у - 1, 1) = ф(у - 2) + ф(у - 3) +
Hence (у, 1) — (з/ — 1, 1) = Ф($ — 1). By the nature of our

Chap. ПЦ PARTITIONS. 109
problem, (у, 1) = 1 or 0 according аз у > 0, у ^ 0. Hence ф(г) = 1 or 0
according as г = 0 or я Ф 0. Hence (y, x) reduces to the single term
Ai¥~x\ so that
Next (pp. 821HI), to find the number (yr x) of ways у is a sum of x
distinct positive integers, we have (y, x) = (y -* x, x) + (y — xf x — 1).
Now <xx = ar*f(l — a-*)* The values of г<щ>, 6(т), A™ are the same as in
the preceding problem* But
Va,**<r* + A'ar'-l+ ••-, (ft*) = ф{у-
^ *(*+!)
Again, Ф(у) = 1 if у = 0, ^(y) - 0 if j/ Ф 0. Hence (y, a;) is derived from
Aim) by replacing w by у — x(x + l)/2. Hence j/ is a sum of x distinct
parts as often as у — x(x — l)/2 is a sura of x equal or distinct parts.
For (pp. 824-7) the number (y, x) of ways у is a sum of x equal or dis-
distinct positive odd numbers, it is stated that {$, x) = (y — 2x, x) + (# — 1,
x - 1). Here a* - o-V(l - <r*)f (y, x) = *{y ~ *) + A'*(jr - * - 2) +
■ -, and ф(у) = 0 unless у - 0, ФA) = 1. Thus {^, x) is obtained from
ji<"° by taking m = (y — x){2, where у and z are necessarily both even or
both odd. If у is partitioned into distinct odd numbers, (y, x) = (y — 2x,
s) + (y *- 2x + 1, 3 — 1), and (у, з) is obtained from Aim} by taking
m = (y — л^)/2. Hence у is a sum of x distinct odd numbers as often as
у — x(x — 1) is a sum of x equal or distinct odd numbers.
The number (y, x) of ways у is a sum of terms chosen from *i, ■ • ■, z% is
the number of sets of solutions p, ■ > >, / of у — рг% + »- * + lz*. Taking
t — 0r lr * * * in turn, we get
(У, s) =» (yf x - I) + (y - г^, x - 1) -h (y - 2zx, x - 1)
+ ft(-fela!- 1)+ ....
Beplace yhyy — zx and subtract- Thus (y, x) = (y — zx, x) + (y, x — 1).
Here <** = 1/A — e^1). Write b{m) for Ше sum of those terms m, m/2t
- • •, m/?re which are integers ^ z^ and are z*&. The formula for Л("*> is
the same as in the first problem. Since A(9l} is the firfet A which is not
zero, we have
(У, 1) ~ Ф(у) + Ф{У - zi) + ф(у - 2*i) + • • •, (ft 1) - (V - *h 1) * *fe)-
Thus *@) = 1, *(y) - 0, v ф 0. Hence (ft x) is given by A^. Ь par-
particular, if ^ = x, we have the number of ways у is n sum of numbers ^ x.
Hence by the first problem, у is a sum of x integers as often as у — z is a
sum of integers ^ x. For *, = nBx — 1), (jf> x) is given by Лс*"° if to
form d(m) we retain only the terms which are integral, odd and g 2x — 1.
For the number (у, я) of ways у is a sum of distinct terms choeen from
*ь 4" *f z*, (V> *) e (ft * - 1) + (У ^ «*, * — !)■ bet y(m) be the sum

110 History of the Theory of Numbers. cchap. ш
of those numbers mt — m/2f m/3y — то/4, • ■>, ± mjm which are integers
Ш z* and are z's. Then АЫ) ia derived from the А<*> of the first problem
by replacing S's by t's, while (у, г) is given by A(»K Let zx = 2^J and
let x increase indefinitely, i. e., use the infinite series 1, 2, 4, 8, ■ ■ •. Then
y(m) ~ 2™ — 2я1-1 — - ■ • — 1 ^ 1, (y, x) - 1, so that every integer is a
sum of terms 1, 2, 4, 8, - ■ ■ in a single way [L. Pfeatio11].
For the number (y, x) of ways у is a sum of x terms of m> m + я, ™ + 2nf
* - - or x distinct terms, (y, x) = {^ — nx, ж) -J- (г, x — 1), where 2 = у — m
or у — nx -\- n — mf respectively. Then (y, x) is given by Л(*° for
л/* — у — mx or у — mx — nx(x — l)/2, respectively. Hence у is a sum
of x distinct terms of the progression as often as у — nx(z — l)/2 is a sum
of x equal or distinct terms»
Finally (pp. 842-5), to reduce the integration of
to that of (]/, x) = Ar(y - ф(х), x) + B,(|^ —/Or), x — 1), substitute
[#**Q = Cf^ ~ ^WIM, 3) into the former and compare the result with the
latter. The condition for agreement is F(x) - F{z — 1) = ф(х) - f
whence, for a constant с independent of x,
F(x) =2{*B + 1} - infix+1)) +c.
Thus, in our second problem, [yt x] = [$ — z, x] + [# — x, x —
while in the first problem concerning zi, • • ■, zx = x,
(ff, x) = {y -x,x) + (y,« - 1).
Hence F(x) = 2(x + 1 - 0) = x(x + l)/2 -f с and с - 0. Hence
2, x),
so that ^ is a sum of x distinct parts as often as у — x(x + l)/2 is a sum of
parts ^ x» Again, for the equation in our first problem, and
&, x] = \jt - 2x, x] + & - 1, x - 1]
of our third problem, we have F(x) = — x, [t/, x] = (jy + ж!/2, л}, so
that у is a sum of x odd parts as often as (y + x)/2 is a sum of x even or odd
parts. Finally, for our first and last problems, F(x) = (m — n)x, so that
у is a sum of x terms of mt m + n, »& + 2n, * • - as often as \y — (m — n)x]/n
is a sum of г positive integers»
G. F. Malfatti19 obtained the general term of a recurring series whose
scale of relation has a multiple root» In the Appendix, he treated the
number of partitions into x distinct terms of the series 1, 2, ■ ■ •, extended
either to infinity (as by Paoli) or to a given number p. Taking first the
former case, he showed how to pass from any of the series
s=l: 111111111--*
x = 2: 112233445 •».
x -3: 11 2345 78 10»»»
" Memorie di mat. e fie. eocieta Italians, 3, 1786, 571-663.

Сват. Ш] PARTITIONS. Ш
to the next. Here the entries for x = 2 give the number of partitions of
3 = 1 + 2, 4, 5, ■ *' into 2 distinct parts; and arc the sums of the units
in the respective columns in the accompanying scheme of units arranged by
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
I
1
I 1
twos. Apply the same process to these numbers for x = 2, taking them
by threes:
112 233 445 ■>■
112 2 3 3 ■-.
112 ...
Summing the columns, we obtain the number of partitions of 6 = 1 -J- 2 -h 3,
7, 8, • ■ • into г ~ 3 distinct parts. Taking these by fours, we get similarly
the series for x — 4. This property shows that
if {tt x) is the Jth term of the series for x.
To pass to the number of partitions into x distinct terms of lf • • -t pt
we must delete the partition (p -\- 1) + 1 oi p + 2, and (p + 1) -f- 2,
(p -\- 2) + 1 of P + 3f etc. Thus the number of terms hi the " first series
of subtraction
"is
X
X
x
= 1:
= 2:
= 3:
1
1
1
1
2
2
1
3
4
1
4
6
1
5
9
1
6
12
any line of which is formed from the preceding line as in the former problem*
Thus (t, x + 1) - (t — x, x H- 1) + (/, x). But for x = 2 we counted the
partition of 2p + 2 into parts each p + 1. Hence we must correct our
subtractive series b}' employing the " first additive series ":
x = 2: 112233*-; z-3: 12 4 6 9 12 -♦;
leading to (/, x + 2) - (t - x, x + 2) + (/, x 4- 1). Then we have a
second subtractive series, etc. The general one of these difference equa-
equations is
(t, x + X) = (t - xyx + X) + (tt x + X - 1).
It has the integral a'll, where IT = n}zTav+A, if a>+A = 1/A ~ я"'")- Thus
П = 1/ДД = A - <^:)A - o-*)---(l -«"*). If
1/7? - 1 -h AV + A"a-2 + - • -,
we find (as by Paoli) that A' = r} 2A" - A'r -f- r\ - ■ *, where rCFfl) is the
sum of the (m -\- l)th powers of the roots of D ~ 0. The general integral
is {*, * + X) = 0@ + AV(e - 1) -h А"ф(Ь - 2) + - ■ ♦. For X = 0, we
find by using x = 1 (cf. Paoli) that (I, z) = A(tw>. For general X, write

112 Histoby of тня Theory of Numbers. [Chap, III
n^ in place of A^. Taking i = 1 in the general integral, we вее that
(«> 1 + X) - (< - 1, 1 + X) = ф(г), which is shown to be Л<*> of PaoU's
caee. Hence фA) = nSJ*, and
(<, ж + X) = т£-1) + ЛЧ} + А"п(;-3) + - ■
He gave the following results for z ^ 4:
ft ъ w + » +17 (-Ц* ■ «^+«^
2C + 24f* H- 8U
288 32 27
27 + 16
where a and orj are the imaginary cube roots of unity, and fi = if fix = — i.
He also gave the general term of the first subtraction series:
' - i - p + 2;
с,
с,
2)
3)
4)
= *' =
+
-р + ]
+ 4/' +
144
1 -
Ш'
(<
- {- 1)'
+ 25
(- IV
16
Earlier щ the paper (pp» 618-26), be gave (t, 5) and the general terms of
the addition and subtraction series; these and various other results given
above occur hi his earlier two articles in Prodromo dell' Encyclopedia
Italiana and (hi more detail) in Antologia Romana, 11, 1784.
V. Brunacci20 reproduced Paoli's1* treatment of his first problem.
S* Vince*1 proved by induction that every positive integer is a sum of
distinct terms of 1,2,4,8, •■ -. For, if true for numbsrs up to « = 1 + 2 +
■ ■ ■ -h 2""*1 = 2* — 1, it will be true for the remaining numbers up to
s + 2n. The proof for ±1, ± 3, ± 31, - - is longer.
S. F. Lacroix* reproduced part of the discussion by Euler.*
Fregjer1* proved that or equals a sum of a terms of the arithmetical
progeseion whose first term is unity and common difference is
Cf. Volpicefli,17 Lemoine,7* Mansion,87 and Candido.511
« ComiS Mmtonaticft 8иЫЬе, FireMe, 1,1801; 55 108-9t pp. 237-248. <5f.
ddOdkxSuR, 18И, |П4.
TcwKj-RaylriahAttd., 12,1815,34-^8. Eoler."
ТЫК Ал CUcal Dxff. Int., 3,1819, 461-6.
" tehd Ge)^, 1818-9, ^

Chap. IHJ PaBTTTIONS. 113
C. G. J- Jacobi*2* gave fundamental applications of elliptic function
formulas to the theory of partitions. He proved tbe identical relation
x (i + floa + flHr^ja +
if | g [ < 1, and another deducible from it by writing gt^foryand multiplying
by qv% viz.,
*Тот this he inferred, through the intermediary of the four the*? r»
the following relations of great importance in tbe theory of partitions:
? - E 9"" = П (i -
2 (- l)"g- = Й A -
For his expansions, as series щ q, of powers of these functions see Chs. VI-
tX on sums of squares. If in the first identical relation above we write + г
and — z in turn for v and multiply the results together, we obtain
A. M. Legendre13 noted that Euler's formula C) implies that every
number, not a pentagonal number Cn* ± n)/2, can be partitioned into an
even number of distinct integers as often as into an odd number; while
Cn2 ± n){2 can he partitioned into an even number of parts one* oftener
or once fewer times than into an odd number, ^:.:,X^ ^ ., _. ^^i» urodd.
'Phis result was implied by Euler1* (§ 46).
С J. Brianchon" noted that the literal part of the general term in the
expansion of {ax + a± -h " • + a^I1* is of tbe type aV • ■ • o?, where
tti + ■ ■ * + «a = »*, ^ ^ w, ж ^ n. Thus the terms form as many classes
as there are values of x, and the terms of a class form as many groups aa
there are partitions of m into x numbers «* In view of EulerV table we
Imow the number of groups of each class.
«•Fundament* Nova Theoriae Func. Effip», 1829, 182-4. Werke, I, 234-6» Of. Jarabi.»
See the excellent report by H. J» S. Smith, Report Britieh Aaaoc. for 1865, 322-75;
Coll. Math» Papers, I, 289^94, 316-7»
* Tbeorie dep пошЬда, «L 3,1830, II, 128^-133.
11 Jout de lJfcole polyt., tome 15, cah, 25,1837, Ш.

114 History of the Theory of Numbers. [Chap, щ
E. Catalan46 proved that a?i + * ■ • + sn = я* bas (*+£^) sets of solu-
solutions Ш 0.
0. Rodrigues56 noted that the number Zn, i of ways of permuting n
letters, such that there are i inversions in each permutation, is the number
of sets of solutions-of x$ + X\ + • - ■ + £«_i — it where Xk takes only the
values 0, 1T - • -, к and where the value of xk for each permutation is the
number of inversions produced by Xk+i- Thus Zn> < ig the coefficient of
t* in the expansion of
A 4- 0A + t 4- P) • • ■ A + t + ■ - + I*-1) = A - t)-nP,
where P = A - 0A -*)... (l - r). Let En, i be the coefficient of &
in the expansion of P. Thus i?np, = En-i, x — E^t ^n, En. i = Eit h and
Here fin, i equals the excess of the number of partitions of г in an even
number of distinct integers < n -f 1 over the number in an odd, the number
of parts being also < n + 1.
M. A. Stern27 wrote nCq (or nCq) for the number of combinations without
(or with) repetitions with the sum n and class q (i. e-, q at a time), meaning
the number of partitions of n into q distinct parts (or equal or distinct
parts), Evidently nC2 *= {jij2~\. Hence, by B), we get
с — v
Т.
Since nCe =* «C^ if m - n — g(^ — l)/2, we get by A),
If C(n) is the number of all partitions of n into distinct parts,
Z (- l)«C(n - v/2) = (- l)r or 0 (у = Зг! Т г),
according as n is or is not of ths form Зг3 ^ r. This follows by expanding
1 - ^ 1 -x* 1 - x6
1 — ar ' 1 - s* * 1 - xz ' "'
» Jour, de Math., 3,1858, 111-2. ^
* Jour- de Math., 4, 1839, 236-240,
*' Jour, fiir Math., 21, 1840, 91-97, 177-9. Farther reeulte were quoted under Stern1»* of
Ch* X Ь VoL I of thiH History-

Chap. Ill] PARTITIONS. 115
Also,
S <- iyC(n - y) = 1 or 0 {у ш № =F £),
according as д is or is not of the form z(z -f l)/2,
A» De Morgan28 considered the number u^v of ways x can be formed
additively from у and numbers ^ y. Adding у to each such composition
of x — yf we see that
Subtracting from this the equation obtained by decreasing x and у by unity,
we get
Regard у as fixed and the second и as a given function, we have a difference
equation of order у whose general integral is of the form
where A^ is a rational integral function whose degree on is the greatest
integer in (n — y)fyf while Pn ia a circuiting function with a cycle of n
values. In particular,
Alto1 -7-9(^ 1)- + 8C*+ 7*)I,
+ 320?*-1 + т^1 - V - т*) + «f + И(- О'»,
where jS, 7 are the imaginary cube roots of unity and i = V— 1, Thus
*,, = x\ x* - 1, ж* - 4, x2 4- 3, s* - 4, x2 - 1,
according as x = 0, 1, 2, 3, 4, 5 (mod 6)» Similarly, г**, 4 has 12 forms
depending on the residues of x modulo 12, Again, ux^ is the integer
nearest x2/12, and uXt 4 ttat nearest to {x* -h 3zJ)/144or (x2 -f Зх2 - 9x)/144,
according as x is even or odd,
A. Cauchy" proved C) and the other formulae of Euler3 and the related
ones involving a finits number of factors:
■•■■■■'
P(- x) ~ + 1 - I * + A - <)A - (•)
_,
- 0A -
" Cambridge Math. Jour., 4, 1843,87-90.
"Comptea Rendua Para, 17, 1843, 523; Oeuvree, (I), VIII, 42-50.

116 History of the Theory of Numbers, [Снар.вд
С. G. J, Jacobi80 stated that if we replace q by qn and set t = =F qr in
his first formula,226 we get
Jj A
- J (±
For m = 1/2, «, *= 3/2, that with the lower signs becomes Eider's C).
Although he** (pp. 185-6) gave two simple proofs of it, Jaoobi here repro-
reproduced Euler*» proof in essential points, but with a generalization. He
gave a proof of Legendre's" corollaiy and proved the following generaliza-
generalization. Let (P, a, ft " ) be the excess of the number of partitions of P
into an even number of the given distinct elements a, 0, •", each =H 0,
over the number of partitions into an odd number of them* Then
Let a, aJf — , a,,^! form any arithmetical progression, and Ы, Ьь
an arithmetical progression with the common difference — a. Set
Then
I/ ^ (bo, a) -h (&i, a, «i) + Fj, e, Cif at)
= д — (Ьж, alf ♦ *
If bo and a are positive and ma > bo, L vanishes except when bt equals
-f- 2&i or 2ty^i + e^, and then equals (- 1)', where
Jaeobi11 noted that Euler# expressed P з A + g)(l + fl*)(I +rt^-
in the form fD*)/f(q)t where /(z) is given by C). Jacob! expressed P in
six ways as quotients of two infinite products and expanded each into
infinite series; the next to the last case is
ExpreseiBg this in the form Xf^Cflt, we conclude that, if d is the number
of partitions of i into arbitrary distinct integers or into equal or distinct
odd integers,
&
where 5 = 1 or 0 according as г is or is not of the form CnJ zk n)j2. He gave
«Jour, fur Math., 32, 1846, 164-175; Werbe, 6, 1891, Э^йУ^Р^иЪ Matt, 1,
345^356. Cf.^yivCTter117, Goldechmidt.*"
« Jour, for Math., 37,1848, 67-73, 233: Werke, 2,1882, 226-233, 267; Opuecula Math., 2,
1861, 73-80, ПЗ.

Chap- III] PARTITIONS. 117
expansions of P* and P*. Only those m-gonsl numbers give the remainder
ly when divided by m — <?bt whose side has the remainder 1 when divided
by aby where a1 is the greatest square dividing m.
H. Warburton*2 considered the number Q7V, p, i?] of partitions of N
into p parts each ^ y, and proved that
IN, p, rtf - ЦЛГ, Pj , + 1J - рт - % P ~ Ii *1
ЦЛГ + p, p, 1] - £ IN, z, Ц СЛГ, *, ,] = £ [W - pn, *, Ц
№ p, 1] = [ЛГ - 1, p - I, 1] + [ЛГ - p ~~ 1, p - 1, 1]
+ [ЛГ^2р- l,p- 1,1] + ...,
to pv7pD terms» He applied these formulas to the construction of a table
of partitions and proved that the number of partitions of x into three parts
is З*2, 3*2 =fc t, 3F ± % 3J2 + Zt + 1 according as ж = 6t, 6i ± 1, 6i ± 2,
6^ + 3 Сш accord with De Morgan]»
J, F. W. HerscheP3 recalled his34 earUer notation sx — s!^, where a
ranges over the 5th roots of unity, so that s» = 1 or 0 according as x is or
is not divisible by s. Then AxSx + Bx8t-i + * • * H- Nj^r-^-i-i will circulate
in its successive values as x increases by units from zero, being Ax when x
is divisible by s, but Bx when x — 1 is divisible by s, etc. If AXJ ete., are
constants, the function is called periodic. He wrote 9IL(x) for the number
(Xj $) of partitions of x into s parts > 0, Starting with
(x, s ^ 1) = ф(х) + &,
where 0(x) is the non-periodic part and Q, the periodic or circulating func-
function, and applying the final formula quoted from Warburton, he obtained
(zfs) = A + Z, Л = ф(яг - 1) + ф(х - s - 1) + -.,
each extending to [xfsj terms» Then A is expressed explicitly in terms of
the numbers Д^Ю", giving the mth order of difference of zn for z = 0, while Z
is expressed in terms of these numbers and the above circulating functions sx*
He deduced explicit expressions for (z, s), s = 2, 3, 4, 5, as (z, 2) — %(x^2^}
(x, 3) - A{*« - 6M - 46^ +3-6^ - 4-6x^ - 6^j,
which, with the expression for (x, 4), are in accord with the results by
De Morgan/* although the lattsr was not treating partitions into s parts.
While the method of Herschel is laborious, it anticipated to some extent
the simpler method of Cayley."
J» J, Sylvester35 quoted Euler's theorem that the number of partitions
of n is the same whether the number of parts is ^ m or every part is ^ m,
and noted that, if we apply the theorem also when the limiting number is
" Trans. Cambridge РЫ1- Sc*^8,1S49, 471-492. ~
u Phil. Тпшв. Roy. Soc. London, 140, II, 1850, 399-422.
" From hU paper on circulating functions, ibid., 108, 1818» 144-168.
e PhiL Mag., D), 5, 1*53, 199-202; Coll. Math. Papers, I, S9S-8.

118 History of the Theory of Numbers.
m — 1, we obtain by subtraction the following corollary. The number of
partitions of n into tn parts equals the number of partitions of n into parts
one of which is m and the others are ^ m* Sylvester credited the corollary
to N. M. Ferrers who communicated to him the following proof. Take
any set A composed of 3, 3, 2, 1, written as
1, 1, *
1, 1, 1
1, 1
1.
Reading it by columns, we get the set В composed of 4, 3, 2. Similarly,
every A in which the number of parts is 4 gives rise bo а, В in which 4 is a
part and every part is ^ 4; conversely, every В produces an A* Euler'e
theorem can be proved by the same diagram» Similarly, the number of
partitions of n into m or more parts equals the number of partitions of n
into parts the greatest of which is ^ m. If we partition each of г numbers
into parts so that the sum of the greatest parts shall not exceed (or be less
than) m, the number of ways this can be done is the same as the number of
ways these г numbers can be simultaneously partitioned so that the total
number of parts shall never exceed (or never be less than) m*
P. Volpicelli* arranged the natural numbers n, n -f 1, • • • in a rectangle
with к -f 1 rows, each with h -f 1 numbers, but in reverse order in alternate
rows. For example,
IS 19 20
23 22 21
24 25 26.
The successive sums by columns are 65, 66, 67 (of common difference unity)
and so always when the number of columns is odd; but, if к -f 1 is even,
the sum of the numbers in each column is constant, being
а = \2n + h(k + 1) + k\(k
and we have special partitions of a. Given a, to find integral solutions
n, h, fc, we note that h = 7/5, where Ь — (к -h II, while 7 and BaJ/5 are
integers. Hence seek those divisors of Ba)* which are squares 5; for each
such 5, we have к and seek integers n for which 7/5 is an integer /l
Volpicelli*7 expressed nk as a sum of numbers in arithmetical progression.
* P. Bonialli58 treated partitions.
T. P. Kirkman" proved that the number of partitions of N into p parts
= a equals the sum of the number of partitions of N — a, N — p — a,
N — 2p — a, — * into p — 1 parts ^ 0. The case a ^_ 1 is the last formula
of Warburton.M He gave an analytic expression for the number (x, к) of
"AttiAccAd. Pont. Nuovi Iinoei, 6,1852-3 A855), 631; 10, l85*-7r 43-51,122-Ш; Annab
di не. mat. e fis>, 8, 1857, 22-27
" Atti AocacL Foot, Nuovi Liaod, 6, 1852-3, 104-llfi. Fregier.114
MFormole algebriche eeprimenti il numero delta p&rtiaioni di qu&Iunque intero. Progr.»
ClugoDe, 1855.
" Mom. Lit. РЫ1. Soc. Manchester, B), 12, 1&55, 129-145.

Сна*. III] PARTITIONS. 119
partitions of x into к parts > 0, tor к ~ 2, • - -, 6, in terms of the circulator
$t, which is unity if e/s is a positive integer, zero if e/s is fractional or negative»
For к ^ 5, his results an? identical with those of Hcrschel,^ but were ob-
obtained by more elementary methods. Kirkman40 corrected his expression
for {xt 6) and found (z, 7), He41 found (r2 - г + 1, г).
J. J. Sylvester*1 called the number of ways of composing n with given
positive integral summands <ц, • •», a* the quotity Q of n with respect to
a^ - - -, a,. Thus Q is the number of sets of integral solutions ^ 0 of
He stated that Q ^ A + Uf the periodic part U (depending on roots of
unity) not being discussed, while the non-periodic part A is the coefficient
of 1/t in the expansion of
Other formulas for A are given. But all these formulas were provisional
and were replaced in his next paper by others more expeditious for com-
computation.
Sylvester43 stated that Q — 2TTgy where Wq (called a wave) is the
coefficient of l/£ in the development in ascending powers of t of*
summed for the various primitive $th roots p of unity. Thus Wq = 0
except for a q which divides one or more of the a*. Thus Wx is his former A.
Taking the a's to be 1, • • >, 6, Sylvester computed Wu * ••, Wt initially in
terms of certain Zpk and finally in terms of НегдслеГв34 circulating functions,
obtaining results agreeing with Cayley's.*
But Sylvester did not give a full account107 of his theorem until 1882»
A» Cayley44 employed P(a, b, • - -)q, in the sense of Sylvester's Q, to
denote the number of partitions of q into the elements a, 6, • ■ •, with
repetitions allowed» As known, it is the coefficient of z<* in ПA — я*).
By decomposing the latter into partial fractions, it is shown that
where к is the number of the elements at bt ■ • -f and I is any divisor > 1 of
one or more of these elements, and the summation extends, for each such
divisor, from r = 0 to r = x — 1, if x is the number of elements a, b,
having I as a divisor. Also
40 Mem. Lit. Phil. Soc. Manchester, B), 14, 1857, 137-149.,
4t Pw>c» and Papere Laoc^ehire and Cheahire Hist, Soc. Liverpool, 9, 1857, 127.
"Qiukr. Jour. Math., 1, 1855 A857), 81-4; CoU. Math, Papers, II, 86~9.
«2Ы., 141-152; Coll. Math. Papers, II, W-99. An Italian imal. of an extract appeared
in Annali di яс. mat. e fis,, 8, 1857, 12-21.
*SyIvesterpB first factor p* has been changed to p~* to accord with Battaglini,4* Brioschi,41
Koberte,^ and Tnidi,"
44 Phil, Trane. Roy. Soc. Londoo, 146, 1656, 127-140; Coll» Math. Papers, II, 235-249.

120 History of the Theory of Ntjmbees, [Chap. Ш
is the " prime circulator to the period a," if aq = 1 or 0 according as q ie
divisible by a or not, and
A, + A*t + . • -h 4(x-d^- ^ 0 (i = 0,1, •. ,1 - 1; X =
He showed bow to evaluate the A'z and then the coefficients A, >
of the non-circulating part. Next, he evaluated the number P(Q, 1, - >)?
of partitions of q into m terms 0, 1, —tk, with repetitions allowed, known
to be the coefficient of яч» in A - z)'l(i - хг)~1 - A - xkz)~\
Finally! Cayley proved that the non-circulating part of the fraction
4>{x)lf(x) is the coefficient of 1/t in
Cayley46 later considered his last formula for ф(х) = 1, obtaining a
formula equivalent to Sylvester's theorem, and applied it to find
)
,, ,)*
Cayley* noted that P@, 1, -., m)'q - P@, - • •, m)\q -I) is the
number of asyzygetic oovatiants of degree 0 and order q of a binary quantic
of order m. Thus it is the coefficient of of in the expansion of a given
function. He calculated the literal parts of covariante by Arbogast's
method of derivatives,85"
F< Brioschi" started with Euler's remark that the number Ct of parti-
partitions of 8 into r parts ^ n is the coefficient of x9zr in the expansion of
Z « A - rj-^l - жг)- 41 - *'«)^
Now Z = 2^(aJ)^, where
Since ^(x) ie unaltered by the interchange of n and r, C9 equals the number
of partitions of в into n parts ^ r. Let «i, «i, • • • be the roots of f(x) «i 0;
Эь №, *" the roots of Ф(х) « 0 and set
Then
G) Ci « «i, 2Gt = Ci^x -f «if • • •, pCv = C^i*! + • - -h Cx^i -h ^
* Phii Tmu. R. Hoc, London, 148,1,1858, 47-52; ColL Math, Рлрепц П, 50&-612. ""
- Ph3, Thuw, R. Soc. London, 146, Шв7 101-126; Coll. Math. Рирепц О, 250-281. ОТ.
F. Bikttehi, Aanali di Mat., 2,1850, 266-277.
* ЛшшН di bc. mat. e 6я.» 7, 1S56, 303-^12. Reproduced by FaA di Brtmo.»

CfcAF. Ш] PAKITTtONS, 121
Hence
«i «i — 2 • - 0
vie*
Set t(h/k) = 0 or kt according as h is not or is divisible by k* Thus
—>+«(sMi)+ +-(?)-(п-:т
(то \ / m \
H ~r л/ \fi -j- т/
G. Battaglini49 proved Sylvester's formula for the wave Wq by means of
the special ease {«*i =• 1, • ••, Or * 1) where the coefficient of x* in A — x)~^
equals the coefficient of Ijt in enl{l — ert)~r. To evaluate the waves, we
need the value of 8:
where, in 8, the summation extends over all imaginary fcth roots Xi of unity*
We can find c>s such that
Since Zx/ — — 1 for j ^ A; — 1, we see that S is у*, ^i^ *' ^ г*-ъ according
aeftsO, 1, - •, к — 1 (mod i), where y, = kc,r — q> — cx — — - —
and 2yy = 0. Hence we obtain Cayley's4* prime circulator [with k4 for
F> Brioechi4t proved Sylvester's* theorem by use of Cauchy^s theory of
residues. He noted that, if aly . • •, Or are all primes,
^1 <_i
according as m is or is not оде of the a's. Here yu - - >, yi are the primitive
with roots of unity^ and ft, A, - • • «re the a's not divisible by m> Applica-
Application is made to2xi + 3x* +5x* = n.
A. Cayley" wrote (p?*< p?) for the partition of n into ni parts pu
Пг parts p2> eto-, where pi > pj > •• •. It is conjugate to the partition
of». For example, F 3* 2*) and E* 3 I1) are conjugate partitions. Given
*»MemoriedellaR. AceadTSc. NspoK, 2r 1&55-7 A857)y 353-363. ~
•» Annali di ac. mat. e fie-, 8,1857, 5-12»
«Phil. Traaa> Roy. Sdc. London, 147, 1857T 480-409; Coil. M*th. Papers, U, 4X7-439.
Reviewed by E. Betti, Atmali di доЦ 1, 1858, 323-6.

122 History or the Theory of Numbers» [Chap, ш
хл _ fjji"^1 -f - • • ± a^ — 0 with the roots Xi, the symmetric function
belonging to the partition (pi - - ■ pm) is Ъх[1 ■ > < z£\ Part of a[a5 -» ' a£ is
the symmetric function
to which belongs the partition (p + • • ■ + I, ■ ■ *> p) conjugate to
(mp- • -2*1')- Thus а\аг} belonging to C1a), contains with the coefficient
unity the symmetric function belonging to the conjugate partition DP)f
and with other coefficients, the symmetric functions belonging to C21),
B"), Cl3), B= I2), B1J), (V), but not C5).
J. J* Sylvester51 stated that the number of ways n can be composed
additively °f the positive integers аъ ♦■-, «if relatively prime in pairs,
differs by a periodic quantity depending on the remainder of n modulo
+ i - \\ ш \/n -l i - 1
where Si, ■"' i 5«--i are the coefficients of z, • • < f x*'1 in
(I + ai - 1)(^ + ая - 1)- - ■(* + ^ - 1).
For sys^ems ^e (аь •••) — Cb 2, 3) or A, 3, 4), the residual periodic
quantity lies between J and — \t whence th$ number of partitions is the
integer nearest to Qn.
Caytey62 proved that the number of partitions into % parts, such that
the first part is unity and no part is greater than the double of the preceding
part equals the number of partitions of 2X~X — 1 into the parts 1, 1', 2,
Sylvester63 gave an explicit expression for 2хлу* - - * w\ summed for
all N s^ts of integral solutions of ax + by + * • • -h lw = n, where a, • • *f I
are positive integers» The case cc = p = • • ■ =X = 0 gives the number
N of sets. Let &{Ft) denote the coefficient of \jt in the expansion of Ft
in ascending powers of L Let m be the l.c.m. of a, ■ * •, L Then hie41
former theorem may be expressed in the form
summed for the primitive with roots p of unity, where Лр = q€Tv\ Then,
for example,
A(o)(l + Ac) - ■ > (г - 1 + Ад)Л(- п)
A - Ла)*Ч1 - Л6) ... A - АО
т. Jour. Math., 1, Ш57, 198-^
w phiL Mag,p {4)p 13, 1857, 245^-8; CollT Math^ Papers, III, 247-9.
Ш*, D)> 1BP 1858, 369^371; O>U. Matt. Papers, Hr UO-2.

. in] Partitions. 123
Sylvester64 cited Euler's14 transformation of the problem of the Virgins
and noted that the general form of the problem is to find the number* of
ways in which a given set of numbers h, * - *, Ь [an r-partite number]] can
be made up simultaneously of the compound elements a^ '■♦,ог; bXi *'',6Г;
etc» This problem of compound partition can be made to depend on simple
partition» Omitting details, he stated the following theorem: Given r
linear equations in n variables with integral coefficients such that the г
coefficients of each variable have no common factor, and such that not more
than r — 1 variables can be simultaneously eliminated from the r equations,
then the determination of the number of sets of positive integral solutions
may be made to depend on like determinations for £ach of n derived inde-
independent systems each in ti — 1 variables» The conditions are satisfied by
Euler's equations
oar + ■ • • ■+■ Iw = m, л -f- • • • + to = jb,
if a, ' • •, I are distinct» Sylvester never published an explicit statement of
the theorem just sketched, nor of his obscure generalization» See the
following paper.
Cayley" called (а, л) + F, 0) -f - ■ • a double partition of (m, /0 if
U a/a, &/£, • • * are distinct irreducible fractions and if <xt 0, • • • are each
< /i -h 2, the number of such partitions is
D(cem — a^; <xb — a/?, ас ~ a% • * *)
where the denumerant107 D(m; a, b, — -) is the coefficient of &* in
He noted that Sylvester apparently eliminated each of the r variables in
turn from ax -\- by + • • • ^ m, ax + fiy + • • • *= pt obtaining r equations
of the form
(orb — aj3)y + (ac — ay)z -f • • • = otm — ap,
from which the above formula follows»
* E. Mortara* treated partitions into three distinct elements.
Sylvester57 delivered seven lectures on partitions in 1850,
G» Bellavitis5* proved that the number \ji, n, p~\ of sets of integral solu-
solutions ^ 0 of aa + «i + • • • + а„ ^ p, Л1 + 2a2 + " * + «л# = Р» equals
the number jji, p, n] of sets of solutions ^ 0 of A, -|- jSi + — * + ^, « n,
^i + 2ft + • - - -h P^p = p,. For, every set of solutions of the first pair
* Number of seta of integral eolutkms ^0 of <цх + Ъ,у + -■r e fc (i ~ I, ♦",■")•
" Phil. Mag.p D), 16, 1858, 371-6; CoU, Math. Papers, II, 113-7*
ы Phil. Mag., D), 20, I860, 337-341; CoiJ, Math, Pai>ersr IV, 166-170.
14 Le paftixioni di un Dumexo itt 3 parti differenti, Pfuma, 1858.
" Outlines of the lectured were printed privately in 1859 and republiehed in Proc. London
Mftth. Soc>p 28, 1897, 33-96; Coll. Math. Paper*, II,
14 AnnalJ di mat., 2, 1869,137-147.

124 History of the Theory of Numbers, [Chap, m
of equations consists of a partition of /* into «„ numbers n, ■ ♦ •, «1 numbers
1, where p is the total number of parts» To such a partition corresponds
as conjugate
(a* + a^i + - •►> «i) + (*■ +■•■+«*)+•■■ + («!• + <*«-i) + ** = P>
which gives a partition of м ^to n parts Ш p. These parts occur in the
second pair of equations as 0P numbers p, - —, ft numbers 1. Again,
bh*, Vl = bh n - 1, Vl + Cm - n, n, p - 1}
[>, n, p] - [np - M, n, p}
There are [jx, я, p] partitions of N into p parts from с, с + d, - • -, с + ж?,
if /i = (N — cp)/d, since if each part be diminished by с and the remainder
be divided by d7 we get the parts 0,1, • - -, n whose &uui 1в ц. Application
is made to seminvariants*
L. Oettinger5* stated and J. Derbes55 proved that (k — l)¥kr"v is the
maxunum of the products of the r equal or distinct integers into which the
positive integer N = тк — v can be partitioned, where v 12 the least positive
integer such that к is integral.
Sylvester*0 noted that Bellavitis'5* first theorem reduces for p infinite
to Euler*s theorem that the number of partitions of p into parts ^ n equals
the number of partitions of >i into n or fewer parts» Bellavitis' theorem,
which is capable of intuitive* proof by Ferrer's*6 method, may be stated as
follows: The number of distinct combinations of oq, • • -, ал figuring in the
coefficient of s" in (oo -f a*r + — ■ + a*zn)* is the same as the number of
distinct combinations of bo, ■k -, ЪР in the coefficient of 3? in F0 + htx + ■ •«
S, Koberte*1 proved Sylvester's'13 formula for waves,
Sylvester82 noted tiiat, if Пп = л!,
у * ^ = 1
where the summation extends over all ways of expressing n as a sum of or
parts each a, £ parts each fr, etc.
E. Fergola** proved the analogous result:
summed for all positive integers satisfying
«1 + • • ■ + <*n - «3 «! + 2as + • - 4- пля = n,
where ДЧ)Я denotes the ath order of difference of хщ for x^O. He evaluated
*» Nouv. Ann. Math., 18, 1869, 442; 19, I860,
и PhiL Mag., D), 18, 1856, 283-1, onder peeud.
л Quer. Jour. Mat4.r 4,1861,15545.
•*Cumptee Rendua Paris^ 53,1861, 644; Phil. Mag 22,1861, 373; CoU. Mbth. Popere, II,
245r290.
••Rendiconto dell'Accad. Sc. Ин, с Mat., Napoli, 2, 1863, 262-8.

Chat. Ill] PARTITIONS. 125
sums in which the preceding summand is multiplied by H(a — l)y* or
Fergola*4 stated that the number of sets of positive integral solutions of
is A/(nt), where
Д =
О п - 2 - (Г!
О 0 п - 3
+ «i 0 0 0 . ' ■ 3 *-o"i
~f ffj 0 0 0 . ♦ ■ 0 2
while <rf is the sum of those divisors of t which occur among the positive
integers аи &ъ ■4 ■- When ъ *= i {i = 1, • • •, л), *v becomes the sum <r(r)
of all the divisors of r. If in Д we change the sign of the second components
in the first column and change the sign before each a above the main
diagonal, we obtain & determinant equal to {— l)hn\ when n is of the form
k(Sk ± l)/2, but equal to zero when n is not of that form.
C. Sardie* proved the preceding theorems,
N. Trudi" proved Sylvester's formula 2TTe for the number /*»(<*, • ■ -, X)
of partitions of n into elements <x, - - -1 \, He also showed that WQ = XF(p),
gummed for the primitive gth roots p of unity, where F(p) is the coefficient
of 1/t in
- (p + I)"^M1 - (P + ОТ1''" (I - (p + ON-
Let Hi, • —, o, be those of the numbers a, - - ••, X which are divisible by Q,
and bt> ■ < •, b# the remaining numbers» Let
upon writiiig the denominator on the left as the exponential of its logarithm
and expanding the exponentials. Laws are given to determine the A's.
From the coefficient of t~x we see that Pn ^ ZFr, <, summed for the various
divisors q of the various elements a, • • -, X, where
summed for all the primitive qth roots p of unity. Simplifications are given
in three cases: m — 1, m = 2t r *= 1. He tabulated results for
^a, 3, 6, 8), Рл<1, 2, 3, 6, 8,10), ЛA, 2,^-^), PnB,3t --,?), g g 8.
11 Giomale di Mat.r lr 1863? 63-64.
1/bid., 3, 1865, 94-99r 377-380.
1 Atti Acc^cL 3c. Fie. e Mat. Napoli, 2, 1865, No. 23r 50 pp.

126 History of the Theory or Numbers. [Chap, ш
A, Cayley,47 denoting by P< the number of partitions of n into % parts,
proved that
1 - Pt+ 12Л- — d= (n- l)IPn = 0.
For, the number of partitions n = o« + bp ~f - - • is
Multiply this by (— l)*^) — 1)! and sum for the sets of solutions of
p = a H- 0 ~f «> •; we get the initial theorem.
A. Vachette" stated that one of n*, n2 - 1, я2 - 4, nJ + 3 is divisible
by 12 and the quotient is the number of seta of integral solutions > О of
x + у + z = n [De Morgan26].
L. Bignon*9 noted that the respective eases occur for n = 6n', 6n' + 1
or 5, 6V + 2 or 4, 6V + 3. For n = 6ny, for example, he separated the
sets of solutions into «/3 seta each with у — x я constant 0,1, - - -, %n — %
and exhibited the solutions of each set.
R Catalan7» noted that xb + • ■ ■ + ж* ^ « has (lit) sets of positive
integral solutions. Subtract unity from each x and apply his* former
result.
Let71 (n, q) be the number of partitions of n into q distinct parts, \nt g]
into q equal or distinct parts. Ftoof is given of theorems of Euler:
(n.q) - (n-qtq~ 1) + (fi - q, q), (n, q) -[»-
and the first written for £ 2 Неге п Ш2д.
Inxl+ — + *ff « n, *! ^ ж, ^ - - - ^ *„ take «i = a ^ [n/ffl and
set x^ « yj -f a — 1 (г = 2, ,, ,f q). Then
2/2 + • • • + »ff « я - 1 - (o - l)g
for y'a > 0, Hence72
- 1 - (* - 1)?, ^ - О "
Taking q = 3, he deduced the result of De Morgan2* and Vaehette/8
C. Hermite73 stated that the number of sets of positive integral solutions
of
x + y + г = N, x ^y + г, у ^ z + z, z ^x + y
»T Math. Quest, Educ. Time», 7,T867, 87-8; Coll. Math. Fapere, VII t
" Nouv» Ann, Math., B), 6, 18G7, 478.
»/Md.,B), 8,1869,415-7,
"Melangee Math.r 1868, 16; Мбш. Soc, Roy. Sc. liege» B)r 12, 1885, No. 2,
n Ibid., 62-65; M&d. Liege, 56^-58.
" Ibid.t 305-12; Mem, Ii&ge, 254r-71. Noav- Ann. Math., B)t 8, 1869, 407.
" Nouv, Ann. Math., B), 7, 1368, 335- Solution hy V. Schlegel, B), 8r I869t

. ni] Partitions. 127
is (N2 - l)/8 or CY + 2)(ЛГ + 4)/8 according as ,Y is odd or even. An
anonymous writer (pp. 93^4) stated that the number of sets of positive
integral solutions of X\ + - - - + Хш = N is {JV( — т{(ЛГ ~- Д/2), wbere
[г] = С*1^) and j = 1 or 2 according as N is odd or even.
K. Weihrauch™ discu3sed the number fn(A) of sets of solutions of
адх» + ••• + о^„ = A,
where the a's are positive integers- Set
P = aja2- * Лп, iSt- — oj + • ■ * + а;, Л *= pP 4- m,
where m is one of the integers 1, - -, P. Then
MA) '
the last being stated without proof, where < is the largest integer ^ r/2,
R = m — ^i/2, and
the 5*s being Bernouilli numbers» Cf. Meissel1^ and Daniels.141
* E. Meissel75 treated the partition of very Urge numbers.
E. Lemoine*6 noted that every power nM of an integer n equals a sum of
nh consecutive terms from 1, 3, 5, 7, • • ■, if ^ ^ 2Л. Cf. Fregier.2**
G» B, Marsano^s77 Table 1 is an extension of Euler's table of partitions
of n into m parts, for n ^ 103, m ^ 102. Table 2 gives the coefficients as
far as &* of the expansions of
and the coefficients as far as xio: of the expansion of the first ten functions.
The results for £/A — x) give the number of ways of partitioning a number
into parts 1, Г, 2, 3, . ■ >. Those for 8/A - ar)(l - x2), into parts 1, ly,
^3
7* Untereucbungen GI. 1 Grv Dies. Dorpat. 1869, 25-13. Zeitechrift Math- Pbye,, 20, I875r
97, 112, 314; ibid., 22, 1877, 234 (n -= 4); 32, 1887, 1-21,
" Notiz Ober die Anzahl alter Zerlegungen «ehr groeser gaxuer poettiver ZahJen ш Summon
ganger poeitiver ZaKJeti, Progt-t Iserlahn, 1870-
T* Nouv. Ann. Math., B), 9, 1870, 368-9; de Montferrier, Jour, de math. &ёац 1877, 253.
77 Sulla legge delle deiivate general! delle funaioni di fiiDZioni e eulla teoria delle forme di
partizione dcJDumeri labeii, Gcnova, 1870, 281 pp. Dteciibcd by A. Cayley, Report
British Амос, for 1875 <1876), 322-4; CbU. Math. Papers, IXr 481-3.

128 HlSTOKY OF THE ThEOHY OF NtJltBERS. £СнагЛ11
G. Silldorf1* considered the number /(*, k) of decompositions of 9 into
h integral summands ^ 0, and the number fr(?7 k) in which r is the least
summand. In the former, 0 occurs in the first place f(s> к — 1) times, 1
occurs/i(* - 1, k - 1) times, etc. But/E, к) = /,(« + rfc, k). Hence
/(*,*) =* f(s, k - 1) +f(a - k, k ~ 1) + *.
Thus/(a, 2) — |(a 4- 2) or J(* 4- 1) according as « is even or odd,
/(*, 3) = (s> + 6* + 12)/12T s m 0 (mod 6),
with similar results for s ^ 1, «»•, 5 (mod 6). Let F(st к) be the number of
combinations without repetitions of к elements with the sum s. Then
[Euler,* § 315]. There are as many partitions in parts < m as into m
or fewer parts. The number of ways * can be expressed as a sum of numbers
^ m, with repetitions allowed, is
2-1 - (a - m - l)o(e - m + 1J~~ + (t - m - 1), ^
F. Gambardella79 noted that ожН-5у^сг
\qBm fa + Hc- abc^) + e + ^
sete of integral solutions if а, Ь, с are positive and relatively prime in pairs,
and *n > 0, m ^= C7 + X, 7-fl=g«bH-T-. Here a is the sum of the
quotients and /ц, ■■-, pr the remainders upon dividing X, X -f c, ^,
X+(r^ I)cbyo6; whilftfc is the number of solvable equations ax + by = p^,
T. P> Kkkrnan,7** counting 5 1 =15-13^1-2-1-3 + 21= --
as partitions of 5, evaluated the sum of the reciprocals of Bе1)-'1Bй)">
- * • will ni^I • • -, for all such partitions mxe\ + m«e* + • • ■ of R*
J- J. Sylvester* noted Oiat a list of all partitions of n may be checked by
2A - x + xy - хуг + • -) = 0,
summed for all the partitions, where in any partition, x is the number of
l's, у the number of 2% eto.
Von Wasserschleben*1 expressed 60& as a sum of four numbers each a
prime or product of two equal or distinct primes, for к = 1, - ♦ -, 16.
* L. Jelinek8* treated a kind of partitions.
"TJdber die ZeHegung g&nwtrZablenm Summanden. Ргокг, Sabwedel 1870, 17 pp.
n Gioroale di Mat., 9,1871, 262-5. Exteneiomj by C, Sardi, 11,1873,123-
*• Math. Queet. Educ. Time*, 15,1871, W-3; 16,1872, 74-5.
•■ Heport Britiah Ajwoc., 41, lg71 A872), 23-5; Coll. Math. Papere, llr 701-3,
л Archiv Math, Phye., 54,1872, 411-8
n Die Wurfebtablen u. dfe Zeriegung ешег Zahl in ganaen %.y der^x Summe gegeben ut,
Pr. Wiener Neuatadt, 1874.

Chap. UJJ PABTITTONS. 129
* V. Boumakowsky8* treated partitionB.
J. W. L. Glaisher54 considered the number P(ay • ■ *9 q)x of ways of
forming x by addition of the elements a, - --, q, repetitions allowed, and
proved that
P(l, 3, 5, • - .)B*) - 1 + P(l, 2)(* - 1) + P(l, 2, 3, 4)(* - 2) + ■ • ■
+ P(I,2, — ,2*^2I,
P(l, 3, 5, • • 0B* + 1) - 2 + P(l, 2, Щх - 1)
l, 3, 5, - - 0* - P(l, 2)(* - 1) + P(l, 2, 8, 4)<* - 1 - 2 - 3) + • •..
Glaisher8* formed the derivations of a4 by the rule of L. F- A. Arbogast r86*
а4; а*Ъ; а% а*Ь>; a4t а*Ъс> <&; -t
omitting coefficients. Each term corresponds to a partition of 4- Thus,
if a = lt b = 2, < • <t aJ6 corresponds to the only partition 11 1 2 of 5
into 4 parts > 0. In general, from the derivations of a* we see that the
number of terms of the zth derivations of a* equals the number of partitions
of x into n parts including zero, also equals the number of partitions of
x + n into n parts > 0, and finally equals P(l, — «, n)x.
Glaisher8* gave formulas for checking the tabulation of partitionB.
The summations extend over all the iy partitions of a given number n,
while in any partition, x is the number of !*s, у the number of 2's, etc.
2A + x + xy + xyz + " -) = T2r7 X(x - 2xy + Sxyz - - • •) =. т(я),
2A - 2y + Zyz - 4yzw +.«.)=T(ftfl)- r(n),
Z{x - 1 - (x - 2)y + (x - 3)yz - - •} =N -h
where r is the number of different elements in a partition, and т(п) is the
number of divisors of n. If Q(o, bt — -)n is the number of partitions
without repetitions of n into the elements а, Ъ, ■ * •, and S(lt • • -t r)n the
number of partitions of n into 1, • • < t r in which all but the lushest r appears
at least once,
2QA, 2, ••■)«- 1 + S(l, 2)n + iS(lt 2, 3)n + - - -,
CA,3,5, ..-)"- Q(l, 3,5, •-•)(«-4)
-<Ц1,г,6, -0(n-8) + Q(l,3,5x .-0(я-»)+ lorO,
according as n is a triangular number or not. The excess of the number
of partitions of n into an even number of parts over an odd number of
parts is (— 1}"QA, 3, 5, • • )n. A partition into a 1% p 3% у 5's, ete.,
is transformable into ir = a + 3$ + by + *-. Express «, ft - ■ in the
binary scale: a = 2* + 2?' -\- — -, 0 *= 2* + . ■». In the new form of *-
no two parts are equal. Hence a partition into odd parts is converted
into a partition into distinct parts, and conversely.
« Memoirs Imp. Acad. Sc., St. Pteterabui*, 18,1871,20; 25,1875 (SuppL), No. l (In Ни*мп)~.
«Phil. Mag., D), 49, 1875,307-311.
» Report British Аввос. for 1874 A875), Sect., 11-15; Comptes Hfindue Paris, 80,1875,255-8,
*- Calcul dee dfenvatiooa, Straeburg» 1800. See papers 46.102, 198.
• Proc. Roy, Soc, London, 24,1875^6, 250-»,

130 History of the Tkeokt of Numbers. [Chap, щ
P. Mansion87 noted that the Ath power of an integer n is the sum of
n consecutive odd numbers (those nearest n*), as 34 = 25 + 27 4- 29.
J. W. L. Glaisher88 stated that, if Cm is the number of compositions of
N into m triangular numbers, and A is the sum of the reciprocals of those
divisors of N whose conjugates are odd, В if even, then
Сг - id + iCz - .. - ± jjCN =A-C.
Glaisher" noted that, if P(x) is the number of partitions of x into 1, 2T
3, • • ■, repetitions allowed, and Q(x) is the number of partitions of x into
1, 3,5, 7, ■«., repetitions excluded, then Q{x) = XP\(x - *)/4}, summed
for the triangular numbers t < z such that t = x (mod 4),
Glaisher**0 used an identity due to Jacobi,*2* p. 185, to show that
P(x) + 2P{x - 1) + 2P{x - 4) + 2P(x - 9) + • • •
if P{x) is the number of partitions of x into even elements without repeti-
repetitions, and Q(x) the number into odd elements without repetitions.
A. Cayley90 denoted by un the number of partitions of n with no part < 2
and order attended to. Then щ = щ — 1, un = w*_! ~f и^3.
E. Laguerre^1 started with Euler's result that the number T(N) of
sets of positive integral solutions ofaz + ty-f • ■ • ^tfisthe coefficient
off" in
decomposed the latter into partial fractions, and called the result Ф($)+4>(i),
where Ф(£) is the sum of the simple fractions whose denominator is a power
higher than the first of one of the factors in the denominator of F($)- Let
GBV) denote the coefficient of £* in the expansion of ф@. Then
T(N) = BIN),
with an error which is independent of N. For example, if ax + by = N
and a, b are relatively prime, 6BV) = (N 4- l)!(<*k), so that T(N) - N!(ab)
approximately [Paoli117 of Ch. II], the error being < 1. For
ax + by + cz ~ N,
the approximation is N(N -f a + b -f c)fBabc)*
F. Faa di Bruno** gave an exposition of BrioschTs47 work and noted
that his linear equations G) are of the same form as Newton's identities if
the sign of Si be changed. Hence, by Waring's formula,
Meeeenger Math., 5,1876, 90, Cf.
• /Wd., 91.
•Math. Quest. Educ. Timee, 24, 1870, 91.
Метепдо of M&th, 5, 1876, 188; ColL M&tK Papem, X, 16.
BtiU» Math. Soc. France, 5, 1876-7, 76-S; Oeuvree, 1, 1898, 218-20,
Tbtori&fobi», 1876,157; Ctenaan tnuwl, by T, Walter, 1881,127.

Chap- Ш] PahtITIONS. 131
summed for all solutions of Xi + 2X2 + • ■ • + P^p = 1?- At the end of
this § 12, he gave other expressions for Cp. He*3 later transformed the
above formula into
where [xp2r denotes the coefficient of xp in r, while, after the expansion,
d{ is to be replaced by i!. Similarly, for the number Wp of sets of positive
integral solutions of a&i + ■ • ■ + а»хл — p,
which is much simpler to apply than Sylvester's" formula. He stated
(p. 1259) the generalization to two variables:
)
F, Franklin94 proved that if, in all the partitions of n which do not con-
contain more than one element 1, each partition containing I be counted as
unity and each partition not containing 1 be counted as the number of
different elements occurring in it, the sum of the numbers so obtained is
the number of partitions of n — 1. Application is made to the distribution
of bonds between atoms»
A. Cayley*6 noted that the partition abc -def of 6 letters into 3's contains
6 duads aby act be, - ■ -, while the partition abed-ef into 2'в contains 3
duads. Hence if a partitions into 3's and /? partitions into 2's contain all
15 duads once and but once, 6» + 3/3 =* 15. The solution a = 1, 0 = 3,
furnishes an answer of the partition problem: abc*deft ad*be-cf, a$*bf-6d>
af-bd-ce. Likewise for <x = 0, £ = 5; but not <x =* 2, £ — 1. Similarly
for 15 or 30 letters.
J. J. Sylvester96 considered the e = (w; i, j) partitions of w into j parts
0,1, ■ * j it the elements of a partition being arranged in non-increasing
order, as 3, 2, 2, Without computrng e and/ ^ (w — 1; i, j) separately,
we obtain e — f = E — F, by counting the E partitions of w in which the
initial two parts are equal, and the F partitions of w — 1 in which one
element is г. Also,
e-f= -(w~i- 1; t\ j-1)+ £(w-2ff; ff,i).
F, Franklin97 proved this rule of Sylvester's by converting each partition
into one consisting of i of the numbers 0, 1, ■■•,,/. Then e — f =? t — ф7
"Comptce Rendus Paris, 86r 1878, 1189, 1259; Jour. f(ir Math., 85, l878, 317-26; Math.
Annalenr Ht 1879, 241-7; Quar, Jour. Math., 15,1878, 272-4.
" Amer. Jour. Math^ l, Ш78> 365-8.
* Meeaenger Math., 7, 1878, 187-8; Coll- Math. Papere, XI, 61-2.
«/Wrf,, 8» 1879» 1-8; Coil, Math, Papers, Ш, 241-8.
»> Amer, Jour. Math., 2» 1879, 187-8.

132 History of the Thxoky of Nuhbbrs, [Ctu?. ш
where € is the ninnber of partitions of w not containing the element 1, and
Ф is the number of partitions of to — 1 not containing 0.
N* Trudi" gave an account of the early history of partitions, made
extensive applications to isobaric functions, and finally enumerated the
eombmations of n letters into at sets each of p letters, 0 sete of q letters, etc.,
first when the n letters are distinct and second for repeated letters.
C. M..Piumaw treated the following problem: From an urn containing В
balls marked 1, - -, B, three are drawn and the three numbers written on
them are added; find the number of times the sum is ^ C. To find the
number £д of Bets of solutions of ф ~f ф ~f x — Я with 0 <ф<ф<х = В.
First, let С < В ~f 4. Then every solution satisfies the inequalities. Of
the six cases Я *= &h ~f j(j = 0, - >-,5),let# = 6^ + 4 and set ф - Ф = xt
x — Ф = V- Then x + y *~ §h — 3^-f4, 0<х<|/. If ф is even,
ф = 2of> there are evidently 3ft — 3a + 1 sets of solutions x, y} and h is
shown to be the largest « giving a solution. Thus there are
sete ^, ^, x» For ф odd, we get h($h + 3)/2 sets. Adding, we get
Stm - h(Zh + 1). Then Tc « 2&_^H is found by treating six cases;
for example, J1^ = cA2c* — 15c 4- 5)/2- Finally, there is treated the case
С ^ В + 4.
P. Boschi100 treated partitions into a parts from 1, • ■ ■, и-. Let
Expand and collect the terms of £«,*; the coefficient of з? is the number
of ways P is a sum of distinct numbers chosen from г, г + 1, • ••, n. It is
proved by induction that
Thus the coefficient of a^ in SUt i = *«^»«йГв11 is the number of ways P
is a sum of в different terms of 1, > • *> n* For и ** 2,
where Л< is the number of ways s + 3 is a sum of two numbers of 1, •■•,».
Then Ar - A
Ат в JBr + 3 + (- I)-} if2^r
Л, = n-2 + r + \\2r + 3 + (-
11АШ R. Accad. Sc Pia. Mat» Napoli, S, Ш79, No. 1^ 88 pp.
" СЙопшк di Mat., 17, 1879, 860-372.
110 Memorie Acead» Be Ы. Bdogaa, 1,1880, Б5ДО71.

p. Ш] PaBTTTIONS, 133
Let Ut be the number of pairs from 1, - ■ • T n whose sums are ^ г- ТЪеп
V, = f{r(r - 2) + tfl - (^ D'J}, 3 ^ r s* + 1;
*/, - i/ifn - 1) - U,^+ly n+Kr^2n-L
Similar applications are made to the cases u = 3, v = 4.
J, W, L. Glaieher101 noted that, if P(i*) is the number of partitions of и
into the elements 1, ■ • •, n, each partition containing exactly r parts, order
attended to and repetitions not excluded, then
p(r + k) + P(t + n + к) + P(r + In + к) + • • * - n1^1
(*-0,l, -,«-1).
E. A. A, David108 noted that Arbogast's8*1 law of derivatives gives
_
(n - 2)! *" (n - 3I + (n - 4)!
gummed for all sets of positive integral solutions of
Pi + 2p* + Зрд -f • - - n.
The latter sets are all given by the exponents of the terms in the left member.
A. Cayley tabulated all partitions of 1, ■«-, 18, wherein each partition
1, 2, — - are designated by а, Ъ, * * -,so as to give the literal terms in the co-
coefficients of any covariant of a binary quantic.
G. B, Marsano1" treated the number of combinations 2 or 3 at a time
of 1, 2, -«', m to give a sum ^ С Simpler and more general results were
given by Gigli.181
b\ Franklin106 proved Euler's formula C)» The coefficient of xv in the
left member is evidently the excess E of the number of partitions of w into
an even пшпЬег of distinct parts over that into an odd number of parts.
To find E, write {a} for a number ш. a, and let the parts of each partition
be in ascending order» Consider a partition with r parts, the first being 1;
deleting I and adding 1 to the final part, we get a partition into r ~ 1
parts, the first being {2j, and without two consecutive numbers at the end,
and conversely. These two types of partitions do not affect the required
E, one being of even order and one of odd order. Hence we need consider
only partitions commencing with {2} and ending with two consecutive
numbers. Consider any one of these with r parts, the first being 2; deleting
2 and adding 1 to each of the last two parts, we get a partition into r —
h., 9, 1880, 47-8.
Comptoe Rcndua Parier 90, 1880, 1344-6' 91t 1S80, 621-2; Jour, de Math., C), 8, 1882,
61^72.
Amer. Jour. Math,, 4,1881, 248-256; CbUL M*th. Papers, XI, 357-364.
4 Giomale di Mat.r 19, 1881, 156-170; 20, 1882, 249^270-
Compile Rendue Pane, 92, 1881, 448-450. Of- Bylv&ter™ 11-13.

134 History of the Theoby of Numbers. [Chap, in
parts, the first one being {3} and without three consecutive numbers at the
end. We may suppress these partitions. In general, consider a partition
commencing with |n] and ending with n consecutive numbers. If the
first term is n, efface it and add 1 to each of the last n numbers, which can
be done unless the number of parts is ^ n, whence w = nCn — l)/2. If
the first term ш л + 1 and if the last n + 1 terms are not consecutive,
reduce by. 1 each of the last n and place n before the first part, which can
be done unless the number of parts is n, whence to - nCn + l)/2. Hence
E - 0 unless w = nCn ± l)/2, and in that case E = 1, there remaining
a single partition into n parts. For an exposition of this proof, with illus-
illustrative graphs, gee E. Netto, Lehrbuch der Combinatorik, 1901, 165-7.
A. Capelli1" considered a matrix (<*.,) of n* integers ^ 0 such that the
sum of the numbers in each row or column is always m:
The number of these matrices equals the number of linearly independent
forms derived from the general form in n sets of variables, homogeneous
and of degree m in each set of variables, by means of the operation S^/dfr,
where the £ and i\ are two of the n sets.
Several"* found the number of ways 34 Is a sum of four distinct positive
integers.
J. J. Sylvester107 gave an exposition of the theory previously only
sketched by him>» Employing Cauchy's term residue to denote the coeffi-
coefficient of \.\x in the expansion of a function of % in ascending powers of x, be
considered any proper rational function F(x)t so that the degree of the
numerator is lees than that of the denominator. Then we may write
The residue of Z\Z{F{a^) is easily seen to be the constant term of — F(x).
Hence if x~*f{x) is a proper rational function, the coefficient of хш in the
rational function f(x) is the residue of 2r~nen:r/(re~*), summed for each
value r + 0 of x making/(x) infinite [as the a's for F(s) J The " denumer-
ant to the equation ax + ■ ■ * -\- U = n/} denoted by
n
is the number of sets of integral solutions ^ 0 of the equation, and equate
the coefficient of xn in the expansion of
Let 54 = 1, B±3 - —, 5^ be the integers dividing one or more of the numbe»
"ИСйотшОе di Mat., 19, 1881, S7-115.
»* Math. Quest, Educ. TSmee, 34,1881, 61.
i"Amer. Jour. Math,, 5, 1882, 119-136 (Евдгвив on rational fractions and partition*)*
Johna Hopkinfl Univ. Ore., 2, 1883, 22 (for the firat theorem). CoU. Mb F*
m 6O5CJ2 658WO

Chap, III} PARTITIONS. 135
a, ■ • ■, I- The denumerant thus equals £{^Г Wit where the wave Wi is
the residue of
summed for the primitive oVth roots т> of unity (or for their reciprocals).
Now make the important substitution » = n+ (a + **• + l)/2. Then
Wi = residue of
the product extended over the similar terms in a, b} - • -, L Expanding the
summands into power series, we see that each wave and hence the denumer-
ant is a sum of products of polynomials in v each multiplied by a quantity
c2(r"+* zh г**), where & is one-half of the number ф(г) of integers < г
and prime to г (since Wi becomes =b Wi when * is changed in sign). Give
to each such term of the denumerant an undetermined coefficient, as
Write s — a -\- ■ • • -f!(« = 6in this case). It is shown that the denuraer-
ant is sero for all values of v from 0 to \* — 1 inclusive if s be evenj and for
all values from i to $& —> 1 inclusive if s be odd. This fact serves to deter-
determine uniquely the ratios of undetermined coefficients. For example, in
the above case, v = 0, 1, 2, and
B-\-C - 2Z> = 0, A+B-C + D~0, 4A+B
whence A = fcy В « - 7<r, С - - 9<r, D =* - 8<r. The value 9A + Я
— С — 2D for у = 3 must be unity. Hence * = 1/72. Since v = n + 3,
the result agrees with that given by De Morgan.28 The case of the elements
1, 2, 3, 4 is treated similarly. The wave Wx is discussed in detail» Applica-
Application is made to the number of sets of solutions of
where aif • • -, <Xi are relatively prime in pairs. For i = 2, the number is
(ага2 - ax — aa - l)/2.
Sylvester"8 noted that there is a one to one correspondence between
the indefinite partitions of n with parts in ascending order and the series
0, -• •>, n such that each term is not greater than the mean between its
antecedent and consequent.
If a and Ь are incommensurable, integers #, у can be found such that
<** -\-Ъу -\- с is indefinitely small. If it be impossible to find integers
\ /*, v such that
\(by — ел) + ii{cot — ay) -\- v(ay — ha) — 0,
we + by -\- cz + d and <xx -\- §y -\- уг + В may simultaneously be made
arbitrarily small by choice of integers x, y, z. Cf. Jacobi2^ of Ch. II.
101 Johns Hopkins Univ. Cuts., 1,1882,179^180; Coll. Matt. Papers, III, 634r-9» Firet theorem
abo io Math» Quest Bduc. Ишея, 37, Щ2,101-2»

136 ШвТОЬТ OF THE ТНЕОКТ OF NUMBERS. [Сна», 111
О. H. Mitchell™ wrote (w; i, j) for the number of partitions of w into j
or fewer parte each ш i. Let ф£п>) be the largest integer ^ (j - l)»/i-
Then
(^; hi) = Z (a; w-xj- 1).
By successive applications of this formula, j can be reduced to unity. Hence
0
1
1
1
1
1
1
1
2
1
2
2
3
1
2
3
4
1
3
4
5
1
3
5
6
1
4
7
«>; *, л = z, z, z- - - z; w*
where the final Z(I) denotes 1 + 4sCj_j) - &**-t - *§-*)> i> e., as many
units as values of the summation index. There is given the long expression
equivalent to the last two signs of summation. This is said to furnish a
proof of the final result by Sylvester.*
G. S. Ely11* noted that EulerVs table of partitions
1
2
3
may be constructed by use of columns instead of rows: To get the tth
element in the jth column, add to the (i — l)th element in the jth column
the tth element in the (j« — i)th column. Euler had noted that the number
(tp; w, j) of partitions of v> into j or fewer parts is given by the number in
line j and column v>. The number (w; i, j) of partitions of to into j parts
^ г can be found from this table when the greater of i and j is Ш (w — 4)/2
by the following rule: Since {w\ i, j) = (w; j, г), let г ^ j. Then to get
(w; itj) subtract from the tabulated value of (to; w,j) the sum of the
first w — i elements in the (j — I)th row and add to the result 0, 1 or 2,
according as г ^ (и? - 2)/2, - (w - 3)/2 or =* (w — 4)/2. Next, the
number of expressions (w; i, j) is
w — n*_^n■ — 1 I
n + 1 JJ
where i = 6 if w is even, t — 5 if w is odd» Let s ^ 24 or 27 in the respec-
respective cases. Then
2n* -3n* + 28n-8 1^
12
. J.
W* P. Durfee111 defined a self-opposite or self-conjugate partition to he
one such that, if exhibited as an array of units (an element n being repre-
"• John* Hopkins Uni^Bre., J, 1882, 2Ж
*4bULf 211 ^n full).
** IbtiL, 2, Dee., 1882, 23 On full).

Сна». Ш] PABTTTIONS. 137
seated by n units in a row), the annas of the columns reproduce the original
partition. Thus 4 3 2 1 is a self-conjugate partition of 10. Evidently
1 1
1 1
i Г
l
4 3 2 1
every such array contains a central square of g* units D in the diagram),
where q is odd or even, according as the partitioned number n is odd or
even, since of the n — g* units outside the square half are at the right and
half below the square. The partition remains self--conjugate under any
rearrangement of the (n — q*)f2 units to the right, provided those below
be arranged symmetrically. The number (£(n — g2); q\ of such rearrange-
rearrangements is the number of ways of dividing ?(n — g2) into q or fewer parts.
In the above diagram we may replace the double row of three dots to the
right of the square by a single row of three dots and derive the only other
self-conjugate partition of 10. In general, the number of self-conjugate
partitions of n is 2{§(и — q*); q], summed for all odd or all even integers
q < V?b according as n is odd or even.
J. J. Sylvester11* noted that Durfee's111 theorem may be expressed in the
following form: The number of self-conjugate partitions of n {tir of sym-
symmetrical partition graphs for n) is the coefficient of xn in
and hence is the number of partitions of n into unrepeated odd integers.
He gave a modification of Franklin's1*5 proof of C).
Sylvester118 proved Brioschi's47 formula Z = 24>(r)zr.
Sylvester114 proved by use of the binary acale Eider's theorem that the
number of partitions of n into odd parts equals the number of its partitions
into distinct parts fGIaisher81]. Of graphical methods in partitions, he
called Ferrers'* method transversion and Durfee's111 method apocopation.
He gave a graphical proof of Euler's C).
F. Franklin115 noted that, since the number (tt; i, j) of ways w can be
partitioned into i or fewer parts ± j is the coefficient of a*z" in the develop-
development of the reciprocal of A — a)(l — ax) * - -A — аз*), the coefficient of a/
in its development in ascending powers of a is the generating function F
in which the coefficient of xw is (w; i, j). To obtain F directly, note that
the number of ways of forming w with % or fewer parts of which at least one
is a number > j, say j + k, equals the number of ways of forming
v> ^ (j + h) with i — 1 or fewer parts; the number of partitions in which
m John» Hopkins Univ. Cii^, 2,1883-3, 23-24, 42-1; ColL Math. Paper*, III, 661-671.
ШЩ, 2, 1883, 46; Coll. Math. Papers, III, 677-9; Amer. Jour. Math., 5, 1882, 271-2;
Coll, Math. Рйретя, ГУ, 21-23.
ш Ibid.f 70-71; Coll. Math. Papera, III, 68О-». Cf. Coll. Math. Paper*, IV, 13-18.
™Jbid. 72 (in full).

138 History of the Theory of Numbebs. [Свар, ш
at least two of the parts are > 3, say j + k, j + A*< equals the number of
partitions of w — {j + к) — (j +■ A') into г — 2 parts; etc. Hence
A -*)-..A -**-*)
) + x»*zm(x**+ + • • ■) + • • •
A-s)...A - z-'-з) +
where 2m(z''+1, - ■ •) is th& sum of the m-ary combinations of sy+1, x*+*y
By induction,
Hence
Euler's theorem that a number can be partitioned into odd parts as
often as into any distinct parts is proved constructively and extended.
The number of ways of forming u> additively with an indefinite number of
parts not divisible by к and with m distinct parts (each repeated indefinitely)
divisible by A; is equal to the number of ways of forming w with an indefinite
number of parts each occurring fewer than к times and with m distinct
parts each occurring к or more times» The proof is made for к =t 10,
though the argument is general. First, let m = 0. Consider any partition
consisting only of parts not divisible by 10 and let the number of times any
such part X occurs be written in the decimal notation, say * • -cba; then if
in place of • * cba times X we write a times X, Ъ times 10X, с times 100X, - * *f
we get a partition in which no part occurs as many as 10 times, and the
correspondence is 1 to 1» so that the theorem is proved if m = 0. Next, if
along with the non-tenfold parts we introduce m distinct parts each divisible
hy lti and at the same time introduce in the corresponding partition of the
other set 10 times these same parts, each divided by 10, the partitions of
the second set will contain m parts occurring 10 or more times, while the
1 to 1 correspondence will not be disturbed»
A» Cayley11* remarked that Franklin's105 theory does more than group
the partitions into pairs. In addition to the existing division E + О of the
partitions into even and odd, it establishes a new division / + D of the
same partitions into increasible and decreasible. There is thus a fourfold
division El, 01 f EDy OD, For instance, if N — 10, the arrangement is
01 : 10, 5 + 3 + 2
SI
ED:
: 8-
94
-1,
7
4
+ 3,
+ 3H
64
-2
-4
f 1
OP : 7 + 2 + 1, 6+ 3 + 1, 5 + 4 + 1
xu Johns Hopkiiie UDiv. Circ., 86 (in full).

Chap. Ш J PaBTTTIONB. 139
where the El and 0Д each taken in order, pair with each other, and
similarly for the 01 and ED. Of course for the exceptional numbers 1, 2, 5,
7,12, ■■ >, there is just one partition which is neither / nor Д and, according
as it is 0 or Et we have in the product a coefficient — 1 or + I.
J. «Г. Sylvester117 called a partition regularized if its parts be written in
their order of magnitude, represented each part p by p points (nodes) in a
horizontal line, and noted that the conjugate partition is obtained by
counting the nodes by columns [Ferrers**]. There is given (pp. 4-7) a
method due to Franklin to construct the partitions which are to be elimi-
eliminated from the indefinite partitions of n into j parts, including zero, so as to
obtain the partitions of л into j parts ^ г, and hence to obtain the generating
function enumerating the latter partitions; also (pp. 18-21) his constructive
proof for the generating functions for partitions into repeated or unrepeated
parts limited in number and magnitude. Sylvester (p. 7) gave his own
construction of partitions of n into j parts chosen from 0, I, - —, г by em-
employing a square matrix M\ of order j in which the diagonal elements are
all г + 1, the elements below the diagonal are all unity and those above the
diagonal all zero. For 1 ^q ^ j, let M9 be the matrix whose (£) rows are
obtained by adding the rows of Mi in sets of q. Denote the rth row of
MQ by (r, q) and the sum of its elements by [>, g]. To each regularized
partition of n — [>, q~] into j parts S 0, add (r, q) term to term. The
partitions of n into j parts so obtained from M4 for all values of r are said
to form the system Pq. If P is the system of all partitions of л into j
parts, the complete system of partitions of n into j parts ш г is ,
S^P-Pt + P, +<-1)'Рл
where the minus sigu denotes cancellation, and the system may involve
duplicates as well as non-regularized partitions. It remained to prove that
a partition of nt in which the number ft of different parts is > г, occurs (;)
times in P« and hence (I — 1)" times in 8; this was proved later by M.
Jenkins.11* Hence the number of partitions of n into j parts ^ г is the
coefficient of x* in
Any integer N can be expressed (p. 15) as a sum of consecutive integers
in as many ways as N has odd factors; Sylvester119 also stated this else-
elsewhere. Cf. Barbette,301 Agronomov,20* and Mason.207
The subsequent topics treated are: generating functions, correspondence
(p. 24, p. 38) between partitions into odd parts and partitions into distinct
parts/18* and graphical conversion of continued products into series. Then
he noted (p. 60) that if in Jacobi's* formula we use the lower signs and take
UTA Constructive Theory of Partition* . . ., Amer. Jour, Math., 5, 1882, 251-330; 6,
1884, 334-6 (for list оГ errata noted by M. Jenkins). CoD. Math. Papers, IV, 1-83
(with the errat* noted by Jenkins corrected in the text), to which the page citations
refer.
X1B/W*,e, 18Й4, 331-3.
"'Cbmptei fondue Рьпн, Wt 1883, 674^5; Coil. Math. Papers, IV, 92. Math. Quert.
Educ. Times, 39,1883,122; 48, 1888, 48-49.
"* CoupUs fondue Pub, 96, 1883,111<Ъ2; ColL Math. Papers, IV, 95-96.

140 History of the Theoby op Numbers» [Chap. Ш
n = \,m ^ \ + «, where « is infinitesimal, we get
a result due to Jacob!10 of Ch. X in Vol. I of this History. Sylvester wrote
Jacobi's initial formula in an equivalent form by setting n — m = я,
n -\- m = ht and discussed at length (here and elsewhere120) the new formula
from the standpoint of arrangements of three kinds of elements. He noted
(p. 53, p. 70) that Eider's formula C) is the special case a = — 1 of
1+ xa+
1 ZC
qg)-»»(! +as') A + aafl) fci
)(lM) l d X "
which was given elsewhere by Sylvester1*1 and proved also by Cayley.1*1
Chr. Zeller1*1 stated Euler's13 recursion formula for P(n) and expressed
the number f(n) of divisors of n in terms of the P(j), j < ra> [See Vol. I
of this History, p. 290, Catalan,42 p. 292, p. 312, Glaisher,55- ш p. 303>
Stem.w]
E. Cesaro1" noted that aiXj + •■ • + a^Xk — »hasn*~7fai" "&№ — 1)!|
sets of positive integral solutions, in mean.
J. W. L. Glaisherl2b noted that Euler's theorem that there are as many
partitions without repetitions as into odd parts follows from the case r ~ 2
of the fact that the number of partitions of щ in each of which a part occurs
at least r times, equals the number of partitions of n in each of which either
r or a multiple of г occurs» In the proof, a repeated term is replaced by its
expression to base r (Glaisher8*). If P(n) is the total number of partitions
of n, and Q»(n) is the number of partitions of n in which no part occurs
more than r times,
P<»)-P(n-r)-P(n-2r)+P(»-5r)+P(n-7r) =Qr-i(n\
Qr(n)-Qr{n-l)-Qr(n-2)-\-Qr(n-5)-\-Qr(n-7) ^ or (~1)",
according as л is or is not of the form Cm2 =h m)(r + l)/2, and
P@) = Qr(Q) = 1.
Write Q = Qu There are given recursion formulas for Q, and
QBm) = P(n) 4- P(n - 3) + P(n - 5) + P(n - 14) + - > ■,
involving halves of triangular numbers; similarly for QBm -\- 1).
M. A. Stern121 proved that the number of variations £with attention to
the arrangement of the parts] with the sum n formed from two elements 1
and m equals the number of variations with the sum n + m formed from
all elements ^ m. This is the analogue of Euler's9 second theorem»
""Сотри» Rendua Paris, 96,1883, 1276-80; ColL Math. Papers, IV, 97-100»
ш Comptea RendiM Paris, 96, 1883, 674, 743-5; Coll, Math. Papers, IV, 91, 93-4.
J» Атеяч Jour» Math., 6,1884, 63-4; Coll. Math. Рарегн, XII, 2I7-S.
m Acta Math., 4, 1884, 415-6.
ш Mem. Soc. R. 8c. de Liege, B), 10,1883, No, 6, 229»
» Messenger Math., 12, 1883,153-170.
*" Jour, fur Math., 95, 1883, 102-4.

Chap. Ш] Partitions. 141
G. S. Ely noted that the partitions of n + 1 can be derived from thoee
of n by adding unity to each of the parts in turn or adding s new part unity.
Hence every partition of n into parts of which v are distinct gives v 4- 1
partitions of n + 1. If the total number of partitions of n be of parity
opposite to that of the number of partitions of n + 1, there has been a gain
in the self-conjugate partitions of n + 1 over those of n, if n > 1.
A. Cay\eyw* wrote the article on partitions in the Encyclopaedia Britan-
nica. Tbe article on combinatory analysis was by P. А. MacMahon.**5*
G. S. Elym called a compound" partition of JV,
аха± - • ■ aa | &i • ■ ■ bff | ■ ■ • | ei • • • еш,
regular if a* ^ 6, ^ ■ * - ^ e^ for every i. A graph is obtained by repre-
representing each portion by an array of points in a plane and superimposing
the planes in order. Thus any compound partition may be read in six
ways. If (to; n\ it j) is the number of regular compound partitions of w,
the number of portions being ^ n, and each portion being partitioned into
г or fewer parts ^ j> the symbol is unaltered by any of the six rearrange-
rearrangements of n, г, j.
G. Cbrystalm gave a recursioD formula which may be used to form
mechanically a double entry table for the number nP* of partitions of г
obtained from 2, 3, -• tn. Since
^((^
= 1 + »PiX+
we see by changing n to n + 1 that
4-
whence
•+Л = nP* (•**!* ■ " "> «)> *+lP»+l = n^M-l + It
n+lPn+r * i^* я+г + n* я—r+1 (Г ^ 2).
He noted that Tait18* had recently communicated similar results.
J. J. Sylvester stated and W. J. С Sharp"*1 proved the double theorem
that, if v [and vf\ is the number of ways n is a sum of i distinct positive
integers [[and ss Д ^nen
M. Jenkins1706 evaluated the number of partitions of ft into three parts*
A. Cayley1* employed non-unitary partitions (into parts > 1) and gave
the developments up to xl* of the reciprocals of B), B)C), *■ •, B)»- »F),
m Johua Норкшв Univ. Ciro., 3, 1884, 76-7! ~~ ""
«* Ed. 9, 17,1884, 614; ed. U, 19, 1911, 865. Coll- Math. Paper», XIr
»"* Ed. Ц, 6,1911, 7fi2-8; ed. 9, Supplement, 3 {*- ed 10, vol 27), 1902,
"*Amer. Jour. Math., tf, 1884, 382-4.
u$ IW Edinbuigh M^th. Soc, 2,1884, 49-50.
u» Math. Quest, Educ. Times, 41,1884. 66-7.
Ud., X07.
r. Jwir. Math., 7,1885, 57-8; Coll. Math, Paper*, XII, 27^-4.

142 History or the Theory of Numbers. [Сна», in
where (k) = 1 — xk, for application to seminvariants.
M. Jenkins131 gave a method to examine bends of a graph of a partition
without actually constructing the graphs (cf. Sylvester117), and discussed
the addition of two regularized graphs, row to row, in order.
J. B. Forney112 wrote АГ for the number of sets of values X» = 0 or 1
satisfying Xi 4- 2\2 4- - * + viK* — n. Then
It follows readily that
Л:= A%*> A: ш А
t
x _
/(a?) 6
summed for all positive solutions of Xi + 2Xf + ■ • - + m\» = i* Thus
C7 is the excess of the number of partitions into an even number of parts
over that into an odd number» Also,
D. Bancroft considered the (w; t", j) partitions of v> into j parts ^ i.
Then
(w; t,j) = A0 -j; i - lfj) + (w; ij - 1).
Taking j = v> — к and summing for & = 0, »- •, fc, we get
(w; г, w) = (w; г, to - к - 1) + £ (x; г - 1, w - x).
9
Hence, if к ^ w/2, (to;*, w — fc — 1) is expressed in terms of r^ = (r; ni)-
К ft ^ $w + a, where w is even and 0 < а ш (w + 4)/6,
(w; г, ^w — a — 1) ^ tr^ — £zx<~i +
+ (a ^ 1){2^2 + 3^,) + --> + Ba - 2)« + Ba -
This and a like formula irclude the rule by Е1у.ш
E. Catalan1" noted that, if (Nt p) is the number of partitions of N into
p distinct parts, and r(k) is the number of divisors of k,
{N, 1) - 2(JV, 2) + 3(ЛТ, 3)
= т(А0 - г(ЛТ - 1) - т(ЛТ - 2) + т(ЛТ - 5) + т(ЛТ - 7) .
г» Jour. Math., 7,1885,
J" Kout. Ann. Math4 C), 4, lSft5, 408^17.
ш Johns Hopkins Univ. Cire., 5,1886, 64.
1M Аваос. frwj9f av. всц 15, 1886, I, S6.

Chap. IIIJ PabttTIONS. 143
E. Meissel11* gave the formulas of Weihrauch74 for n = 3, 4, 5 and noted
that a synthesis of these cases gives
provided the final term of the derivative be omitted.
P. A. MacMahon1* called a partition perfect if it contains one and only
one partition of every lower integer; sub-perfect if, when each part is
taken positive or negative (bat not both), it is possible to compose every
lower number in only one way. Thus, 3 + 1 is sub-perfect since 2 = 3 — 1,
3 = 3, 4 =* 3 + 1. Аду factorization
leads to the perfect partition (XV"---)oiu; then
Formulas involving the number of partitions of и are given. For sub-
perfect partitions, use
instead of ф, and divisors of 2u -(- 1 Instead of those of и + 1.
E. Catalan*" noted tbafc
logA + x + a*.+ ...)«_ logA -x) = x + j + j
Developing each exponential, we get Jacobi's result (Jour, fur Math., 22,
1841, 372-4)
• Г(а 4- 1)Г(& + 1)Г(с + 1) - - -
where the summation extends over all solutions ^ 0 of
Since the denominator equals 1-2 -a-2-4-6- --2&-3-6- 3c- -f we see
that if n is partitioned in all ways into parts or, 0, % - - - belonging to pro-
progressions with the differences 1, 2, 3, ■• -, the sum of the fractions l/(o07- ■ ■)
is unity.
W. J. C. Sharp stated and H. W. Lloyd Tanner* proved that, if PH
or QH be the number of partitions of n without or with repetitions, then
Q* = P+
ш Cber die ЛвпЫ der Diuvtelhiiieen aner доеЬепеп Zahl J dunk die Form A p,
in welder die Я gegebtme, enter rich verechieden<s РхшздЫоо emd, Progr. Kiel, 1886.
Шв Д_! Ьав been changed to/, to conform to Wethrauch's notation,
m Quar. Jour. ВЫЬ, 21,1886, Зв7-373.
w Шт. Soc Hoy. №. de Liege, B), 13,1886, 314-8 ( - Mebuigee M&tk П)
"* Math. QnesL Edoe. Timee, 45,1886,123L

144 History of the Theory of №швенэ. [Снлр.Ш
and two similar relations. There is a list of unsolved questions on parti-
partitions by Sylvester.117*
P. G. Tait138 considered in connection with knots of order n those
partitions of 2n with no part > n and no part < 2. After the largest part
is removed, the numbers left form the partitions pi, pl^[, • ■ >, pi*-*, where
p\ is the number of partitions of s with no part > r and none < 2. If
г > 9t pT4 = pU If r < s, the above argument shows that
p: = pr*-r + pT-\-v + - ■ + ?;_,.
There is a table of values of p[ for г ^ 17, s ^ 32.
E. Pascal*" used n numerical functions /,(x) which increase when x
increases. Let the difference of two values of/x for two successive integral
values of «i be unity. If x*_i < xk and
/4(x -{- 1) - /*(*) > /*-i(x), /*-i(x*-i) < U(xk + 1) - /*(**),
every number is expressible in the form fi(xk) + •• • -r-/«(z») in one and
but one way. As corollaries, every number N can be expressed in one and
but one way as a sum of n decreasing binomial coefficients:
N *= fe)i + (*l)l + • ■ • + (Xn)n, Xk
also as a sum of n increasing binomial coefficients:
ДГ - PIU + [8J, + • • • + [n + 1\, x
E. Sadun1** considered the number &(n, r) of sets of integral solutions
of the pair of equations, in which г Ш nt
Xi + ^2 + ■ ■ ■ + XA = r, \i + 2X3 + ■ ■ ■ + nX» « «.
^ n, the pair of equations have as many solutions as the equation
<*i + 2ar« + • ■ • + r«r = n
has integral solutions ^ 0 with ar > 0, or as the system
*i + 2*, + ■ ■ • + (r - l)^_t = n - tr (* := I, 2, • ■ -f tn/r])
has solutions. Hence we can compute s(n, r). For r = If the equation
is a* = n, ai > 0, whence s(n, 1) = I. For r = 2, the system is
«! =»-2, ftX - n - 4, ...f ttl
whence s{n, 2) ^ [n/2]. FinaUy, he identified s(n, r) with a function
connected with a linear differential equation of order n.
P. A- MacMahon1<x employed symmetric functions as an instrument for
the study of partitions and other problems of combinations. He considered
n objects specified by (pqr- - ■), p -f- q + ♦ •• = n, meaning that p objects
»*» Math. QueaL^duc Timee, 45,1886, 133-7. One is proved by Sbup,Vi 1887^ 139-140
»*Тткп&. Roy. Soc. Edinbur^, 32,1887, 340-2.
U1 fflomale rfi Mat., 25r 1887, 45-9,
i* Annali di Mat.r B), 15, 1887-8r 209-221.
ш Ptoc. London Math. Soe., 19,1887-8, 220-256. Cf. 28, 1896-7,

Chap. Ш] PARTITIONS. 145
are of one kind, q of another kind, etc. The general problem of cooibinatory
analysis is to enumerate, under various imposed conditions, the distributions
of the n objects amongst the m parcels specified by
(prfi'--)> Pi + tfi 4- •" = m,
when the arrangement of the objects in a parcel is immaterial, and when the
arrangement is material. The solution is effected by identities between
symmetric functions. To pass to the special case of partitions of n into m
parts, consider the distributions of n similar objects (n) into тп similar
parcels (m)f it being allowed to place more than one object in a parcel. In
the partitions of multipartite numbers, we distribute objects (pqr- • ■) into
parcels (m).
G. Plainer141 found for г ^ G the number ф(г, n) or ф(г, п) of ways of
forming a sum n or a sum ^ n from r terms of 1, 2, 3, ■ • ■. For r = 2,
the result is q + x ~ 1 or g2 + (я — l)qf respectively, if n = 2g -f- xt
x < 2. In the second paper, ha expressed the results as functions of n.
For example, the number of pairs with the sum n is (n — k)/2} к *= 2 or 1
according as n is even or odd; the number of pairs with a sum ^ n is
fa* — 2n -f- 0/4,1 ^ 0 or 1 according as n is even or odd. For r = 3, 4, 5, 6
the formulas involve a parameter with listed values for the least positive
residues of n modulo 0, 12, 60, 60, respectively. It is proved that
the resiilte for ф are due to I>e Morgan,*8 Herschel,18 Kirkman,8* etc.;
while the results for ф follow readily from those for ф.2
Schubert148 noted that 10m Pfennige can be made up of 1, 2, 5 and 10
Pfennigs coins in 1 4- 10mi + 19*n2 + 10mj ways, it mi = G)> and treated
two similar problems.
G. Chrystal141 collected theorems on partitions and introduced various
notations.
Bellene and Vemiory14* found the number of sets of solutions of
s-f-y-f-z = я -f- 2, xfy> z chosen from I, • ■ •, n, by grouping the solutions
corresponding to a fixed x, and separating the cases n ^ 0, • ■ -, 5 (mod 6).
M. F. Daniels1* obtained the results of Weihrauch74 another way.
P. A* MacMahon147 enumerated the perfect and sub-perfect partitions.
For example, if a is a prime, there are 2" perfect partitions of cf — 1. If
a, hy — - are primes, a*lf > ■ ■ — 1 has as many perfect partitions as the multi-
multipartite number (or, 0, - ■ ■) possesses compositions (partitions with attention
to order).
8. Tebay147* found the number oi ways * is a sum of i distinct integers,
also when each part is — q.
шКвм^«дайК, let. LombardodiSc. Let,, B), 21 r 1888, 690-^5, 702-8.
10 Mitt. Math. Ge«JL Hamburg, 1,1889,269 Cf. сГОодкР* ff С* П
ш Algptmn, 2,1889, 527-537; ei 2t voL 2,1900, 55W565.
«• Mfttbeeis, 9, 18S9r 125-7.
"• Uneaii« Co^groenties, IXn^ Amrterdam, 1890,120-135.
*" Meeeengw МиЫ, 20,1Й91, 103-119, Cf. MMb"*
"* Math, QuttL Educ. limee, 66,1892, 34-37.

146 History of the Theory of Numbers. [Сна*, hi
L. Goldschmidt14* gave an elementary proof of Jacobi's*0 theorem on the
excess (P, a, 0, • ■ 0 of the number of partitions of P into an even number
of the a, 0, - • - over those into an odd number of them, and showed that
(P,l,2, ...,m-l) = (P, 1,2,3, .-0 + (P-m,2,3, '■>)
+ (P-2л, 3,4, -••}+■'••
His proof of Euler's formula C) is essentially the same as Franklin's,18* as
admitted, ibid., 39, 1894, 212.
J. Zuchristian1" proved, by means of Euler's recursion formula for the
number nk of partitions of n into к parts, that щ is the integer nearest to
(n + 3J/12, while
according as n is congruent to an odd or even number к modulo 12, while
if*8 lif
f
K. Th. Vahlen150 wrote N(s *= 2a^) for the number of partitions s - 2a^
Consider a partition 8 = Ze%<n where the v elements o,- are distinct. If we
select X of these a% say d, — >, ^, the partition may be written
(8) з ^£5^ + 2*^,
Consider all possible partitions (8). The excess of the number of those
for which X is even over the number for which X is odd is denoted by
and is proved to be sero. It suffices to prove this for the partitions (8)
which arise for any one а = Ze&i. From the latter we get Q partitions
(8) for each X; since X has the values 0, 1, - - -, v,
He proved analogous formulas. Next (p. 10), from the theory of elliptic
functions, we have
which, if R(n) denotes the absolutely least residue of n modulo 3, may be
written
^C- 1}ь),1ог2й(»<) - Л,equals0unless*-(ЗА1-)/,
and then equals (— 1)\ Or, in words, among those partitions of s into
wZeitecbrift Math, Phye., 38, 1893 i2i-«; P^ogr. <l hfiheren Handebchnle, Goth», 1803.
«♦Mraatdiefte Math. Phye.,4, 18ЭД 185-9. Cf. Gloeel.1»
w Jour» fQr Math,, 112, 1Й&3,1-36. Cf. von Schiuti»,1"

, 1Щ Partitions. 147
distinct positive summands in which the sum of the absolutely least residues
modulo 3 of the sutmnands equals a given positive or negative number A,
there occur as many partitions into an even number of summands as into
an odd number, except only when s is the pentagonal number (ЗА* — h)/2t
for which there exists an additional partition into an even or odd number of
parts according as h is even or odd- Also a purely arithmetical proof is
given. If we employ this theorem for each of the permissible values of
h and add the results, we get Legendre's11 result:
^; (-1)*).
These theorems are extended (pp. 16-17) to тп-gonal numbers.
T. P. Kirkman1*1 took all partitions of x into k parts ^ 0, as 0 0 5,
1 1 3, 2 2 1, 0 1 4, 0 2 3 for x = 5, fc - 3f formed their permutation
symbols, 3o*b -f 2abc, counted their permutations 3-3 -f 26 — 21 = ($),
and stated that the result is always ('t-Г1)- There is a question on the
partition of & polygon of г sides into к parts, treated later (ibid., 8, 1894,
109-129); cf. Cayley.1"
P. Bachmann1" gave an exposition of the work by Euler.
P. A. MacMahon1** considered compositions, i. ev partitions in which
the arrangement of the parts is essential. ТЫ number of compositions of
n into p parts > 0 is the binomial coefficient £lj). The total number
of compositions of n is 2*~l> If the parts are ^ s, the number is the
coefficient of z* in (x 4- x* + ■ • ■ -f- sO*. A multipartite number pipi7^
specifies p\ + Pi H~ ■ ■ - numbers (or things), px of one sort, pt of a secotid
sort, ete. The number of its compositions into r parts is the number of
distributions of the pi -f- p* -f- — - numbers into r parcels and is the coeffi-
coefficient of а1Ла1Л» • * in the expansion of (ht + &t 4- ■ • •)*> where К is the sum
of the homogeneous products of degree & of <*i, a*7 * -. ТЫ graph of a com-
composition B, 1, 4) of 7 is given by placing nodes at pointe P, Q on the line
AB divided into 7 equal segments, so that in moving from A to Б by steps
proceeding from node to node, 2, 1 and 4 segments of the tine are passed
over in succession. The graph of a composition of a bipartite number pq
is derived by placing nodes at suitable points on q + 1 ymibi.r graphs of p
placed parallel and equidistant and with corresponding points joined by a
second set of parallels. Let A and В be opposite vertices of the resulting
total parallelogram [see figure, MacMahon*"^ P*88 from A to В by
successive steps, each consisting in moving a certain number of segments
parallel to AK and then moving a certain number of segments parallel
to KB. The successive steps are marked by nodes, which define the graph
of a composition. An essential node is where the course changes from the
«Mem ft»d Proc. Manchester lit. Phfl. &*., D), 7, 1893, 211-3. Math. Quo*. ^
TSmo, 60,1894, 98-102.
ш Pew. London Math. Soc., 22,1891, 237-262; CoIL Mmih. Ржрехв, ХШ,
ш ZdU«itbeorie, 2, 18M, СЪ. % 13^15.
*" PhiL Тглвди Roy. floe London for 1893,184, А, 1ЯМ, 835^901.

148 Histohy of the Theory of Numbers. [Chap, ш
KB direction to the AK direction. The number of different lines of route
with exactly s essential nodes is (C)(!)- Each of these tinea of route
represents 21J+«~f~1 compositions. For tripartite numbers, we need three
dimensions. Generating functions were found for the number of all
compositions of multipartite numbers; helb**1M treated this topic also
later.
K. Zsigmondy1" partitioned m into distinct parts each unity or a product
of distinct ones of the first s primes; for example, the parts may be 1, 2, 3,
5, 2-3, 7, 2-5, 11. If the partition has an even number of parts, consider
the excess of E of the number of parts with an odd number of prime factors
over the number of terms with an even number of prime factors, unity
being a possible term. Thus for 11 =2-3 + 5, IS = 0; for 25 + 1,
E = - 2; for 5 + 3 + 2 + 1, E = 2. But if the partition has an odd
number of parts, let E be the excess of the number of parts with even
over that with odd number of prime factors. Thus for 2-3 + 3 + 2,
or 7 + 3 + lor 11, E = — 1. The sum2n of them's for these 6 partitions
ofllisO~2 + 2-l-l-l--3. Next, <rn - 3 - 3 = 0 is the
excess of the number of the partitions of 11 into an odd number of parte
over those into an even number of parts. He proved that, if m > 1,
2« + *m-i = 1 or 0, according as m is the (a + l)th prime p or is < p.
For example, if p « 13, m ** 11, we had 2Ж = - 3, while «r^i = 3 since
the partitions of 10 into an odd number of parts are 2-5, 7 + 2 + 1,
2-3 + 3 + 1 and 5 + 3 + 2, while 7 + 3 is the only partition into an
even number of parts.
W. J. С Sharp stated and H. J. Woodall16*1 proved that, if Fft is the
number of partitions of n without repetitions and Q* is the number of par-
partitions into odd parts, then Ря = Qn + Q^iPi + Qfu-*F> + ■ • ■, and that
the same formula holds when Pn and Q» denote the number of such parti-
partitions with repetitions.
L* Eamonson1** expressed the number of partitions of 2n into two
primes in terms of the number of odd primes ^ к for various values
oik.
L. J. Rogers18* established the important identities
i . " . L _*_ " ♦
» PhiL Trane, Roy. Soc. London for 1894,185, А, Ш-НЮ.
1» Monatahefte Math. РЬуя», 5,1894,12ДО.
"" Math. Qiicet, Ednc. Times, 60,1894, 41.
"»JW&, 63,1895,116-7.
c. London M*tb 8oc.? A), 25, 1894, 328-9, formulae A), B). (X. papers 226-8.

Chap. Ill} РавппОКЗ. 149
where, on the left, the exponents in the numerators are n* and n(n -f- 1).
G. Brunei167 considered two sets of n points such that from each point
of each set issue two bonds connecting it with two points or a single poirt
of the other set. Each such configuration can be considered as the result
of the juxtaposition of polygons of 2ku ' • ■, 2kr sides, where
ki + • • ■ + kr = n.
Regard two configurations as identical if, after a permutation of the points
of each set, the bonds are in the same order in the two. For the number
Л*, г of configurations relative to n and rf
J. Hermes1*8 noted that the number of compositions \j& by Mac-
Mahon1**] of m into к parta ^ p is ("+*Ii"*0" There are 2m~1 compositions
of m\ each defines the elements of a Gauss Klammer [V*, . - -t pj, occurring
щ continued fractions {GaussM of Ch. II); they give the 21Я~2 Farey numbers
of the (m — l)th set, each taken twice |]вее Vol. 1, p. 158 of this HistoryJ.
Hermes1" generalised Euler's9 formulas on the number of partitions.
If 8, tt n are integers ^ 0, let E$, t{n) = Etm ,(n) be an integer such that
E(Q) = 1, En(n) *= 0 if n > 0, and
E$I,(«) = E9i t(n - I) + ЕШь ^(п).
For ^ = 0, -ff*. o(n) is the number of partitions of n + 5 into a positive parts.
Several recursion formulas are proved, including
The niunber of partitions of n -f- x — 1 into it — I terms chosen from
1, ■■-,* + 1 is
unless n > sx — s, when the sum is aero. Properties of the A's are given.
A. Thorin140 asked for the integer к for which the number of sets of
positive integral solutions of diXi -f- • ■ ■ 4- <*nXn ^ bt xx + * ■ ■ + x» ** к
is a maximum.
UT Proc6^-v«rfaftux dee aeancee eoc, dea ec. phys. nat. de Bordeduxr 1894-5, 24-7.
^ Math. АддлЗеп, 4$, 1894, 370-80.
«• 7Wrf.p 47,1896, 281-297.
119 L'intenttediaire dee math,, lt 1894, 181-2.

150 History of the Theory of Numbers. [Cka*. Ш
"Itotdv" tinted the last question for л = 3. Take the greatest integer
Xi ^ (b — ax — oj)/ai. In the first of the pair of equations, replace x%
by Xz- Then if a\X\ + QtXt = b — а*Хг has integral solutions X%t X?, the
required к is X^ -f- X* + X*.
M. Kuschniriuk182 proved that, if I\(wi) is the number of partitions of
m into A parts > 0, then
Л! "
R. D. von Sterneck1" considered the number {n\ of ways of obtaining
n additively from alt alt ■ ■ ■, using ui at most кг times, as at most кг times,
etc. The number of these representations of n in which the element a<
occurs at least once is
where k\ = &, 4- 1* This is used to prove that the number of representa-
representations of n as а вит of an odd number of distinct summands is odd if and
only if in the decomposition of 24n -f- 1 into primes either a single exponent
is odd and of the form 4£ -f- 1 or no exponent is odd and there is an odd value
to the half sum of the exponents of those primes which are = lf 5, 7, 11
(mod 24). He also found the condition that there be an odd number of
those representations of n by distinct summands whose number is an odd
multiple of 3 (or of 5 or of 7). Finally, he drew similar conclusions from a
general theorem due to Vahlen.1*
A. R. Forsyth1*4 expanded the reciprocal of the product
of n pairs of factors, suppressed every term with a negative exponent for
any of the symbols a, b, • • >, and in the surviving terms replaced each a, b7
... by unity, and proved {in accord with a conjecture communicated pri-
privately by MacMahon) that the sum of the resulting series is the reciprocal
of
He gave a similar theorem when each pair of factors is replaced by т -f- 1
factors.
G. B. Mathews1" showed that the problem of multipartite partition
is reducible in an infinitude of ways to a problem in simple partition. For
example, every set of integral solutions ^ 0 of
ax-\-by + cz-i-du> = int a'x -f- Ъ'у -f- cfz 4- d'w = m'
ш L^ntermediaire dee math.r 3,1896, 249-250. ———
^Progr., МаЪгУГгЩиш, 1895. Quoted from Netto*1" 128-130.
"•SiUuneeber. Akad. Wise, Wien (Math.), 105, Ila, 1896, 875-899.
«ftnc London МдОь fioc., 27, 189ОД
Ш1Ш.Р 28,1896-7, 486-490.

Сялг-
PaBTITTONS.
151
ш a set of solutions of
(\a -f
■ + ■ • ■ 4- (xd +
Conversely, if \f p are suitably chosen positive integers, every set of solu-
solutions S 0 of the latter is a set of solutions of the pair of equations.
K. Gloseliw considered the number C-(o-) of ways of expressing a- as a
sum of г distinct positive integers, gave a new proof of De Morgan's2*
formulas for г = 2, 3, and, for r =* 4, simpler expressions than Zuchris-
tian's*14* If («} is the integer nearest to a,
C.Bfc + 1) =
2k{k - Zf
1 36
which may be combined into
i.
The complicated expression for Сь{о) was simplified on page 290.
P. A. MacMahon1** gave a report on combinatoiy analysis and parti-
partitions. He suggested (pp. 30-1) a method of enumerating multipartite
partitions.
MacMahon"8 noted that a partition (pi ■ • >p&) has the " separations "
(рфг)(РяР*)(Рб), (piP2p*)(l><Pb), ete., the numbers in any parenthesis being
considered as a partition with those parts» It is easily proved that the
number of separations of the partition (рГрГ* ■'), where тгх indicates the
number of repetitions of the part pl7 is identical with the number of parti-
partitions of the multipartite number irinv ■ ■ Sylvester's method of graphical
representation of partitions can not be simply extended to multipartite
it
6
•
с
•
•
•
•
partitions. But there is a correspondence between m-partite partitions
and (m + l)-partite compositions. For example, let m = 1 and consider
the graph of the bipartite number 76. Each composition has a line of
route through the lattice Qas MacMahon164} a, b, с being the essential
nodes of the line of route shown in the figure. The principal composition
lm Monatebefte Math. Phys., 7, 1896r
w Pwe. London Math. Soc., 28, 1S96-7, &-32.
*• Phfl. Тлпя. Boy. Soc. London, for 1896r 187, Ar 1897» 619-673. Меиммг Г on Partitioned

162 History of тнв Thbort or Numbers. ССнлр. Ш
is D1 12 11 12), since 4,1 are the coordinate» of a referred to the origin A,
1, 2, the coordinates of b referred to the origin a, and of В referred to the
origin с The nodes in the lower portion Ca- • -cDK form a Sylvester
regularized graph of the partition C 2s 1); similarly for the nodes in the
upper portion.
Again, we may think of Sylvester's graph : : ■ *, not as representing
the partition C 2), hut as representing the multipartite number 4, 2.
Then consider the partition D2,31) of the multipartite number 4 + 3,2 4- 1.
By placing the graph of 3, 1 upon the former graph, we obtain a three-
dimensional graph of the partition. Such a graph can in general be read
in six ways. At the end of the memoir are conjectures as to the generating
functions of partitions whose three-dimensional graphs are limited in height,
breadth and length.
R« D. von Sterneck1" proved Legendre'e21 theorem and deduced from
it in a simple way Vahlen's160 extension. He proved also that, if £ is not a
triangular number and if we represent & as a sum of integers so that the
same part is not used oftener than 3 times in the same representation, then
among the representations which contain p distmct parts less often than 3
times there are as many sums of even as of odd parts. If \{n — h) is not
triangular, among the representations of n by distmct summands for which
the sum of the absolutely least residues of the summands is — k (mod 3)
and in which occur p pairs, each pair being two of three numbers of the
form 3m — 1, 3m, 3wi + 1, there are as many sums of even as of odd parts.
Corresponding to the last two theorems there are more complicated ones
for triangular numbers,
J. Franelm stated that, if a, b, с are positive integers, relatively prime
by twos, and if n is a positive integer,
(9) ax + by + cz = n
has n{n 4- л 4- b 4- c)/Babc) sets of integral solutions ^ 0, if we neglect a
quantity whose absolute value remains, for every n, less than a fixed number.
E. Barbette171 considered (9) for a, &, с positive, a and Ь relatively prime.
If a, fi are particular solutions of ax + by = 1, then
x ~ a(n — cz) -t-Ъв, у = 0(n — cz) — ав
are the solutions of (9). Let к and h be the quotients obtained when n
and с are divided by <xb\ then the number <o of positive integral solutions is
*[2& ~(q + 1L - 2Hj,
where q is the largest integer ^ nfc If л is divisible by bf and с is divisible
by ab, set Я = c/ab and call К the largest integer ^ n/ab; then
- (q
wr. Abbd. Wi». Wien (Math.), 106, Щ, 1807,115-122.
"* L'mterm6diaire dee math., 5, 1898, &L
tn МмЪееиц 0), 5,1906,125-7.

C&af. Ш] PARTITIONS. 153
P. A. MacMahon17* found the number of ways и is a 011m of 8 numbers
nt n, m n*
f»i 7П% Pit fH4t
two sohitione being identified if one can be derived from the other by a
permutation of the two rows or of the four columns. This question of bi-
partition is solved also when the number of columns is arbitrary.
H. Wolff17* evaluated the number F» (n) of partitions of n into p positive
integers X( arranged in order of magnitude, Xq ^ % Ш *i ^ - - - = x+-i,
and proved that
where the summation extends over all decompositions /* *= /? 4- ffv + * - •,
while, for each, Ф(п) is the number of partitions of rt into/ sets of ^successive
equal parts, followed by g sete of 7 successive equal parts, etc., the various
groups not being arranged according to the magnitude of the parts. Thus,
for example, n=*4 = 0-f-0-f-2-f-2 and 2-f-2-f-0-f-0are counted as
distinct in computing ф(п).
The number of decompositions of n into A equal parts is evidently 1
or 0 according as л is or is not divisible by X, and hence is
if R(nfk) denotes the least positive remainder on the division of n by X.
If X is the g.c.d. of £, 4, ■ • -f the above ф(п) equals the product of p(nf X)
by the number Ф(п/\) of partitionB n = /$/X + ffv/Ь + 4 • • ■ Again, the
number of decompositions n = /f -f- ^17 is p(n, ?y) + C^tVfJ ~ Lnit'/vl> ^
£f 17 are relatively prime And fa' — i|f' •= т 1. Recursion formulas for the
Ф*8 are found and the FJj%) evaluated for ^ ^ 6 as explicit functions of я.
By means of ВетоиШап functions, FJji) is expressed as a polynomial in
n whose coefficients are linear functions of the coefficients of ^^(л).
* G. Csorbal7Se made an addition to the theory of partitions.
P. A. MacMahon174 generalized the concept of a partition into parts
<*i, otit - - ж7 <*ш by replacing the conditions <*i ^ аг ^ * - * by the conditions
А»*ж + АТаш+ •■■ +A?o.ZQ (i-l, -^,r),
where at least one of the integers A is positive. There is a finite number of
fundamental solutions (a?1, - ■ -, a^O for j — 1, * - -, *»of these conditions,
such that every solution ie of the form a< *= Xio^^ + ■ • • + ^dF for i = 1,
•< ■, *, where the Xfs are positive integers.
MacMahon171 treated the generating functions for the enumeration of
tbree^limensional graphs possessing either xy-eymmetay (when each layer
"* ВиШ вое. bUtb, Fr*zicef 26,1898, 57-64; M. d'Ocagne, p> 19, fот = зД
1ЛиЬвг die Anxahl der ZeriecaneeQ «нет ganxen Zahl in Summanden^ Div., ШОе, 1899,
"»• Matb бв tenn&j ertntft ^tm^iian Ac&d. 8c.)t I7t 1899, 169.
14 Phil. Тгада, Roy. Soc London, 192, A, 1800, 351-Ю1. Memoir П on Partitions
m Тпшв. Cambridge PniL So^, l7t 1899, 149-170.

154 Histobt of the Тнххяст of Numbers.
of nodes is symmetrical in two dimensions) or ,xyr-symmetjy (whai the щ
forms obtained by rotations about the various axes are identical).
MacMahon,17* to enumerate the oombmationB defined by certain laws,
would find an operation and a function such that the result of performing
the operation on the function grves the number of combinations* Thus,
operating with (djdx)* on x* we get the number n! of permutations of n
distinct letters. Again, let dx = d/daj 4- a^d[da± -J- thdfdai + ■ ■ -, where
the a's are the elementary symmetric functions of oti ■• -, <*.< Using
symbolic multiplication as in Taylor's theorem, write D, = dH&t Then,
operating with Dw% - - - DWm on fat + - - - + <*„)* we get the number of
permutations of «Г1 • * • «I" where Zr» = n. finally, if we apply DJ^\
to the symmetric function (I4)(l*)(l)f where (Iя) denotes a* = Xcti ■ - - a*
in partition notation, we get the Sylvester-Ferrers9 graph of the partition
C 2* 1) or its conjugate {4 3 1), according as it is read, by rows or columns»
The method is successful in solving the problem of the Latin Square1** in
its most general aspect, Cf. Hammond.217*
R. D. von Stenieck,177 to extend УаЫеп'в1* work from modulus 3 to
modulus 5, considered the excess \n\kof the number of representations of n
by an even number of summands over the number by an odd number of
summands, where the summands are distinct and the sum of their absolutely
least residues (— 2, — 1, 0, 1, 2} modulo 5 has the value A» He proved the
recursion formulas
jfc}A *= ik-2h + 3}»-\ {k\h = - I* - 5ft + 15)*^.
By successive applications of the second, we get
Hence its value depends on certain iJJ'forj = 0, dr 1, =h2- By Lagrange's
theorem, 2{k\k ^0or{- 1)< for к + or k « CF ± i)/2,
where h ranges over the integers = к (mod 5). This gives a recursion
formula for {k]*,j = 0, ± 1, ± 2- Hence we can compute any {k\h*
M, d'Ocagne178 found the number of ways s francs can be formed with *
French silver corns E, % 1, £, i francs), also when the number of smalfeftt
corns is fixed.
R. D. von Stemeck"9 gave an elementary derivation of the number of
decompositions of л into six or fewer equal от distinct positive integral
summands, distinguishing 29 types like п = <х + а + р + Р, often with
various sub-cases. Thus the results are expressed by many formulas,
E. Netto1" employed eight symbols for the various types of combinations
and variations, with a prescribed sum, of given numbers taken k at a time,
"Trans. Cambridge Fhfl. Soc,, 16, 1898, 262; FbiL Trims. Roy. 8oc, London, 104, A»
1900,361.
i" Sitmngeber. Ak&d. Wke» Wfea (Math/), 100, Ш, 1900, 28-43.
"■ Bull. Soe. Math. Fiance, 28, 1900, 157-168.
J" АгсЫт Math. Phys.r C), 3.190lF 195-216.
** Lehrbueh der CombiiLatorik, 1901.

Свлг- ШЗ Радтптокй. 155
with or without repetitions. In Ch~ $, he gave an exposition of Euler'a
work on partitions and Sylvester's theory of waves, illustrated by examples*
In Ch. 7 it is noted that any relation between two partitions of n leads to an
identity between two infinite series.
A. S> Werebrasow** noted that if а, Ъ, - •, h,l are positive integers and
if \n] denotes the number of sets of positive integral solutions of
f^ax-\-by -f -• -\-Н^щ
the nmnberof sets for/-f^w =* m is {m — l\ + \m — Щ 4- \m — 31} 4- • • •.
IX Gjgji15* considered the number ЛГ, of combinations of 1, •.-, m taken
n at a time with the sum *. The least * is L = л(я -f l)/2 and the greatest
is G — отл — nfw — I)/2. It is shown by induction that NLy NL+U * • •, JV^
are the coefficients of the powers of or in the expansion of
С. Г. Gauss182 bad treated this function without reference to partitions and
noted that
•i-i
(wi, n) = (m, m - n), (от, j» + 1) *= £ ^"M(t", /*).
Gi^i tabulated the iVs for ш =^ 10, n ^ 2, 3, - -, and proved that
0», n) = "^'«-^«(да - p, л - 1).
T. Muir18* noted that there are C^kr+h.r combinations of n elements
taken т at a time such that no element ш taken along with any one of the к
elements immediately following it in the initial set. The number of sets
of r things obtained from n by omitting n — r of them so chosen that they
form(n — r)/£flefeof fr consecutive thing* is Ca.r, where 8 *= (n -f кг — r)jk.
EL Landau1M discuBsed tbe maximum order of literal substitutions on
a given number n of letters. It is a question of the maximum of the Lc.m.
of at, * - *, a, in all decompositions n — fli + — - + о,, of n into positive
integral stunmands. Of. Landsu.1M
E. Netto115 found the number of cyclic decompositions obtained by
arranging in a circle each of the (p|) decompositions of n into p summands
with attention to order.
L, Brufiottt1M proved tbe result of Catalan'^»
F. H. Jackson117 wrote P* for p? - p^T and [рг]* for
A + i*-^)(l + M) •••(!
гт. ВршяавекГв Bate, Ockas^ 1901, N^ 298-0, Pp. 224-9,250-1.
"> Eendkoati Ore. Mat. Ffcfermo, I6> 1902,28О-6.
«Coram. Sac, Gottb&, IT Mil; Werte, П, 1И7.
'« Proc. Roy, jSoe. £dmbtfTKbr 24,1Q01-3, 102-4,
i" Arcbiv Math. РЬув^ C>f 5,19ОД, 92-103.
«/Ш, 185^186.

156 Ншгонт of the Тняовт of Ntjmbebs. C^haf, ш
which reduces by cancellation to A + P-z)(l -f P^V) > - A -f i*«-i>)
if л is a positive integer. The simplest of the general formulas proved is
which includes as special cases formulae of Euler*-' and Cauchy."
A. 3. Werebrusow"* gave a recursion formula for the number of sets of
positive integral solutions of а&г + • • • + a*^« = A, where the positive
integers a have no common factor. Then he considered the number of
sets when at least one x is ^ 0.
P. A. MacMahon1** treated a " general magic square," consisting of n*
integers (zeros and repetitions permitted) arranged in a square such that
the rows, columns and diagonals contain partitions of the same number
(whereas in an ordinary magic square the n* integers are 1, 2, • • -, n').
The treatment applies to all arrangements of integers which are defined by
linear homogeneous Diophantme equations or inequalities such that the
sums of corresponding elements of two solutions give a solution Qcf. Mac-
Mahon174!
0. Meissner noted that to decompose n into positive integral sum-
mauds whose produet is a maximum, the summands must be equal or differ
at most by unity, and must include as many threes as possible.
G. Mijmosi1" wrote с for the number of sets of integral solutions Ш 0
of atXi + • • • + a*^ = ny and <r{j) for the sum of those of au - —, a*
which are divisors of j, and proved the recursion formula
Taking t =^ 1, - -, n, we obtain п]ся as a determinant of order n. If each
a< = 1, then <r(i) = m and с» ю the number of combinations of m -f n — 1
things taken n at a time.
S. Minetola115 wrote Rm. „ for the number of different ways m distinct
objects can be separated into n groups, where n ^ m. For example,
^4pi = 7, the separations being Oi — а*ал, - -, 04 — a^^at> а±а% —
ata» — ^гОч^ ai<h — л^дз. We have
*•
( n _ jt )
^ЛШе^. Sbonuk (Math. Soc Пока»), 24,19M, G62HSffi!
JWPhil.Tram.Roy.Soc.London,205,A, 1906,37-68. MemoirШa*Partitions, Abstract
in Proc Roy. Scx^ 74,1905, 318.
1>v Math. Naturw. BUtLerT 4,1907, S5.
»« Periodico di Mat., 23, 1900, 173-6.
l-Giornak di Mat, 45, 1907, 333^366; 47r 1909,173-200. Сапх^янц ввпегаНш^шв and
eimplificatiom in Д BoH_ di MateMtieA Gmt. Sc-DidmJ., Rome, llr 1913,34-Ю, with
«rrat& corrBCtod pp. 121-2.

р. mj Рдвятпокгз. 157
which for k ~ 1 becomes
The number Rm. » of ways of separating m like objects into n groups is the
number of partitions of m into n parts > 0. Let k — m — n. Then
There are as many partitions of m as partitions of 2m into ro parts. Recur-
Recursion formulas are found for the number N of ways of separating into n
groups ш = I + ofi 4- -b- -f«* objects* of which I are distinct, but one
is repeated at times, and the last ecu times» Thus if the objects are a, a, a,
by bf c, d, then I = 4, «i = 2, a» — L There are JV factorizations into л
positive integral factors of a number which is a product of m primes not
necessarily distinct.
MmetoIalw proved by use of Bn -f l)Bn' -f 1) = 2k + 1, ete,, that if
2k + 1 is decomposed into a product of h primes, the h — 1 equations
2nn' + Zn = kt 2fain\n\' + 2Хпгп\ + 2nt = k,
admit Д*р i, ДА, 3t ■ ■ ■ sets of positive integral solutions, respectively»
P. A. MacMahon1" used the example of a permutation 3, 1 | 4 | 5, 2 of
the first five integers separated into compartments with the numbers in each
arranged in descending order; the succession of numbers 2, 1, 2 giving
the size of the compartments is a composition of 5. He found the number
i^(a,6, • ••) of permutations of 1, • —,n having as the descending specification
(corresponding to 2, 1, 2 in the example) a given composition (a, 6, • • •)
of rt. He proved that
= N(ax *
and similar formulas. He found the number of permutations of 1, - - •, n
whose descending specifications contain a given number of integers. He
treated the analogous problems for permutations of numbers not all dif-
different, and problems on packs of cards. The number of permutations of
of1 • • • or? with descending specifications of m parts is the coefficient of
Х'^а?1 • • • op in the reciprocal of
1 - Sffa + A - XJ«№ - A - X1)^^^ + ■ • ••
His1** study of this generating function is continued here.
, 7,1909, 43^54, for the number of «даЫмЛюпй of tbeee m objects n
At л time.
«Ibid., 47,1909, 305^320.
^ PhiL Tram, Key. Soc L^idon, 207, A, 190^45-134. Afstnot, Proc Key. Soc., 78,
O74S96

158 History of the Theory of Numbers. [Chap, m
MacMahon1M applied his1** second memoir to find the probability that
in the election of P by m votes to Q's n votes (m > n) the order of the ballots
is such that P has at all times more votes than Q, and similarly for n candi-
candidates.
Start with any Ferrers* graph of an ordinary partition and place the
parts of the partition at the nodes so that the numbers in a row, read
from west to east, and in columns, read from north to south, are in descend-
descending order. We obtain a two-dimensional partition of 19:
3 2 2 2
2 111
2 1
2
E. landau1*1 considered the maximum value /(ft) of the Lc.m* of <*i,
• • -, a, in all the partitions of n into positive parts, n = ai 4- • • • -1- a,
(p Ш n). Thus, for» = 5^4-fl = 2-f3, /E) =* 6. He proved that
lim
ft. W. D. Christie197 noted that, ifl^
partitions into parts ^ 3, and v + 1 partitions if M =* 0.
J. W. L. Glaiaher1" treated various questions of partitions by solving
equations in finite differences which were constructed by means of L. F*
A. Arbogast's8** rule of derivations. The capital letters А, Я, С, •••
signify any distinct numbers in ascending order of magnitude, while Greek
letters denote any distinct numbers. The only partition of 8 of the form
A2BC is 1, 1, 2, 4; the only one of the form AB*C is \, 2, 2, 3; while either
partition is of the form c?0y. Denote by Pn(i, j, k, - - * • AvB** - -)z the
number of partitions of x into the elements i, j, к, • - •, each partition con-
consisting of n parts and being of the form A*Bq< * -. When the elements are
0, 1, 2, *- •, this number P h the number G*(x, A"B*<< 0 of terms of that
form in the xth derivation of a"; its values for n = 2, 3, 4 and all possible
forms are tabulated (p. 67), and by simple additions, we deduce
The latter are computed for n ^ 7; liJkewise Gn(x) = P(l, 2, • >*t n)x
and РяA, 2, >..) for n ^ 9, and Pn(l, 2, - ; «^- - -)s for n ^ 7. It is
"■ Pbfl, Tmna Roy. Boc London, 209, A, 1909,153-175. Memoir IV on Partitrone.
i* Handbuch . . . Vertefluug d^r Prioutahlen, 1 1909. 222-9. СГ Landau.1»4
™ M*tb. Quert. Educ. Thnee, C^), I* 19W lOt'
1M Quar. Jour. M&th., 40,1909, 57-143.

Coat. 1Щ Partitions. 150
proved that the last circulator of Gn(x, a9flQ~ • *) is the same for all forms
ctpfi*~ * -, and hence need be computed for the form a* only, which case is
treated at length.
Glaisher1*8 proved Sylvester's theorem on waves, developed the formulas
for waves of periods 3, 4, 5, 6, and treated the non-periodic terms.
Glaiaher205 noted that his1'8 formulas for the number P(l, • • ■, n)x of
partitions of x into 1, * - -, nt repetitions allowed, are greatly simplified if
expressed in terms of $ = x + \n{n + 1) instead of x and gave the simplified
formulas for n ~ 9, and also those in terms of X = 2f for n = 2, 5, 6, 9.
He proved (p. 104) that
(-l)^P(l, •••,*)(-*) =PA, •••, *Mr - |n(n +1I «в*<1, 2, *,
where Q is the number of partitions of x into elements I, 2, * - >, unlimited
in number, each partition containing exactly n parts without repetition.
He proved (p. 106) that, if in the circulators occurring in the {-formulas,
the order of the elements be reversed, the original circulator is reproduced
except as to sign. Finally, he gave the leading circulator in each wave
Wm(l,2, -..,«ft + r).
E. Barbette*1 noted that there are exactly 2BI~2 —> 1) ways of partition-
partitioning x + a into distinct parts the greatest of which is xt where
« = Я*-Д, 1^Д^ \х{х - 1) - 1, & = 1 -f 2 -f - • + £.
In fact, such a partition of x -\- a corresponds to a partition of a into distinct
parts each < x. Next, to find all the partitions of N into distinct parts, let
x be the least integer for which 8* ^ N, and convert Sx, £*ц, • * •, 8ж_м
into sums of distinct numbers of which the greatest is N and such that all
the other parts are less than z, x -\- 1, — -, N — 1, respectively. Suppress
the parts in common to two members of the resulting equalities* Finally,
to find all sets of consecutive integers whose sum is N (as 8 + 9 = N),
write lt 2, 3, * * * along the diagonal of a square; above x in the diagonal
write the Bum 2x — 1 of x and the preceding term x — 1 • above that sum
write the sum Bx — 3 of it and the number x —- 2 preceding it in the former
list; ete,, until 1 is added. Cf. Sylvester.119
P. Bacbmann20* gave an extended clear account of the literature on
partitions* He inserted (pp. 109-110) a theorem communicated to him
by J. Schur: If S is any set of positive integers not divisible by r, and R is
the set of numbers obtained by multiplying the numbers in Shy lfrirti —}
then any positive integer can he partitioned into equal or distinct parts
chosen from S as often as into parts chosen from R, each occurring at most
r — 1 times. The case г — 2 gives Euler's theorem that any integer can
be partitioned into equal or distinct odd integers as often as into any distinct
parts.
"*Qu«r. Jour. Math., 40, 1909, 275-348. ™
"tlhii^ll, ШО, 94--И2.
991 Lee soxnmee de p-ifenes ршшымев distinctee egaks & пае p-igroe puieeauce, liege* 1910,
'« Ntedere Zohlentbecne, 2, 1910,102-283.

160 History of the Theory of Ntjmbkrs. qchaf, in
R. D. von Stemeck20* proved De Morgan's28 result that the number of
partitions of n into 3 parts is the integer nearest to n*ll2 by use of three
coordinate axes every pair of which make an angle < 60° and counting the
lattice points inside or on the triangle cut out of the plane x + у 4- z = n
by the coordinate planes* Similar use is made of 4-dimensional space to
show that the number of partitions of n into 4 parts is the integer nearest
(n* + 3ns _ 4)/144.
N. Agronomov204 noted that N = 2*pF* • 'P? is representable as a
sum of consecutive integers in (aL + 1) ■ >>(<** -1- 1) ways [Bylvester118].
P. A. MacMahon206 noted that his three-dimensional graphs of plane
partitions admit not only of 1, 3 or 6 readings, but may admit just two
readings if the weight be ^ 13» Let each part be ^ I and be placed at a
node of a two-dimensional lattice with m rows and n columns. The
generating function giving as the coefficient of x18 the number of partitions
of weight w is expressible in six ways, one of them being
and the other five being derived from this by permuting lt m, n. A general
proof is here first given. The theory of generating functions, especially
for I = со 7 is developed further here and in his next paper.**
T, E. Mason207 proved that 2*pV- • *p?, where the p's are distinct odd
primes, can be represented as a sum of consecutive integers not necessarily
positive in 2(ax + 1) • • • (ar + 1) ways. In just one half the representa-
representations there is an even number of terms, and in just one half are the terms
all positive [Sylvester118].
W. J. GreenstreetMe proved that x + 2y + Zz = 6n has 3n2 + 3n -f I
integral solutions ^ 0.
MacMahon208 showed that the enumeration of partitions of multipartite
numbers may be made to depend upon his141 theory of distributions and
symmetric functions of a single system of quantities,
A. J. Kempner210 proved that, if 1, Ci, Cj, • • • form a set of increasing
positive integers such that every positive integer is a sum of к or fewer of
them, the radius of the circle of convergence of 1 + ax + CtX* + • • • is
unity» Let every positive integer be a sum of at most к terms of a given
set au atf — •; let оц, fa be integers such that 0 < л» ^ R, \ ft | ^ S,
where R and S are any given positive integers; then every positive integer
is a sum of fewer than R\BkS + к + 1) terms of the set 1, a&i + ffu
«jftj + &, — *- Finally, the known theorems that any positive integer n
is a sum of four squares and that z* = l-l-3 + 5-h -— -f Bx — 1) imply
'"iRendicontt Circ. Mat. Иегаю, 32, lfill, 88^94. ^ "^^
■» M*th. Unterr. 2r 1912, 70-2 {Ruaeian).
*" Phil. Trana. Roy, Soc. London, 211, A, 1912, 75-110, Memoir V on Partitions.
*»/WdM 345-373. Memoir VI on Partitone,
r, Math. Monthly, 19, 1Й12, 46-50- Cf. Sylvester,**
501
яо» Ттвав. Cambridge Phil. Soc.y 22,1912,1-13.
*»Uberda* Warinesche Problem . . ,t Diae. G6ttiikgen, 1912.

Chap. 1Щ PARTITIONS. 161
that n =* «i*l + x*t-3 -1- «,-5 -1- • — is solvable in integers such that
4 ^ lii ^ %h = ut • • • ^ 0» A generalisation is noted.
S. Minetola211 investigated the number R(t; «j, • • •, a,>; n) of ways of
separating into n groups m =? £ -h <*i 4- * • • -h «P objects of which t are not
repeated, while p further objects, distinct from each other and from the
preceding t, are repeated aiy - • -, <xp times, respectively. After finding
recursion formulas for R, he proved theorems on the тйлНтитп value of R
when m and n vary, but so that m — n remains constant* Finally, he
studied /2A; m\ n), so that one object is taken single and another is
repeated m times. It is the coefficient of x^1 in
Its recursion formula is
J2(l; m; n) = R(l; m - 1; n - 1) 4- Щ1; m - n + 1; n).
G. Scorza211 evaluated sums of reciprocals of products, each summation
extended over all the partitions of a given arbitrary integer.
G. Candido*" noted that a* is a sum of a consecutive odd integers.
For m = 3, this was also proved by J. W> N. le НеихЯ* Cf. Fregier.2*1
G. Csorba215 stated that all questions concerning partitions can be
reduced to a single one, viz., the question of the number of ways Л can be
obtained from <tlr - • •, an by addition, repetitions allowed. Cayley*4
had expressed this number of partitions of Л in the form
where c{(A) is a periodic function of A; but essentially proved only the
existence of such a representation* СгюгЬа gave for c<(A) an explicit
formula involving Bernoullian numbers and the g.c.d. d of all the a's except
^4> '' *> <К>) and involving summations extended over all solutions ^ of
the congruence S'^Joi^ — A (mod d).
*СзотЪьш treated multiple partition.
P. A. MacMahon217 has given an extended account of the theory of
partitions as a branch of combinatory analysis. A small part of VoL I and
nearly the whole of Vol. II are taken up with theories more or less connected
with the partitions of numbers. The theory is investigated from the
standpoint of a new definition of a partition. A partition is defined
as a set of positive integers <*i, <x±, »»», ан, whose sum is n, such that
at = «* Ш • - - ^ лж. ТЪ.е importation of linear Kophantine mequalities
leads to a syevgetic theory and thence to the determination of ground
forms connected fay various orders of sysygies as in the theory of algebraic
invariants. A generalization is made by considering one or more general
ш Periodic* di Mat., 29,1913,67-ЗД ™ ^ ^^ " '
« Rendiconti CSrcob Mat. jPalenno^ 36,1913, 163-170.
*" SuppL ftl Periodico di Mat., 17, Ш4, 116-7.
w WTakm>d« TljdBchrift, 12r 1915-6, 97-в.
ш Math. AnnaJen, 75F1914, Б45-56».
ш Math. 6a termee erteeitd (Hungarian AesuL ScJf 32,1914,
m Comlriiiatoiy Апл1уя», Cambridge, 1,1915; П, 1916»

162 History of the Theory of Numbers. [Сна*, m
linear inequalities connected with a number of linear relations» Such
theories are grouped under the title a partition analysis." As regards
the simple partition of numbers the idea results in laying foundations deeper
than the intuitive generating functions which served Euler and his successors
as points of departure. There is an extension in the direction of two dimen-
dimensions in such wise that the parts are laid out in the compartments of a chess
board of any dimensions, a partition being defined as a distribution of
numbers such that in every row and in every column of the board a de-
descending order of part magnitude is in evidence. The complete enumerative
solution of this question for a complete or incomplete lattice Or chess board
is reached. The solution depends upon the idea of a lattice permutation
and of an associated lattice function. An assemblage of letters a*la? ■ • • a?
is said to be a lattice assemblage when the repetitional exponents satisfy
the condition at ^ a^ ^ - • - ^ a,; and of this assemblage a permutation
is a lattice permutation ii the first к letters (k being any number < s)
of the permutation constitute a permutation of a lettice assemblage
а?*а*1* ■ * ■ af'» These permutations have been enumerated, but the theory
of the derived lattice functions is not yet complete. The theory of parti-
partitions in three dimensions is completed in this book only as far as the simplest
case when the parts are placed at the angular points of a single cube. The
enumeration of the partitions of multipartite numbers is investigated
principally by means of J. Hammond's217" differential operators ^Mac-
Mahon17tJ. The problem of enumerating partitions which do not involve
sequences of parts is considered in Vol. I.
* К von Schrutka21* gave an extended account of methods employed
to further develop Vahlen'slb0 results.
R. Goormaghtigh21* noted that, if N is the sum of the consecutive integers
comprised between v + 1 and nf then 2N ~ (л — v)(n + v + 1) and the
number of couples n% v is the number of odd divisors > 1 of N»
G. H- Hardy and S. Ramanujan220 proved that the logarithm of the
number p(n) of partitions of n is asymptotic to тг V2n/3, and the logarithm
of the number of partitions of n into distinct positive integers is asymptotic
to xVny3» Tbey221 developed a general method for the discussion of these,
and analogous problems of combinatory analysisr by means of the methods
of the theory of functions of a complex variable. This method is, within
limits, applicable to the study of all numerical functions which occur as
coefficients in power series possessing the unit circle as a natural boundary.
In this particular problem it leads to the result that
f- * + 0(e*«), к < ir/V6,
dn Vti- 1/24 Л
London Math. Soc., 13, 1882P 79; 14, 1883, 119,
Jour. fQr Math., 146, 1915-6, 24S-264. Sitzungeber. Akad. Wise. Wien (MatJb), 126, IU,
1917, 1081-1163.
I/mteftDediaire бея math., 24, Ш7, 95.
»PkDCL London Mnih, Soc, B), 16, 1917. 131.
GtRdPH 164,1917,35-38- Froc, London Math. Soc., B), 17,1918,75-115.

Свар. IIIJ PARTITIONS. 163
and to still more exact results in which the sum of a number of approximating
functions appears on the right hand side. Six terms of the series thus
obtained give pB00) = 3972999029388, with an error of .004, a resnlt
confirmed by MacMahon by direct calculation. Here O(g(t)) denotes
a function whose quotient by g(t) remains numerically under a fixed finite
value for all sufficiently large values of L At the end of the paper occurs
a table, calculated by MacMahon, of the number of partitions of n for
n ^ 200.
P. A. MacMahon222 proved that, ft pu ■ - *,Pt are integers in descending
order of magnitude and {mi • ■ • m*) ia the partition conjugate to (pi • ■ • p<)>
the number of ways of distributing n objects of specification (n) into boxes
of specification (mi • • - тш) is the coefficient of x* in the expansion of
MacMahon*23 established a A,1) correspondence between combinations
derived from m identical sets of n distinct letters and general magic squares
of order n in which the numbers in any row or column have the sum m
[MacMahon188].
S. Ramanujan*24 proved that, if p(n) is the number of partitions of n,
pEm + 4)*0 (mod 5), pGm -f 5) = 0 (mod 7),
+19) = 0 (mod 35), pB5m + 24) ss 0 (mod 25),
47) =0 (mod 49);
which imply the first two congruence theorems.
H. В. С Darling^ gave elementary proofs of the first two of Ramanu-
jan's224 congruence theorems.
L. J. Rogers226 gave a new proof of his1M* two identities» J. Schur227
gave two proofs. Finally, Rogers228 and & Ramanujan*28 each gave a proof
which is much simpler than all earlier proofs.
P. A. MacMahon239 solved the problem of multipartite partition,
mProc, London Math. 8oc.t B), 16, 1918, 352-i ~ ™
ШШ., B), 17,1918, 25-11
Fe. Cambridge Phil. Soc., 19, 1919, 207-210.
f27e
,pp
«IYoc. London Math. 8oc4 B)r 16, 1917, 315-7.
*7 Situngsber, Akad. Wias, Berlin (Math.), 1917, 302-321.
m Proc. Cambridge PhU. Soc.y 19> I919y 211-6.
* Phil, Traw. Roy, Soc. Loodoo, 217, A, 1916-7, 81-113. Memoir VII on Partttons.

164 Histoby or the Theory of Numbbbs.
A. Tanturri280 gave expressions for the number of partitions of n into 2,
3, 4 or 5 distinct parts, and recursion formulas. He231 investigated the
number Dn of partitions of n into powers of 2 and the number DBP} n) of
partitions of n into powers of 2 of which 2p is the maximum. The first
function can be computed from the second* In the second paper occur
recursion formulas for the second function, and expressions for DBP, 2*Ж)
and DC2*, 2ph + 2*~l) m terms of binomial coefficients.
On the number of positive integral solutions of ox -\- by ~ nf see papers
117-142a of Ch. II. Cesaro, VoL I, p. 306, gave relations involving the
number of positive integral solutions of & + 2£j + • ■ • + v%T — n*
Von Stemeck, VoL I, p. 427, used partitions mto elements formed from
the first s primes.
*» АШ R. Aocad. Sc. Torino, 52,191&-7,9Q2-918. In Peano'e symbolism, with & translation
of mogt of the results.
vLIbid.y Dec. 1,1918. Continued in Atti R. Accad. Lincci, Bendiconti, 27, II, 1918, 39&-
403. In Peano's symbolism with partial translation.

CHAPTER IV.
RATIONAL RIGHT TRIANGLES.
Methods or bolting x* + yr =^ z* in integers.
According to Proclus,1 Pythagoras represented the smaller leg by
x — 2at 4- 1, the larger leg by у = 2o* + 2a, and the hypotenuse by
г ^ у -h 1- Plato1 took the difference z — у to be 2 (instead of 1) and
obtained2 x — 2a, у ^ c? — ly z — a* + 1,
The Hindus Baudh&yana and Apastamba,1 about the fifth century B.C,
obtained independently4 (?) of the Greeks the solutions C,4,5), E,12,13),
G, 24, 25), which are cases of the rule of Pythagoras, and (8, 15, 17),
A2, 35, 37), eases of the rule of Plato.
Euclid* gave the set of solutions
<ФЪ WF - 7s), **<? + 7*),
ae well as (II, 5; X, 30) the related set
Marcus Junius Kipsue/ at least a century before Diophantus, gave
two rules to find right triangles with integral sides, one leg being given*
Expressed algebraically, his rules give, as solutions of z* — у* ^ x*,
г = J(x* + 1), у = J(x* - 1), for x odd;
s =^ {x* + 1, у «« }я* — I, for x even,
formulas equivalent to those of Pythagoras and Plato, respectively.
Diophantus7 took a given value (in fact, 4) for г and required that
z2 — x* shall be a square of the form (mx — z)K Thus
x =Z~rT^> у ^tnx-
Неге яг is any rational пгхшЬег; replacing it by mfn, and taking & = m* + ^2>
we get
A) x ^ 2?nn, у = wi2 — л2, г =* тп5 + »*-
1Ргос1шз Diadochus, pritnum Euclidw elem. ]Ibr. comm, Eth ceot.), e*L by O. Friediein,
Leipzig, 1873, 428. Elements d' Euclide ave<? lee Comm. <te Proclusf 1533, Ш; Latm
trantfl. by F. Baiodtifl, 1660, 269. M. Cantor, Geschichte Math,, ed. 3,1,1907, 185-7,
224. G. J. AHm&n, Gfcek Geometry from ТЬЫея to Euclid, 1889, 34.
1 Cited by Heron of Alexandria, Geometric, p. 67; Boethiue Fth cent.), Geoinetrie, lib. 2.
* Sulbaeetxa, pubL by A. Burfc with German tranal-, Zeitecbrift der deutechen mot^enland-
iechen GeeeH.r 55,1001,327-91, 543-91.
'Bflik.» H. G» Z^rthen, Bibiiathec» Млйь, (В), 5, 1904, 106-7. M. Cantor, Gcecbicbte
МгОЦ «L 3, 1,1907,636^5; 96 for 31 + 4s ^ 5s in Egypt-
1 Ekmenta, X, 28,29, lenim* 1; Opem, ed. by J. L. Heibexi, 3,1886,80. M.G&ntor, Get*
cJuchte bUtb, ei 3,1, 1907, 270-1, 482.
• Cf. J. B. BSot, Jour, dee Saywite, 1849, 250-1; Comptee Rendue Рлпй, 28, lS49f 576-81
(Sphinr-Oedipe, 4, 1909, 47-8). M. Cantor Die ifimidchen Agrim . > . Feld
1875, 103, 113, 165. C. Henry, BuU. BibL Storift Sc. Mat. Кв., 20, 1887, 401-2.
7 Arith., H, 8; Орезя, ed. by P. Turnery, l, 18»3, 90; T. L. Hertfa, 1910,145.
1в5

166 History or the Theory of Numbers. [Chap. IV
Diophantus (III, 22, etc.) referred to the right triangle with these sides as
that formed from the two Dumbers то, п,
Bralunegupta8 (bom 598 A.D.) gave explicitly the solution A).
An anonymous Arabic manuscript9 of 972 stated that in every primitive
right triangle (i. e., with relatively prime integral sides), the sides are given
by A). Necessary conditions that A) give a primitive triangle are that
mf n be relatively prime and w-fnbe odd» The hypotenuse of a primitive
right triangle is a sum of two squares and is of the form 12fc + 1 or 12k 4- 5,
though not all such numbers are sums of two squares. But 65* is a sum
of two squares in two ways: 63' + 162 = 332 + 562. To find a triangle
with a given hypotenuse A, we need an expeditious method to find two
numbers the sum of whose squares equals h. If the last digit d of A is 1, the
two squares end in 5 and 6 or in 00 and 1. If d — 3, they end in 4 and 9;
if d = 7, in 1 and 6; if d = 5, in 00 and 5,1 and 4, or 6 and 9; if d = 9,
in 00 and 9, or 4 and 5; with similar rules if d is even.
The Arab BeD Alhocain10 (tenth cent.) gave a geometrical proof that
A) give the sides of a right triangle, and noted that if the hypotenuse is
even, also both legs are even. Rules equivalent to that by Pythagoras are
given; also false theorems on triangles formed from several consecutive
numbers.
Alkarkhi11 (end of tenth cent.) derived the solution 3,4, 5 of x7 + y* = z*
by setting у = x+ \,z — 2x — 1.
Bhascara1* (born 1114) gave A) and employed it, as had Brahmegupta,
to find the second leg (m2jn — и)/2 and hypotenuse, (vtf/n + n)/2T given
one leg ml Given the hypotenuse hp the legs are12* I = 2кЬ/(№ + 1) and
fo - Л or h - g and bq, where q - 2Л/(^ + 1). To find (p. 201) a right
triangle whose area equals the hypotenuse take 3z, 4x, 5z as the sides.
Leonardo Pisano1* employed the fact that the sum 1 + 3 + ■ ■4 ot n
consecutive odd numbers is n2 to find two squares whose sum is a square-
First, if one square a2 is odd, take the other to be 1 + 3 + — + (a2 — 2);
their sum 1 + 3+ • • • +a2isa square. If one square is even, as 36, add
and subtract unity from its half, ob taming the consecutive odd numbers
17 and 19; then 1 + 3 + - - ■ + 15 = 64 and
64 + 36 ~ 1 + h 15 + 17 + 19 - 102.
* Brahme-epbut'n-ekktTijiot*; Algebra with Arithmetic and Mensuration, from the Sanskrit
of Bmhmegupt» and BMscanv, transl, by H. T. Colebrooke, London, 1817. 306-7,
363-72.
* French transl» by F. Woepcke, Atti Accad. Pont, Nuovi lined, 14, 1860-1, 213-227, 241-
269 <M. Cantor, Geechichte Math., ed. 3,1,1907, 751-2).
"/MA,30l-2ir 34^56.
11 Extrait du Ibfchrt, French traosl. by К Wocpcke, Paris, 1853, 89.
* Colebrooke,8 pp. «1-63. Jofcn Taylor's tianel. of Brahme . . . ,* Bombay, 1816, p. 71.
** Same in Ladies1 Diary, 1745, 14, Quest. 254; T. Leybourn's Math. Qae«t- ladiee* Diary,
1, 1817, 366-7; C. Hutton'a Diarian Miscellany, 2, 1775, 200.
w liber quadratorum L. Pisano, 1225, in Tre Scritti medi^i, 1854, 56-«, 70-5; Scritti L.
Pisano, 2, 1862, 253-4. Cf. A.Genoochi, AnnafiSc.Mat. Fis., 6, 1855, 234-5; P. Vol-
picelli, Atti Accad. Pout. Nuovi lined, 6, 1852-3, 82-3; P. Coesali, Online, Traeporto
in Italia . . . Algebra, 1,1707, 97-102,118-9.

О* *Ч-1Г* = А 167
[ correspond to the rules of Pythagoras and PUto.^
obtained rational solutions of z* + У* — e* by It method quite different
from that of Diophantue; starting with any known rational triangle for
which ** + 0* ^ У, he took z = оа/у, у = <tf/-y.
F. Vieta* A540-1613) used the method of Leonardo, last cited, and
that of Diophantus.
M. Stifel" called abb diametral number if a* + Ь2 — с2 and stated in-
incorrectly that a - b is a diametral number if and only if a/6 belongs to one of
the series l£, 2#, 3^, • • ■ and l£, 2H, 3£$, • • - , and hence in effect that
a : b = 2n2 + 2n : 2n + 1 or a : b - 4n* + 8n -f 3 : 4n H- 4 fcf. Meyer*J
which correspond to the solutions of a2 + Ь2 -= & by Pythagoras and Plato,
These diametral numbers are not those defined by Theon of Smyrna2 of
Ch. ХП.
The Japanese manuscript of Mateunago17 of the first half of the eigh-
eighteenth century contains three proofs of (I).
T. Faotet de Lagny1* replaced m by d + n in A) and obtained
x ~ 2n(d + n), у = d(d H- 2л), z = x + & = у + 2n*>
Taking d — 1 or n — 1, we obtain the rule of Pythagoras or that of Plato,
C. A. Koerbero19 proved that the sides of any rational right triangle are
proportional to the numbers A).
L. Euler5* expressed the hypotenuse с as Ъ -f anfm* By a* -f- b8 = &,
b\ a = m* — n2: 2mn. Hence a, bt с are proportional to the numbers A)
with m > n > 0.
Euler21 noted that the sum of the squares of x + 1/яр and у -\- 1/y is a
square if
the latter being true if (p* — l)ar = 4p.
J» P. Gruson22 noted that n -+■ 1 and та generate a triangle f of Pythagoras'
type] whose larger leg у — 2n* -\- 2n and hypotenuse у + 1 generate a
new triangle whose least side is a square [2y + 1 — B?i + IM}
L. Poinsot23 noted that every set of integral solutions of z* — #* = x2
is given by г = (j> -f з)/2, у = (р — g)/2, where ac2 has been expressed
in every way as a product of two integers p and qt both odd and relatively
prime or both even, but with no common factor > 2.
" liber Abbaci, Ch. 15 (Scritti L. PfeaDo, Home, 1,1857)~ " ~ ~ "
u Frandficua Vieta, Zetetica, 1591,1V, 1; Opera M*th., 1646, 62.
w Aiitb. Integra, Nuii»beFs^ 1M4, f. 14v-f. I5v. Copied by Joeeppo Voioomo, Dc l'Arith.
TJoiveraale, Yenetia, 1598, 62,
*» Y. Mikami, AlA. Geach. Math, Wbs., 30, 1912, 329. Keport by K. Yanagjhara, T6hoku
Matbu Jcrar., 6t 1914-5, 120-3; continued, 9, 1916, 80-7 (by use of progressions).
fl BkL Acad. Sc. fine, 172», 3l8.
11 Nova tnonguti recrtanguli analyau, Hatae Magd., 1738, S,
w Comm, Aead. Fetrop., 10, 1738, 125; Camm. AHth,, 1, 1849, 24.
n Opaac. anal., 1, 1783,329; Comm. Aritb, П, 4&
& EnthttIHe 2kub^i:yen u. Gebeimiim der Ariih., Berlin^ 1796, 104r-6.
» Comptee Rendns Faiie, 28, 1849, 581-3; abo p. 579 by X B. Biot.

168 Histoby of the Theory of Numbers.
P, Volpicelli* noted that* = a' + b*-al + /92 imply that
x = d= {aa =F ЪР)> у - ±(afizbba)
are solutions of я* 4- у5 — z2 and stated incorrectly that they give all the solu-
solutions, whereas formulas (I) do not. As to J. IAouville's26 remark that, for z
given, я* + y5 — z2 has relatively prime solutions if and only if z is a product
of prunes 4n + I, the solutions x = 1020, у = 425, z = 5-13-17 are not
relatively prime»
Volpicelli" distinguished к types of solutions of x* + уг = z2, where
г — hi-- hkf Лу — aj + b). The к solutions of the first type areg(a, — b*),
2qa,hjt where q = яг/А/. The fc(Jt — 1) solutions of the second type are
where q « г/(А,-А/), the quantities я2, y8 in brackets being such that
a^ + j^ - Aft). From
we obtain the 4(*) solutions qxh дуг of the third type, where q
Thus the total number of solutions is
pli57 noted that all solutions of s2 + #* = s* depend on the solu-
solutions of я5 + y5 = ^ 0* = 1» * • s fy, where «i, — •, zk = z are the products
of the factors of z taken 1, 2, •••, к at a time. For г2 = (а2 + Ь2)*, а
solution is
x - a* - (J)a^ + (J)a-^ , у = фа*"* - Gia^V + ■ • ••
For, if (a + *)fc =* A -fiB, (aJ + ¥)k « A2 + ^, which was verified with,
out using £ - V-Л. Also a2 — b* is a factor of В if fc = 4A, but is a
factor of A if к = 4Л + 2.
С- A. W. Berkhan*8 gave nineteen methods of finding two numbers the
sum of whose squares is a square, with references on several proofs.
E. de Jonquieres** discussed Volpicelli's* topic.
A. J, F- MeyP noted that, according to an argument by de Jonquieres,2*
(s + 3J + (x + 4)' = С(У + IJ -i- (У + 2JJ
has only the solutions x + 3 = 3 or — 4, whereas s + 3*=0or — 1 also.
C. de Polignac" used a rectangular lattice to prove that A) gives all
integral solutions of x* + y* = z2.
* Giomale Arcadico di Sc- Let. cd Arti, Rome, 119,1849-50,27. Annali di Sc. Mat,Fi»., 1,
1850,159-166,369,443.
* Comptee Rendua Гапв, 28,1&49, 687.
« Atti Aecad. P6nl. Nuovi lined, 4,1850-1,124Г-140, 346-377, 508^510-
17 /Wd., 5,1851-2, 315-352; Comptee Rendiw Paris, 36, 1853, 443-6. Extract in AnnaU di
8c. Mftt. Rg.f 3, 1852, 130-3; 4,1853r 286-297.
« Die metkwflrdig^n ЪввшхЬаЛел der FyUiftg. Zahleo, Eieldben, 1853.
» Nouv. Ann. МдИь, B)f 17, 1878, 241-7, 289. <X papers 26-31 of Ch. XVII.
mIbid^ B), 18, 1879, 332-3.
« BdL Math. Зое. France, 6t 1877-8,162.

Chap. IV} SOLUTION OP ^-f^^Z2- 169
С. М. Piuma" quoted the known result that all relatively prime integral
solutions of x* + y5 *= z2 are given by
у
, z = 2,
where *a and n are relatively prime odd integers, and proved conversely
that then these three expressions are relatively prime in pairs, by showing
by use of congruences that no two are divisible by the same power of a prime.
D. S. Hart81 proved for n £ 4 that, if г is a product of n primes each a
gum of two squares 21, z2 is a 21 in C* - l)/2 ways [Volpicelli2*].
L. E. Dickson" obtained, as a solution equivalent to A), т + &, r + t,
r 4- s + t, where r* = 2st is a square. The same rule was given latsr by
P. G. Egidi,* D. Gambioli» A. Bottari,'» and H. Schotten.36-
Graeber17 noted that if the point of tangency of a circle inscribed in a
right triangle divides the hypotenuse г into the segments к and mt while
n and m are the corresponding segments of leg yt then
(Jfc + my - (m + ny + {n + кJ, к - (n8 + mn)/(m - n).
Thus x, y, z are proportional to A). The sides if integral are shown by a
long proof to be A).
L. Kronecker3* proved that all positive integral solutions of xz + y* = zx
are given without duplication by
x - 2pqtt y = t(p* -?), г = 1{р* + <?), p>q>0, t > 0,
p and q being relatively prime and not both odd. The reason why every
solution is obtained once and but once is due to the fact that the circle
f* + 7j2 = 1 is of genus zero, all its points being expressible rationally in
т = tan w/2:
1 - t2 . 2t
$-coSu>-j-^, V -smoj *=Y+^±-
A. BottariSfl proved that all integral solutions of з3 + у1 = г2 are given
bys = w + tfl, у = r/ f ш, ?^w-f v+w, where и = p% v = 2*^lq%
to = 2'pqk, p and q being relatively prime odd integers. Ibus xy is not
a square»
P. Cattaneo*0 gave a simple proof of Bottari's theorem.
P. Keutzel41 noted that, if a > 2, we can solve c2 — У = aK Set
с = Ъ -\- v. Then & я: (fl! - ^)/B«) is an integer if v *= 1 and a is odd^
or if v = 2 and a is even. We may take v to be any divisor a/n of a; then
Ь » (n2 - 1)ip/2, с = (n^+ I)»/3.
* Mith. Qaeet. Educ. Timea, 39f 1883, 47-Z.
uAm«r> Math. Monthly, 1, 1894. S.
* АШ Acced. Ptont» Nuovi Iinoei, 80,1897,103.
11 Periodic* di Mat., 16, 1901 r 151-5.
^Zeitechrift Math. Nature. Unterricht, 47, 1916,181-2.
" Arehlv Math. Pbya, B)r 17,1900, 36.
"<Vorlt»ungen Qber Zahlenlbeorie, 1,1901, 31-36.
" Periodico di Mat., 23, 1908,104-110. Cf. Dickeon.1*
"ТШ.,218.
41 Zetteehiift Vejrmewingeweeeii d- Deutechen Geometervereme, Stuttgart, 38, 1909,208-11.

170 History or the Theory of Ntjmbers. [Chap, iv
J. Gediking" noted that, for relatively prime solutions of z* — у* = я2,
we may take as x — у any number of the form Bn + IK or 2n*t but no
other. Then x + у - Bm + IJ or 2m\ with 2m + 1 and 2n + 1 or
7» and n relatively prime. [^It was overlooked that we may restrict to
one of the two cases»] AH solutions < 1000 are given. J. C. Milborn
(pp. 167-9) erred in saying that this method does not give all solutions.
T. Boelen (pp. 238-40) noted that we may take as z any integer > 2, if
solutions with a common factor are allowed.
C. J. van der Burg° gave an incomplete proof of A).
Fitting44 discussed the relatively prime solutions of x* -f y* =^ z* by
setting г = x + a, whence уг = a{2x + a). Without loss of generality we
may take a to be an odd square 1, 9, 25, • - •, and equate 2x + a to the
successive odd squares.
W. Jfluge45 noted that x* -f- y1 — z1 is satisfied by
f — d Г -h d
z* «df, d <xf y = -Y~> *~~2*
and gave recursion formulas for computing successive solutions»
E. Meyer46 noted that Stifel's1* formulas for diametral numbers do not
give all, for example not 33-56, and that he should have used
a : Ь — m2 — n2 : 2mn>
He compared many known ways of solving x* -f y2 — z2.
Fl Lambert4* solved x* -f y1 — z* by use of numbers a -f Ы.
N. Gennimatas411 would solve x* + y2 — a2 by setting 2a = с Н- d, where
Ы is a square x2, whence у = a — d.
*E. Haentzechel48* noted that from one rational right triangle we can
derive an infinity by use of the formulas for sin na and cos па £cf. Vieta^
Ch. VIJ. From two right triangles whose hypotenuses are primes of the
form 4£+1, we can derive an infinity by use of the addition theorem for
sine and cosine. By means of these theorems we can arrange in order the
proper solutions of x^-^-y1—^-.
P. Quintili4* attributed to F. Klein (!) the solution A) of x* + уг = **.
A» E. Jones50 discussed right triangles whose three sidee are of the form
xt - 1.
C. A. Laisant51 noted that MQ, 2PN, P1 + N1 are sides of a right tri-
triangle if Af, Nt Pt Q are four consecutive terms of Fibonacci's series (Vol. I,
Ch. XVII of this History), so that P = M + Nt Q = N + P.
« Vriend der Wlskimde, 25, 1910, 86^96»
u Ibid., 26, 19llr 188-191.
44 Umtermediaire deft math., 18, 1911, 87-90 B334).
tt Verhandluxigen der VereamrD. deutecher PhilolageD u* Scbulmanncr, Leiptigt 51,19ll7137.
Unterrichtebl&tter Math. Naturwiea., Berlin, 19, 1913, II.
* 2eit49chrift Math. Naturw, Unterridit, 437 1912, 281-7.
*7Nouv. Ann^Matb., D), 12, 1912, 408-421.
"Zeitechr. Math. Naturw. Untenicht, 44, 1913, 14-15.
"• Blatter fiir d. Fortbildung d, Lehrere u. Leherin, Berlin, 6, 1913,395-6.
"HBoll. Mat. Sc. Fis. Nat., 16,1915,69-71.
w Math. Quest, and Solutions (contm. of Math. Quest. Edoc. Tunee), 2, 1916, IS.
u Comptes Kendus dee So. Soc. Math. France, 19l7, 18-19.

Chap. ГУ] DlVlSOBB OF SlDES OF A RlGHT TRIANGLE. l7l
Paper» without Novelty.
G. Oughtred, Opuacuh Math., Oxonii, 1677,130-8.
А. Tnacker, A Miecellany of Math. Problems, Birmingham, 1, 1743, 171-S [Proof of AI
Anonymous, L&diee' Diary, 1752, 39, Queet, 344 [Proof of A)].
A. D. Wheeler, Ашег, Jour. Arts, Sc» («L, Silliman), 20,1831, 295 [Plato's rule].
J. A, Granert, KlUgeTa Math. Wflrterbuch, 5, 1831, 1Ш-3 [Euler"].
C, M. Ingleby And S. Bills, Math. Quest* Educ. Time*, 6,1866, 39-40 [Proof of (l)].
M. A» Gniber, Amer. Math. Monthly, 4, 1897, 106-8,
H, Schubert, Niedere Analysie, 1, 1902, 159-162 [Proof of A)],
F, Thaanip, Nyt TidBekrift for Mat., 15, A, 1904, 33 [Proof of (l)].
A. Aubry, Matbeste, 5, 1905, 6-13 [historical].
A. Holm, Math. Quest. Educ. Times, B), 9, 1906, 92; 10, 1906, 56 [Troof of A)J.
V. Vaiali-Thevenet, Rivista Fig. Mat. Sc. Nat., 8, I,1906, 422-3-
C. Botto, Giomale di Mat., 46, 1908, 297-8 [Poinsot^J.
P. Riehert, Untemchtsbl&tler Math., 14, 1908, 55-7, 87.
C. Botto, Suppl. al Periodic© di Mat,, 12, 1908-9, 68-74.
T. 8. Rao, Jour. Indian Math, Club, Madras, 1,1909, 130-4.
School Sc, and Math., 10, 1910, 683; 11, 1911, 293-4; 13,1913, 320-2»
Sides of a right triangle divisible by 3, 4, or 5.
Frenicle de Bessy52 (f 1675) noted that if the g»c.d» of the integral
sides of a right triangle is a square or the double of a square, the sides are
of the form A), and that one of the sides is divisible by 5, one of the legs
by 3 and one by 4» If the sides are relatively prime, the sum and difference
of the legs are of the forms 8fc ± 1.
P. Lentheric53 noted that the product xyz of the numbers (I) is divisible
by 60, since mn{m2 «— и2) is divisible by 6 and if no one of m, nt m ± n is
divisible by 5, wi2 -f n* is. К Paulet added (p. 382) the remark that
m4 — n4 is divisible by 5 if neither m nor n is, since m* ^ 1G& + 1 or 10k + 6,
L. Poinsot23 stated, as if new, that if x, yt z are relatively prime solutions
of it2 + y2 = z*, 3 is a factor of x or y, 4 a factor of x or yr and 5 a factor of
x, у or г. This was proved by E. R. Grenoble" by considering the residues
modulo 3> 4 or 5, and by J. Binet (pp. 686-7, 755) by use of Fermat's
theorem. J. liouville remarked (p. 687) that x> у, х + у or x — у is
divisible by 7. Bourdat56 stated that he had found these facts in 1839 and
added that, ifz^ + y2— z4, 5isa factor of x, у or z, likewise 7 and 24.
If x* + y2 — z*, one of the numbers has the factor 2**3*7.
A. Vennehrenw proved that xyz is divisible by 60.
A, Levy*r noted that in a* + № = c2, 7 divides « -f fc or a - Ь if 7 is
prime to a, ht c; 11 divides one of 5a :£ b, 5b ± a if II is prime to a, ht с
dee triangles rectangles en nombres, I, Paris, 1676, $$24-25, pp. 50-61. Re-
Reprinted with part II in 1677 at end of РгоЫётев d'Architecture de BloudcL Both part*
in Mem» Acad. R. Sc» faiis, 5, 1666-99; 6d. Para, 172», pp. 146-7» С Henry, Bull.
Bibl. Stori* Sc» Mat- Fis., 12,1879, 691^2, gave a list of Freniele'e writings; cf. Nouv.
Ann. Math,, 8, 1&4», 364-5.
* Annalee de Math» (cd., Ger^onne), 20,1829-30, 376-582; 21,1830-1, 96-98. Cf» Jour, fur
Math», 5, 1830, 3S6; Jour, de math. 616m. spec., 1880, 261.
N Comptee Rendus Fans, 2S, 1849, 665-6.
« Bull, de 1'Acad. Detphinate, Grenoble, 3,1850, 37-43.
* Die Pythagoraiacnen Zahlen, Progr, DomBcboie, Gttetiw, 1863.
" BuU. de math, €\6mtp 15,1909-10, 277»

172 History of the Theory of Numbers. [Cuap.iv
Number of right triangles with л given side.
Report has been given above of the papers by Volpicelli,* Hart"
and de Jonquieres.* See Fennatwand Frenicle17 of Ch. VI and papers
19-32 of Ch. XIIL
К Gauss6* noted that to every hypotenuse composed of к distinct
primes belong
different pairs of legs, where {jx~} is the largest integer ^ x* The legs are
relatively prirne for 2k~1 pairs.**
D. N. Lehnier*1 proved that the number If of right triangles whose
sides are integers with no common divisor, and whose hypotenuse is ^ nt
is asymptotically п/Bт). But, if the sum of the three sides is ^ n,
N = n (log 2)/**, asymptotically.
(X Meissner*1 stated that the number P of integral right triangles with
one leg x = 2?p? • • - p*n (p's distinct primes) is:
^ ». то — 1
1
where \jaT\ is the largest integer ^ a. Also P + I ia the number of sets of
positive integral solutions z, у of z2 — y2 = a? (x given).
E. Bahier*2 noted that if A, Bt - ^ P are distinct odd primes the number
of right triangles one of whose legs is A*B? - - P" is
2a -f-
И А — 2, we bave only to replace a by a — 1 in the last result.
Right triangles of equal area.
Diophantu£> V, 8, required three rational right triangles of equal areas.
If, as in V, 7, ab -f a% + b2 ^ c*, the right triangles formed7 from с, а; с, Ь;
c, a + b have the same area abc(a -f- 6). The chosen example has a = 3,
6-5, с = 7. This solution was given in general form by F. Vieta,
Zetetica, IV, 11* Fermat*2*1 observed that if z is the hypotenuse and b, d
the legs of a rational right triangle, we obtain a new right triangle of the
same area by forming the triangle from z\ 2bd and dividing its sides by
2z(b* — d2). From this new triangle we may derive similarly a third, etc*
Apart from notation, this method is the same as the " construction " in
* Tiber die Pytbvg, Zabfen, Progr. Bunxlau, 18»4, p» 15.
11 If the hypotenuse ie 65f the Jega are 25, 60; 16, 63; 33, 56; or 39, 52.
•• Amer. Jour. Mftth,, 22> 1900, 327-8,
* Arcto Matb JPhys., C), 8, 1904, lSl.
«* Recherche Methodique et Propriety dee Trianglea Eectaoglee en Nombne Entiere, Paris,
1916, 21-27.
* Oeuvree, Ш, 254^5; S. ^ermat'fl Diophanti Alex. Arith,, 1670, 220.

Chap.ivj Right Triangles op Equai^ Ahea. 173
the second part of JYenicle's52 Traits; this process has been summarized
by A. f?i iri nrngh ятп .**
Fermat" stated that he could give five right triangles of equal area and
had a method to find as many as one pleases, whereas Diophantus, V, 8,
and Vieta, Zetetica, IV, 11, gave ooly three.
J. de Billy" noted that the right triangle with the legs 3r, (x + 4)r
will have the same area 6 as C, 4, 5) if f (x + 4)r* *= 6. Thus x + 4 and
9 + (x + 4J must be squares, which is the case if x ^ - 6725600/240560L
John Kersey*6 discussed the problem to deduce a rational right triangle
with the same area as a given one, and stated many problems on areas.
L. Euler" discussed the solution of
noting the case p = 11, r = — 35, q = — 23, s = 33. Hence the right
triangles formed' from 11, 35 and 23, 33 have equal areas.
Euler** noted that if we take q = p, p2 = r2 + re + «*, we get
9
Take p =P + 3g*t s - 4/g. Hence the values x = P + 3g*t у = 4fg or
Zg2 -P ±2fg give the sides 2zy, ^ — y5, s* + yJ of three right triangles
with tile same area xy(a? — y5).
GnisonM (pp. 109-114) and Young134 of Ch. XIX discussed the deter-
determination of three right triangles of equal area.
J. Collins48 employed the three right triangles with the legs
t»2 — x*t 2vx; if2 — y5, 2vy; z2 — y3, 2zv,
The first two are of equal area if у2 = я8 + xy -f ^ Set у = x — i.
Then z —. (? — y2)/^ + 2Q. The first and third have equal areas if
if* « x* - хг + г1. Set» = s - *. Thenx - (e2 - e*)/Ba - г). Tomake
the values of x equal, take t = m + n, у = m — n, * = p + q, z = p — q.
Then mn/Cm + n) = pqfCp + g) determines w in terms of p, g, n.
J. Cunliffe70 treated the problem to find к rational right triangles of
equal areas. For к = 3, let тг ± пг, 2тп be the sides of one triangle. In
mn(m* - л2) = pg(p2 - g2),
set p = m H- r, g = n — r, and solve the resulting quadratic for r. Thus
4r « ± VE — 3(w — n), where Д « ш2 Н- Umn + nJ. Set
^___ R => (m + w + 8J>
» Math. Quest. Edue. Times, 72r 1900, 31-2,
« Oeuvres, II, 263, letter to Mersenne, Sept. 1,1643, He had aaked (p. 259) for four.
• Inventum Novum, I, $ 38, Oeuvree de Fermat, III, 348, In his Dioph&atus Geomctra,
Pane, 1660, 108, 121, de Billy treated tho probleme of Diophantue V, 8, VI, 3,
* The Elemente of Algebra, London, Boolu 3 and 4, 1E74, 94, 124-142.
» Nova Acta Acad» Pfetrop., 13,1795 A?78), 45; Comm. Arith., II, 285,
м Opera poetuma, 1, 1862, 250-2 (about l7Sl),
M The Gentleman's Math. Companion, 2, No. Ц, 1808, 123.
" New Series of the Jtfatfc. Repository (ed., Th. Leyboum), 3, П, 1814, 60,

174 History op the Theoby of Numbers, [Chap. IV
thus determining m rationally. Hence we get two new rational right
triangles. For any k, let at b, A be the legs and hypotenuse of one right
triangle; another of equal area has the sides
2abk 26*-ft* h* + 4УА* - 4fr
26*- A5' 2h ' 2ABU*-A*) "
From this, we obtain a third, ete. To find any number of rational squares
h*, k^t • • • and a number N which if added to or subtracted from each of
the squares yields sums and differences which are rational squares, use
right triangles of equal area and take A2 = a2 + &7 h" « a" + У1, - ■ >,
N = 2ab = 2a$' = .... Cf. Ch. XVI.
D. S. Hart71 repeated the method of Diophantus V, 8.
A, Martin,** using the 3 triangles of Collins,6* concluded that the
conditions reduce to x — z — y, i? = z* — гу + y2, which is satisfied if
у = тг — n*, z = 2mn + m2, v = m? + ran + пг,
С. E. Hillyer73 noted that equal right triangles are formed7 from
Jt2 + hi + Л, k* - P; ifc* + *! + P, 2A;i + P; Й?1 + 2^i, *?• + « + P.
С Tweedie,* to find all rational right triangles of area A, discussed
a2 + /3* - 72, o0 = 2Л, whence zj + Й =* 1, T^iVi =* 2Л. Thus
Write x = ?n, у = A 4- *tt*)/-y. Hence we seek the rational points on
B) x(l - x2) = Ay\
To apply Cauchy's tangential method (papers 287, 296, ete» of Ch. XXI),
start with any right triangle with sides or, /3, 7 and derive the corresponding
rational point (x, y). The tangent there cuts the cubic at a new rational
point, which corresponds to a new right triangle with the legs 2офу({с^ — /3*),
(л* — 0*)/B7). From it we get a third right triangle. The problem is also
treated by Cauchy's second method (the line joining two rational points
of a cubic determines a third).
E. Bahier," pp. 149M68, treated the subject.
TWO BIGHT TRIANGLES WHOSE ABEAS HAVE A GIVEN RATIO.
Diophantus, V, 24, asked for three squares x\ such that x\x\t\ + x\
are squares for г = 1, 2, 3. A solution will be *> * sbi/pi if three right
triangles {pi7 bir hi) are found such that ргргр$ — s26i&2&3, since
» Math. Viator, 2, 1882,17-18.
" M&tb. Quest. Ednc. Times, 48,1888,
" Matb Quest. Edac. ТЬпев, 72, 1900v 30.
"Proc. Edinb, Math. Soc,t 24, 1905-6, 7-Ш. He quoted from "life aed Lettere of Lewie
OurolV' p, 343, thai the trmn^ee B0, 2l, 29) and A2, 35t 37} are equal, but failed to
find three.

Свл*.тг] Problems on Areas of Right Triangles. 175
Diophantus took C, 4, 5) as one triangle and stated that it is easy to find
two triangles such that the product of the legs of one is 12 (or 3) times that
of the other, as (9, 40, 41), (8, 15, 17). С G. Bachet74' chose an arbitrary
triangle (pif b:, hi) and the two triangles formed7 from bif hi and ph hlt
obtaining * = pijBh\)* Fermat75 gave general rules for finding two right
triangles whose areas are in a given ratio r/з, where г > s, vizv form the
triangles from 2r dr st r =F & and 2s ± r, r 4= $; or from 6r, 2r — & and
4r _j_ Sf 4f — 2s; or from г + 4$, 2r — 4s and 6$, r — 2s. Thus to find
three right triangles whose areas are proportional to given numbers r, sf t,
such that r + t =f 4a, t > tt form the triangles from r + 4st 2r — 4s;
6$, r — 2s; 4s + t, 4s — 2t. The areas of the triangles formed from 49, 2;
47, 2; 48, 1 are themselves the sides of a right triangle.*4
L. Euler*7 found ten types of pairs of right triangles whose areas
A ^ P?(p2 ~ Я2) and В = rs(r* — s2) have a given ratio a : b. He equated
r and s to two of the numbers p, 2p, qt 2q, p ± j. For example, r = p,
s = p — q give p + g : 2p — q = a : 6, whence p:g = aH-b:2a — 6;
taking r = p = a -|- 6, we get g = 2a~&, в — 2Ь — a» He gave (pp.
222-3) several methods to make A jB a square (cf. Euler** of Ch. XV, Euler81
of Ch. XVI, Euler18'19 of Ch. XVIII and Euler2" of Ch. XXII).
A. Homi78 noted that the problem leads to a cubic curve with two given
rational points, whence the chord determines a third»
Отжвв problems involving only area.
An anonymous78 Greek manuscript, probably dating between Euclid
and Diophantus, found the sides of a right triangle with the area 5 by
seeking a product of 5 and a square 36, divisible by 6, such that the product
5*36 is the area of a right triangle with the sides 9, 40, 41, and reduced
them in the ratio 1 :6,—which shows a knowledge of the fact that the area of
a right triangle with integral sides is a multiple of 6 (L, Hsano," Ch. XVI)*
Diopharjtus, VI, 3, required a right triangle whose area increased by a
given number g yields a square. Take g = 5 and denote the triangle by
(hx} px, bx); we are to choose % so that %pbx2 + 5 = nV. bet (h7 p, b)
be formed from m, 1/m and take n = m -\-2 >5fm. Then \pb = m1 — l/m\
When this is subtracted from n*, the difference shall be 5 times a square.
Hence 100m2 + 505 * D, say A0m + 5)*. Thus m = 24/5, n - 413/60,
x - 24/53. F. Vieta (Zetetica, V, 9) took g = r2 + a*7 formed the triangle
from (r + aJ, (r — sJ, and divided its sides by 2(r H- s)(r — sJ; the area
is now 2ra(rs + s2)f(r — s)*, which added to g yields the square of (r* + &*)/
(t — s). C.G.Bachet7** remarked that g need not be the sum of two squares
7<e Diopbanti Alex» Aritb- . ♦ , Cbmmentariie . . . Avctoie C. G» Bacheto, 1621,333^
7J Oravree, I, 319; French tianel, Ш, 259. Cf., П, 224-6.
"Otber solutions, Oenvree de Permit, II, Ш, 250, 277^ Oeuvt«8 de Dteecartee, Щ Ift5.
De Billy e^e the tn&Dgles formed from 6, 1; 7, 6; 8,1; Oeuvres de Fermat, IV, Ш2,
139; Bull. BibL Storia Sc. Mat. Ffa., 12, 1879, 517,
77 Opera posturaa, 1, 1862, 234-7 (about 1773).
7' Proc. Edinburgb Math. Soc., 22, 19<Я-±, 48.
79 With German transl. by J. L. Heiberg and comments by H. G. Zeuthen Bibliotheca Math^
C),8, 1907-8, 121-131.

176 History of the Theory or Numbebs. ГСкар. iv
and solved the problem when g — 6. Fermat (Oeuvres, III, 265) pointed
out the probable origin of Vieta's unnecessary assumption on g. Let the
triangle be formed from ax2t a; its area xV(x* — 1) increased by bz* shall
give a square» Siace 5 is a sum of two squares, we can determine у so
that 5y2 — 1 = □. Take у ~ x + 1; then x4 — 1 + &У2 c*11 readily be
made a square» But Vieta did not observe that the problem can be solved
when x4 — 1 is replaced by 1 — x4 since we can solve gy* + 1 = □. Fermat
found the triangle (9/3, 40/3, 41/3) whose area 20 increased by 5 gives 5\
The history of the theorem that the area of a rational right triangle is
never a square or double a square is given in Ch. XXII, where are given
Backet's and Vieta's comments on the problem to find a right triangle
with a given area.
Fermat80 stated that the area of the right triangle with the sides
2896804, 7216803, 7776485 is of the form 6u*; likewise for the triangle with
the sides 3, 4, 5. E. Lucas81 obtained these triangles and that with the
sides 49, 1200, 1201 and area 6G0)*. He noted that the area of a right
triangle is never a square, nor the double, triple or quintuple of a square.
Fermat's problem to find three right triangles the sum of whose areas
by twos are sides of a right triangle was solved by Gillot at the request of
Descartes.82 The triangles
24 35 337\
' 12' 60 /'
have the areas 7, 14, 21, whose sums by twos are the sides 35, 28, 21 of a
right triangle. Gillot gave also the areas 15, 30, 45 and 7 more sets.
Miscellaneous problems involving the area and other elements.
In an early Greek manuscript7* there occurs the problem to find the
integral legs а, Ь and hypotenuse с of a right triangle such that the sum of
the area T and perimeter 2s is a given number A. The solution given for
A *=* 8-35, 6*45, 6*20, 5-18 is made clear if we introduce the radius r of
the inscribed circle, whence T = rs = «Ь/2, г + s = a -\-bt с = s — r.
Separate A into two factors s, r + 2 such that (r + $)* — 8rs is a perfect
square n2. Then 2a, 2b = r + s ± n. Cf\ E. Bahier,*2 pp. 190-9.
Diophantus VI, 6-9 relate to right triangles whose areas increased or
diminished by one leg or by the sum of both legs shall be a given number g.
To solve the first two problems, Fermat formed the triangle from g, 1
and divided the sides by g + 1 or g — 1; he enunciated the problems to
find a right triangle such that one leg or the sum of the legs diminished by
the area is a given number. Cf. E. Bahier,M pp. 170-X90.
Diophantus, VI, 10 pi], found a right triangle B8z, 45x, 53s) whose
area increased ^diminished] by the sum of the hypotenuse and one leg is 4.
«•(китов, 1П, 256, 348; comment on Dioph&Dtue V, 8 and Inventum Novuib, 1, §38T
Of. A. Genocchi, Annali Sc. Mat. Fie,, 6, 1855, 319-20.
«BuIL BibL Stori* 5c Ma*., 10,1877, 290.
•«Оеаугея, П, 179; letter from Descartes to Mer&eDne, June 29, 1638. Of. Oeuvres de
Fennst, IV, 1912, 56.
\ /8 21 65 \ / 7 25\
/' \3' 2 ' 6 /' \ ' 21 2 )

Chap. IV] PROBLEMS ON" AREAS OF RlGHT TRIANGLES. 177
Fermat asked that the sum of the hypotenuse and one leg, diminished
by the area, shall be 4; the answer A7/3,15/3, 8/3) is given in the Inventum
Novum, III, 33 (Oeuvres de Fermat, III, 389). Bachet found a right
triangle whose area increased (or decreased) by the hypotenuse is 4.
Diophantus VI, 13 relates to a right triangle (px, bxT hx) whose area
increased by either leg is a equare. Let A *= pb/2. From Ax* 4- bx = m*x*7
x = Ь/(тг — A). Then Ax1 + px = П requires that
pbm*-\- АЬ(Ъ - p) = D.
As in VI, 12, we may choose (pt b, k) similar to C, 4, 5) so that the greater
leg Ъ, b — p and p + A are all squares, say b — p ^ m2. The preceding
condition is thus satisfied. Fermat's method (Oeuvres, III, 267) yields an
infinitude of triangles not similar to C, 4, 5).
Diophantus, VI, 15 [17], gave a right triangle (&r, 15z, 17s) whose
area diminished [increased] either by the hypotenuse or one leg is a square.
FermatM required that on subtracting the area from, the hypotenuse or
one leg each difference be a square-
Diophantus, VI, 19 [20], required a right triangle the sum of whose area
and hypotenuse is a square [cube], and perimeter a cube [square]. His
solution and various related papers are considered in Ch. XX.
Diophantus, VI, 21 [22], required a right triangle the sum of whose
area and one leg is a square [cube], and perimeter a cube [square]. Use
a triangle given by the rule of Pythagoras,1 after dividing its sides by
a -h 1. The perimeter 4a + 2 is to be a cube. By the other condition,
2a + 1 = П. But 8 is the only cube which is double a square* Hence
a = 3/2.
Diophantus VI, 23 [24]M relates to a right triangle the sum of whose
area and perimeter is a cube [square], and perimeter a square [cube].
Use a triangle given by the rule of Plato.1 The perimeter p = 2a2 + 2a
is a square for a = 2/(ти* — 2). Then a(o* — 1) + p and hence 2m is to be
a cube for 2 < m* < 4, which is the case when m = 27/16.
Bh4scara12 found a right triangle whose area equals the hypotenuse.
C. G. Bachet, at the end of book VI of his edition of Diophantus, added
22 problems. In the first 13, we are given the perimeter, or hypotenuse or
area of a rational right triangle and seek the maximum or minimum of some
specified function of the sides» In 14-18, we seek ths sides, given the sum
of the legs or perimeter pt or p and the area A} or p and the product of the
sides. In 19, p and р±Л are to be squares. In 21 and 22, we are given
p or A and the perpendicular from the right angle to the hypotenuse.
J. de Billy*6 found a right triangle in which one leg, the sum of the legs,
and the excess of each leg over double the area are all squares. If x and
у = 1 — x are the legs, the conditions are that у and x* + y1 be squares,
as is true if x = 40/49» If we formulate the problem algebraically and then
* Hie solution is in Inventum Novum, I, 26, 40; Oeuvres, Ш, 34l, 340.
u for VI, 24, see T. L. Heath, Diophantuu, iS85y 256-7; l910y 244^5; P. Mansion, Mathews,
D), 4, 1914, 145^9.
* Inventum Novum, I, \ 52; Oeuvres de Fermat, Ш, 359.

178 History op the Theory of Numbers» [Chap.iv
interpret as the hypotenuse the letter which stood for one leg, we have a
new problem solved by A. Cunningham.**
Fermat87 proposed that St Martin find two right triangles whose areas
are in a given ratio and such that the two legs of the larger triangle differ
by unity.
Fermat" noted that if in B05769, 190281, 78320) we add the area to
the square of the sum of the legs, we get a square.
Frenicle*9 stated the last result without comment; also that the sum
of the area and hypotenuse of A7, 144, 145) is a square; while the first
three right triangles in which the sum of the area and smaller leg is a square
are C,4, 5), A6, 30, 34), A05, 208, 233).
J. de Billy90 treated a large number of problems on rational right tri-
triangles.. In the first 44, a prescribed multiple of the area when added to
or subtracted from certain sides gives squares. The next five involve
the perimeter. In Prob. 58, the cube of the sum of the hypotenuse and
one leg when increased by a given multiple of the area shall be a cube,
while 55-67 are analogous. In Prob. 68, the areas of C0-23*, 18-23*,
24-28n) are seen to form a geometrical progression of ratio 23, while 69-73
are similar. The 120 problems of Ch. 2 do not involve areas, but make
certain functions of the sides squares and cubes.
J. Ozanam91 found that in the right triangle whose sides are the ratios
of 2264592, 18325825 and 13465217 to 20590417 each side exceeds double
the area by a square. This problem was proposed in obscure verse in the
Ladies' Diary for 1728 as Question 133; a modified uninteresting problem
was solved in 1729.
C. Wildbore*2 took x and 1 — x as the legs; they exceed the double area
by s2 and A — x)\ Equating the hypotenuse k to v(l -i)|x, we get
x *= A - t?)/(l + 2v — \P). The condition h - (x - хг) = D becomes
1 + 4t^ - y* = Q. First, take v = Ъ/а, Ь - d - 3, a - d + 5; then
4^ -|- ... = □ *= B<F - 260d - 2J for d = 4223/66, which yields Oza-
nam's answer. The next value of v is said to be 491050/555466, which
gives x = 8426546832/76616941657. Elsewhere" he took
By the radical in the solution for v,
2A - n)B + n + л2) = □ = 4r»(l - n)\
say. Solving for n, we see that 4г* + 12т2 - 7 ** Q. Take r = ajb,
и Math. Quest, and Solutions, 3, 1917, 79^80.
v Оешга, II, 252y letter to Meraenne, Feb. 16, 1643.
" Оецдггея, II, 263 B60, 3°), letter to Meraenne, Sept, 1,1643.
"Methode pour trouver la solution dee probl&mea par lee exdueiooe, Ouvmges de Math.,
P&rie, 1693; Mem. Acad. R. Sc. Pane, 5,1666-99 A676J, ed. 1729, 56.
•" Diophanti Kedivivi, Lvgdvui, Ш0, Pane Prior, pp. 1-302,
« Nouveaux ettmens d'afcebre, 1702, 604.
*I*dieer Шигуу 1772, 40-1, Quart. 638; T\ Leybourn'a Math. Quest, from Ladiee7 Diary,
2, Ш7, 342-5; С Hutton'a Diarian МщсеШюу, 3, 1775, 356-7,
»a HuttODrs Miscellanea Math-, London, 1776, I63r4; LeybournTe Math. Qiieet. L. D.,
2,1817, 342^5.

Сяар. IV] Problems on Areas of Right Triangles. 179
a = d+l,b-d—1 and equate the quartic in d to the square of 3 +
22d/3 - 43^/27; thus d = 202752/179200, which gives the last answer.
A longer analogous discussion led to the new value r = 50929/46200, which
yields an answer involving numbers of ten digits»
T. Leybourn*1 took xj{x + V) and уКя 4- У) && the legs» since each
exceeds double the area by a square» Take x = m1 — n*yy = 2mn. Then
the hypotenuse exceeds double the area by a square if m* + 4mn* — л4 - □.
Take m — 1 + v, n = 4, and equate the quartic in v to the square of
r2 - 130» + 1? whence v = 4223/66. Or take m = v + 5, n = v - 3, aod
equate the quartic in v to B^ - 236v — 2K, whence » =* 7619/176.
Malezieux9* proposed to find two right triangles the sum or difference of
whose perimeters is a square; the difference of the areas a square; the
difference of the least side of the first and the least side of the second equals
the difference of the two largest sides of the first or of the two largest sides
of the second, the difference being a cube; the difference of the largest leg
of the first and the least leg of the second is a square; and the sum of the
least side of the first and the medium side of the second is a square-
L4 Euler* discussed the problem proposed by Fermat (on the margin
of his Diophantus VI, 14): Find a right triangle such that each leg exceeds
the area by a square. Euler denoted the legs by 2x/z, yfz, where £ = aby
у = a? — h2. Subtract the area xyjtz. Hence 2хг — xy and yz — xy are
to be squares. Let their product be the square of xy — yzpjq. Hence
z-x = x*y(p - qYlk, 2z-y = x{2qx - pyYjk, к = 2^ - p*yx.
It remains only to make к a square, say rV. Thus x : у - p2 : 2q* — r2.
Taking the proportionality factor with z, we may set x - рг, у = 2ф - г*.
Then г = р* + (р - q)\2q* - г*Iт\ The condition 4x2 + у* = □ be-
becomes Е = 4р4 + Bд* — г2)* = П. Special solutions are obtained by set-
setting Vff = 2p* =F r2, 2p* ± 2^ or тг + 2q2 =fc 2ргш Returning in § 20 to
the general case, Euler expressed к — rh? in the form
a2hz
ab(a* - V) = ~ Bq* - r*) = 21* - u\
V
Every product of primes 2, &m db 1 and a square is of the form 2t2 — u2
and only such products. Moreover, if a product of two numbers whose
g.cd. is 1 or 2 is of the form 2t2 — uz, each factor is» Hence at b, a 4- 6,
a — Ь must each be of the form 2J2 — u\ Conversely, when this is the
case, solutions of the initial problem can be readily found» Euler tabulated
the permissible values a < 200 for each permissible b < 100, and gave
formulas for p, q> rf z~
To find a right triangle whose area increased by the square of the
hypotenuse is a square, J. Whitley*7 wrote rs(r2 — s2) + (r2 + s2J = a2
« Math, Quest, from Ladle*' Diary, ly 1817, 173-5,
•• Elements de Geom^trie de M. ]e Due de BouiBogne. par deMftlezieux, 1722. Solved by E.
FauquemberKue, Sphinx-Oedipey 2, 1907-8, 15-16.
« Novi Comm. Acad. Petrop., 2r I749y 49; Comm, Arith., Iy B2,
w The Gcntieman'ia Math. Companion, 2r No. Ш, 1807, 69.

180 Hibtohy of the Theory op Nuhbers. £Cha>. iv
and took r = t - 8s, a = & - mts + 61s*, and found t = 3S39s/488,
г = — 65s/488. J. Wright took a =* r2 + я2 + Jre aad found г = - 8$,
which does not give positive answers. Hence set r = / — 8$.
" Calculator " " found three right triangles of equal perimeters and
areas in arithmetical progression. The areas are proportional to the radii
r of the inscribed circles; for the sides 2amn% a(m2 ;£ n2), r — an(m — n).
A long computation yielded triangles ell of whose sides involve eight digits:
A8601944, 13951458, 23252430), A8559223, 13999464, 23247145),
{18515584, 14048388, 23241860).
W. Wright" found a right triangle whose perimeter is a square and
area a cube by taking rn2 ± n*t 2mn as the sides. Let the perimeter equal
д*т*> whence m = 2n/(g* — 2). Then the area is a cube if
Ы - 2n(q2 - 2I = s3,
which gives n. " Epeilon " took p(m* d= n2), 2pmn as the sides. The
perimeter is a equare if p = 2m(m -\- n). The area is a cube if 4n[m — n)
is, whence either n is a cube and the double of m — n is a cube or vice versa.
To find a right triangle the sum of whose sides equals the area, many
solvers noted that 2s2 + 2rs = rsjs* - r2) implies - 2 - r* - sr. The
root t involves the radical V<s* — 8, which is equated to s — x, giving
s = (8 4- x*)fBx). For integral solutions we have X < s, whence X = 4,
* = 3, r - 2 or 1 and the only triangles are A3, 5, 12), A0, 8, 6).
J» Baines,101 to find two right triangles the differences between whose
bases, perpendiculars, hypotenuses, perimeters and diameters of inscribed
circles are all squares, and difference of areas a cube, took 257»2 — nl,
\0mn and 25?n2 + n3 as base, perpendicular and hypotenuse of one, and
9m2 — ri*, Ьтп, 9m2 + n2 for the other, so that we have only to make Amn
and A = 32m2 + 4mn squares and В =* 98m*n — 2mnl a cube. Take
mn ~ a2. Then A = DifSa2 -\-n2 = П = Bar/s + nJ. Taker =* 8 = 1,
whence n = a = m* Then В = 96a4 is a cube if a = &V96. G. Heald
took the triangles (Ш2, 24z2, 26s2) and (fa*, 8x2, 10x2). All but the last
condition is satisfied identically. The difference 96х* of the areas is a cube
if* = pVi2.
J. Daveyiw found a right triangle whose perimeter is a square p2 such
that p8 equals the area. Take pr, pst pt as the sides. Then r = p — $ -~ t,
s = 2pft, and r* - s* + **, which gives p - 2*(* - 2)/(< - 4).
Many108 found the sides a, b and hypotenuse с of a right triangle such
that af с Н- Ъ, с — b are integral cubes, say p3, m8, n*. Then с2 — Ъг ^ a2
gives m/i =* p2»
«« The Gentleman'a Math. Companion, 4, No. 22,18X9, 861-1. Cf. Perkine.l«
w Ibxd.t b> No. 28,1826, 371-3.
"■ UdW Diary, 1828, 34, Queet. 1466.
i« Ludie»J Diaiy, 1830, 37, Quest. 1500.
1в» ТЪе Lady's aad Gentleman'* Diaiy, Londoo, 1841,58 (Quest. H16 of Gentleman7» Diaiy,
1840).
l*Ihid.t 1845, 51-2, Queet. 1722.

Chap, iv] Problems on Areas of Right Triangles. 181
G. R. Perkins101 noted that the triangles D0, 30, 50), D5, 24, 51),
D8, 20, 52) have equal perimeters and areas 600, 540, 480 in arithmetical
progression*
V. J. Knisely108 found the'same result as had Perkins, by taking as the
sides
{p% + 2pq)a, Bpq + 2q*)a, (p2 + 2pq + %f)a>
(p2 ~ <f)b, 2p^6, (p2 -\- q*)bt
(p* — 4#*)c, ^VQC* (p2 H~ 4^)c.
The conditions reduce to
(p + 2q)a = pbt (p + q)a « pc, 2(p - q)b = pa + 2pc - 4qc.
Substitute into the third the values of &, с given by the first two conditions;
we get p =* 4?j whence b =* 6o/4, с = 5a/4. For q = 1, p = a ~ 4, с = 5,
b = 6, we get the answer cited. A. B. Evans gave a long discussion said
to give the complete solution; but his numerical example involves very
large numbers.
E. Lucas proposed and Moret-Blanc*06 solved the problems to find a
right triangle such that the square of the hypotenuse increased or diminished
by the area (or by double the area) is a square.
Lucas107 showed that the method of descent leads to a complete solution
of the second (double area) of the last two problems.
C. de Comberousse108 discussed rational right triangles whose area and
perimeter are equal. Eliminating z between x2 + y2 = & and
x + у + z - xy/2,
we get у = 4 + S/(x — 4). Thus x — 4 is a divisor of 8, and the only
solutions are (x, y} z) = E, 12, 13), F, 8, 10).
A. Holm109 discussed a problem including the cases of Diophantus VI,
6-11, and the additions by Bachet and Fermat: Find a rational right
triangle such that the sum of given multiples of the area and three sides
shall be a given number. Taking (x2 db l)fy, 2xfy as the sides, the con-
condition is
The discriminant of this quadratic for у is a quartic function Q(x) in which
the coefficient of x4 and the constant term are squares. There are many
known methods of making Q(x) a square.
Right triangles whose legs differ by unity,
A. Girard1Ma gave fourteen such triangles in which the least leg is 3, 20,
119, №, 4059, 23660, 137903, 803760, • > ■, 31509019100.
^ The AnaJyety Dee Moines, 1,1S74, 151-4. a, Calculator.»
1Л Math. Quest. Edue. Times, 20,1874, 81-3.
i»Nouv. Ann. Math., B)r 14P 1875, 510; B), 20, lS8l, 155-160.
"*BulL BibLStoria Sc. Mat,, 10, 1877, 291-3.
1M Algebra euperieure, 1, 18*7, 190-l
lwProc. Edinburgh Mslth. Soc., 22, 1903-4, 45-8; Matb. Quest. Educ. Time», B), 10,
1906,47-^8.
"*■ L'aritL de Simon Stevin - ■ pax A, Girard, 1625, 629; Oeuvree, 1634, 158, col. 1.

182 History of the Theory of Numbers. ССлаг. rv
From one right triangle (x, x + 1, г) whose legs are consecutive integers,
Fermat110 deduced the second triangle (X, X + 1, Z)t where
X = 2s + 3* + 1, Z = 3z + 4z-\-2.
For example, we have the series C, 4, 5), B0, 21, 29), A19, 120, 169), -
The alternate triangles give solutions of the problem to find right triangles
whose least side differs from the other two sides by equares. He noted
later (pp. 232-3) that such a triangle is formed from r2 + #*, 2s(r — «)*
Fermat111 noted that the sixth such triangle is B3660, 23661, 33461).
From the first such triangle C, 4, 5), we get the second by taking the double
(viz», 24) of the sum of the three sides and subtracting separately the legs
and adding the hypotenuse.
J. Osanam11* gave the first six such triangles. If one is formed (Bio-
phantus7) from mt n, where m > ny the next is formed from m, 2m + n.
In the edition by J. E. Montucla, 1, 1790, 48, the triangle is formed from
any two consecutive terms of 1, 2, 5, 12, 29, 70, • • •, fc, where к is such that
one of the two numbers 2k? ± 1 is a square. The same rule was given
by Gruson.22
C. Hutton111 noted that, if prfqr is the rth convergent to "^2, then
PrPr+i and 2grtfr+i are consecutive integers the sum of whose squares is a
equare qlr+*
Du Hays114 gave triangles the difference of whose legs is 1 (or 7).
L. Brown115 gave the first six and the eleventh such triangles.
G. H. Hopkins and M. Jenkins116 reduced the problem to xl — *hf — rh 1,
and gave recursion formulas for the solutions. A. B. Evans used the con-
continued fraction for V£ Cf. Moret-Blanc154 of Ch. XII.
Judge Scott117 gave the first eight and the eleventh.
A. Martin118 employed the legs Hx =Ь 1), whence x2 — 2уг =* — 1, and
the odd convergent^ xjyn to the continued fraction for <$. Thus
xb = 6sR^i — £„_2 and likewise for the y'$> Also
2xnt 2 <2yn = A 4- &)*л+г ± A - iS)a-+J.
The eightieth such triangle is found.
T. T. Wilkinson stated and J. Wolstenholme119 proved a rule equivalent
to a recursion formula for the solutions of x* — 2y2 ^ 1.
D. S. Hart120 took x and x + 1 as the legs. Then
2s* + 2x + 1 = П = (xp/g - XJ
«• Oeuvres, II, 224^5. Reproduced iu Spbinx-Oedipe, 7,1912 103-4.
ш Oeuvre», II, 258- letter to St, Martin, M»y 31,1643; reproduced, Sphinx^Oedipe, 7t 1012,
104.
1U Recreations Math,, 1,1723; 1724; 17S57 51; etc. (firet ed., 1696).
i" English tranal. of Оадоатта Recreatior», 1, Ш4, 46.
i" Jour, de Math-, 7, 1&42, 331.
ш Math. Monthly (ed., Runkle), Cambridge, Maas>r 2t 18бОу 394^
i" Math, Quest. Educ. Timeey 12, 1869, 104-6,
117 Of commensurable right-angled triangles » ■ » , Bueyrua, Obbr 1871, 23 pp.
"• Math. Quest, Educ, Times, 14,1871, 89-91; 16,1872,107; 19,1873,89; 20,1874, 21y
™ Ibid>, 2Oy 1874P 0799
a/№634

Сват, nrj Right Triangles With a Given Difference of Legs. 183
gives x = Bpq + Зд2)/^, where d = p* — Зд2. He made d = =b I by use
of the theory of Pell's equation»
A. Martin121 gave the nth triangle for n =* 80 and 100.
P. Bachmann122 proved that the only integral solutions of я2 + у* = z*
in which z > 0, while ж and у are consecutive integers, are those given by
z + jr + zV2 = A + V2)-C + 2V2)* D = 0,1,2, - >)-
Several writers123 obtained the first six triangles.
K. W. D. Christie124 noted that the solution of x2 + (x + 1)* = Уг ш
integers is
ж = 2<> + 2i + > - - 4- 2,r-i, У = 2gr,
where 2r is the simple continuant of order r all of whose diagonal elements
are 2. This was proved by T. Muir,125 who cited Fermat'flua rule.
A. Martin12* noted various methods. The first three methods are based
on the solution of 2&* ± 1 = □ [Ozanam,312 Hutton,111 Bachmann].
Fermat's method wag used to compute a table (p. 322) of the first forty
such triangles.
A* Levy117 found when two of the numbers A) are consecutive. Evi-
Evidently z — у = 2п2 Ф 1. Next, г — x = (m — nJ = 1 for m ~ n + 1.
Finally, у - x = =h 1 gives (m - n)* - 2л2 = =b 1. Write A - V2)* in
the form a — bV2; then a, 6 are integral solutions of a2 — 2b? =* {— 1)',
and all solutions of «* — 2if - ± 1 are said to be obtained in this way by
using all integral values of p. Or we may compute the solutions of the
latter by the recursion formulas of G. Fontene2* of Cb XII. We get
C, 4, 5), B1, 20, 29), A19, 120, 16»), F97, 696, 985), D060, 4050, 5741).
G. A. Osborne1*8 discussed the problem. Cf. Barisien100 of Ch. IX.
Several129 made use of zl—2}f ^ — 1. F. Nicita180 employed recurring series.
Right triangles the difference d or вим of whose legs ie given.
Frenicle1*1 stated that every number is the difference of the legs m an
infinitude of ways, every prime 8n + 1 or product of such primes is the
difference of the legs of an infinitude of primitive triangles. To find all
triangles with d = 7, start with E, 12, 13) formed from 3, % and take that
formed from 3, 2-3 + 2, etc» A second series is found similarly from
(8, 15, 17), formed from 4, 1. He discussed right triangles the sum of
whose legs is given.
>u The Analyst, Dee Мошеау 3,1876, 47-50' Math. Visitor, i, 1879, 56,122 (егтопеоия value»
forn = 5, 6 occuroq pp. 55-6)»
«* ZahleotheOrie, 1, 1802, 194-6; Niedere Z-, 2, lftlO» 436.
« Amer. Math. Monthly, 4r 1897, 24-28,
i* Math. Gaaette, ly 1S96-19OO, 394.
ш Froc. Roy. Soc, Edinburgh, 23, 1899-1901, 264-7.
м Math. Magazine, 2, l9l0y 301-24.
itf Bull, de math^ elemeotaifea, 15, 1909-10, 165-6.
u« Amer. Math. Monthly, 21, I9H, 148-150.
m b'mtennediaire dea math., 22, 1915, 139-144, 185-8.
»« Periodico di Math., 32y 1917, 2О0-2Ю.
™ Oeuvreu de Fermat, П, 235-6, 238-41, letter to Fermat, Sept. 6,1641.

184 History of the Theory of Numbers. [Chap, iv
Fermat1*2 noted that d = 7 for E, 12, 13) and (8,15, 17); from these we
get all by his111 rule.
Frenicle*' examined the 16 triangles with hypotenuses < 100 and found
that d = 1,7, V, 17, 23, 31, 41, each of the form 8n d= 1, The triangles
formed from n, m and m} 2m + n have the same difference of legs.1*2*
T. T. Wilkinson1" would start with a solution of a* 4- Ь* = с1 and form
a — a -h c, 0 = 6 + 0, *y — a + & + c; repeat the process; we obtain
a' =* a _|_ 7 = 2a + Ь + 2c, V = /3 + т = a -f 26 + 2c,
c' = a_|_0_|_7 = 2a + 2&-|-3c,
which are sides of a new triangle with a' — b' — a — Ъ. H. S. Monck
(pp. 20-21, 76) failed in his attempt to prove that if we start with Cn, 4n,
5n) and apply the process repeatedly we obtain all triangles with the same
difference of legs. J. W. L. Glaisher (p. 54) noted that the proof is
inadequate. Proof was given by S. Tebay (p. 99) and P. Mansion.1"
T. Pepiniss considered the problem of Fermat (Oeuvres, II, 231) to find
the number of right triangles the sum of whose legs is a given number A.
To the resulting condition x* — 2jf = A we may apply the theory of
quadratic forms and show that, if A = cf- - >су, where a, * • *, с are primes
SI ± 1, the total number of primitive triangles whose sum of legs is A is
i(Ba+l)..-<2T+l)-l|.
J, H. Drummond and M. A. Gruber136 found solutions when d is given*
Several137 treated the case d = 7.
E. ВаЫет,*2 pp. 72-120, treated the problem at length by recurring series.
TWO RIGHT TRIANGLES WITH EQUAL DIFFERENCES OF LEGS, AND LARGER
LEG OF ONE EQUAL TO THE HYPOTENUSE OF THE OTHER.
Frenicle138 proposed the problem to J. Wallis. Wallis (Aug., 1661) took
two overlapping triangles ВАС and ВСЕ with the respective hypotenuses
ВС = 5 + x and BE. Take BA =- 5 - x. Then ВС2 - BA* = 20x is a
square if 5x is; take 5 — ba*, x = he2. Then
ВС = Ъа* + be2, BA = ba2 - Ье\ АС = 2Ш.
On AB lay off AD = AC; on ВС lay off B6 = BD. Since
ВС - СЯ = AB - AC = BD
ш Oeuvree, II, 258^9; letter to St. Martin, May 31, 1643.
"^Oeuvrefl de Fertnat, II, 235-7.
w Math. Quest, Edue. Tiraee, 20,1874, 20,100. G. H, Hopkins, p. 22. On the proof flh«eta,
E. B, Escott noted that "this process can be applied to other trianglee than right-angled
triangles. Under this transformation, & — 2ob as weQ as a — b is invariant. Cf.
Dickson.'
ш Mathems, C), 6,1906,113,
>* Mem. Pont. Aocad. Nuovi Lincei, 8. 18S2, 84-108: extract» Oeuvree de Fermat, 4, Ш2,
205-7; cf. 253.
"•Amer. Math. Monthly, 9, 1902, 230, 202-3.
117 Math. Quest. Educ, Times, B), 7, 1906, 88-9,
wCf, C. Henry, Bull. Bibl. Storia Sc. Mat. Fis., 12, 1879, G95: 13, 1880, 446: 17, Ifc84,
351^-2.

Сяар. IV] Problems ok the Sides of Right Triangles. 185
by hypothesis, CE = &C = 2be2 + 2bae. Hence
BE1 = ВС2 + СЕ1 - &'/, / = a4 + 5e* + 6W + 8ae*.
It remains to make / a square, which Wallis suspected to be impossible,
Frenicle (Dec. 20, 1661) took a = 2, e = 4, whence / = 52* [whereas we
desire a > e~\- Fermat139 formed the first triangle from N + 1 and 2.
Then the legs of the second triangle are N* + 2JV + 5 and 4N + 12; by
their sum of squares,
jV* _|_ 4№ 4- 3(W* + 116JV + 169 = П - f 13 + ~ JV - ЛР V,
say. Thus JV = — 1525/546» Hence we use as the first triangle that
formed from 4- 979 and 2 >546. The resulting triangles are
B150905, 2138136, 234023), B165017, 2150905, 246792).
If we had used the sum of the legs instead of their difference, we would
obtain the simpler solution A517, 1508, 165) and A525, 1517, 156).
T. Pepin140 noted that the initial problem is equivalent to
C) x2 + y2 = z2, u% + у2 ^ x*, u~v = x-y>0.
We have u, v ~ a1 — e*f 2ae; x = a* 4- e1. According аз и is odd or even,
у = 2e{a + e) or 2a(a — e). Then the first condition becomes
*2 = a< _j_ 5^4 ^. foV ^f 8a^ Or г2 = 5a< -h ^ -f 6aV - 8tffe,
according as the larger leg of the smaller triangle is odd or even. Contrary
to Frenicle's solution a = 2, e = 4, the geometry requires a > e. But
we can satisfy the first condition by taking x = d(m* — n2), у = 2dmn,
z ^ d(m* + n2), where d = 1 if x, у are relatively prime, and d ^ 2 if
are even, while m, n are relatively prime and one is even. Then
— 2e(a + e), which is completely solved by
D) m = o)9, n = hkf e = &kt a -h e — оЛ, or e = аЛ, a-|-e* 20fe,
according as ^ = 1 or d = 2, where a, ft Л, ft are relatively prime in pairs,
the first three being odd and к even» Whether d = 1 or d = 2,
d{m2 - n2) = a2 + ^
gives
E) k\kz + 2?) ~ 2opAA + «*(A5 - £*) - 0.
Solving this for к/а or hffl, and making the radicals rational, we get
2/5* — A4 =5 D, a* — 2A4 = D, which have been completely solved by
Lagrange*4 of Ch. XXII, so that we know all solutions under a given limit.
Then D) give solutions of the proposed problem. We may also solve E)
by a method equivalent to that of Eiiler"*-145 of Ch. XXII. Set h{p = f,
к/а ^ Vt then
(e + 2w-2tv+?-i = o.
Я» Inventum Novum of de Billy, in S. Fermat'fl Diophanti Alex* Arith., 1670, 34-35. Oeuvree
de Femmt, 3, 1896, 393-4; 4, 1912, 132.
w Atti Accad. Pont. Nuovi linoei, 33, 1879-80, 284-9- extract in Oeuvras de ibrroat, 4,
1912, 219-220.

186 History of the Theory or Numbers» [Chap. IV
Call щ tj' the values corresponding to the same £; and f, £' the values
corresponding to the same 17. Hence
Hence all solutions follow from the primitive solution 7 = 0, £ = 1:
t _ 1 _? >__! _ _ *i 1343
€ - ii V 3, €1 - 13i *i n3, fc -
The second set is said to furnish the least positive solution of C):
x « 2150905, у -= 246792, z = 2165017, и = 2138136, v - 234023.
M. Martone1*1 satisfied the first equation C) by taking x — 2abf
у = a* — Ъ*, г* — а7 -\- &*. From the square of the third given equation,
we get x? — 2uv = гг — 2xy. Thus we have uv and и — v expressed in
terms of a, b. Thus
2v = a2 - 2ab - b2 ± r, r2 = 8a*&* - Ba& - a3 4- ^)г-
Taking a = 5b, we get r2 = 464, (f, u) ^ Ei2, - 8i?) or (8b*, - 662}.
MlBCELUJTEOtTS PROBLEMS INVOLVING THE SIDES, BUT NOT THE AREA»
Diophantus V, 25 relates to x\x\x\ — x) *= D for г = 1, 2, 3. A solution
will be %i *= ibijhi if three right triangles (pi} h^, hi) are found such that
kihjki — ^>iW»i. He took C, 4, 5) as the first triangle and ^ = 4» From
the triangles A3, 5, 12) and E, 3, 4), the ratio of whose areas is 5 : 1, we
can find two triangles such that the product of the hypotenuse and base of
one is 5 times that of the other» Indeed,142 he knew how to deduce from
a right triangle {a, 0, 7) a triangle (a, 6, c) with ac = ?yj2> where a and a
are the hypotenuses. He took a = a/2, b = @* — у2)/BаO с ^ pyja.
From A3, 5, 12) and E, 3, 4) he thus deduced F£, 119/26, 60/13) and
B£, 7/10, 12/5), the product of the hypotenuse and final leg being 30 and
6, respectively. Fermat14* gave two such triangles for which the ratio in
question is 5 :1, the sides being numbers of 10 and 11 figures (Oeuvres, I,
325; 111,263).
Fermat,144 to find two right triangles {p, bf h), (p', b\ hf) for which
p — b = bf — hf and b — h = pf — h\ took three squares r2, s2, t2 in arith-
arithmetical progression and formed the triangles from r ■+■ s, s and s + *, s.
From r = 1, s = 5, / = 7, we get A1, 60, 61), A19, 120, 169). We may
also take r = 7, s ** 13, t = 17.
10 Sopra un problema di analiai iudetemiixiata, Catjmsaro, 18S7,
«• Restoration ol the obecure text by J. O. L, Schuls, ** Diophantus/11822, 546-61.
'eP. Tanneiy, Boll. Math, Soc. France, 14, 1885-$, 41-5 (reproduced in Sphinx-Ocdipe, 4,
1909j 186-7), concludes that Fermat was aided by chance in obtaining his solution,
which is not genera] and contains aa error of eign. S. Roberte, Aseoo. frang. av. ac.f
15, IIj 1886, 43-9, discussed the problem. Both papers are reprinted in Oeuvrca de
Fermat, 4, 1912, 168-180. Thie problem of Fennat'a hae been treated by A. Holm
and A. Cunningham, Math, Quest. Educ. Times, B), 11, 1907, 27-29; special casee
by К J. Sanj&na and Cunninghamл ibid., B)f 13, 190B, 24-26; E Fauquetnberguej
|JintenD6diaire dee mstih., 24t 1917, 30-1; cf. 25, 1918, 130-1.
"• Oeuvree, U, 225, letter to Fienicle, June 15» 164L Cf, IIj 229j 232.

. IV] Problems on the Sides of Right Trianoles. 187
Saint-Martin asked how many ways 1803601800 is the difference of the
[larger2 sides of a right triangle whose least side differs from the other
sides by squares. Ferrnat145 replied that there are exactly 243 such triangles.
Fermat14* asked for two right triangles such that the product of the
hypotenuse and least leg of one shall have a given ratio to the corresponding
product for the other triangle.
Under 2x* — y* = □ in Cb XXII are discussed right triangles whose
hypotenuse is a square and either the sum of the legs is a square or the
least side differs by a square from each of the remaining sides»
Fermat117 gave A56, 1517, 1525) in reply to Frenicle's question to find
a right triangle in which the square of the difference of the legs exceeds the
double of the square of the least leg by a square» A. AubryHS obtained an
infinity of solutions by descent.
Frenicle89 noted (pp. 71-8) that if the hypotenuse and perimeter of a
right triangle are squares, the perimeter has at least 13 digits.
J. Ozanam149 gave a rule to find a right triangle whose hypotenuse exceeds
the larger leg by unity [Pythagoras1]. From the lengths of its legs form a
new triangle; its hypotenuse is a square. He found right triangles whose
base and hypotenuse are triangular numbers and altitude is a cube.
Wm. Wrightlb0 found a right triangle the sum of whose perimeter and
square of any side is a square. Let the sides be ax, bx> ex, where аг + Ьг ^ c*.
Then x2 + px, x2 + gxt x* + rx are made squares in the usual way (Ch-
XVIII), where p = $/a2, q =* s/bs, r = s/c2, s - a + Ь + с. Не and others
gave a similar treatment to find a right triangle such that the square of
any side exceeds that side by a square.
Several152 found a right triangle whose perimeter is a square, also the
sum of the square of any side and the remaining two sides, also the sum of
any side and the square of the sum of the remaining two aides. These seven
conditions are satisfied if the sum of the sides is 1/4. Take/(p2 =F g2), %fpq
as the sides. Equating the sum to 1/4, we get/.
R. Tucker and S. Bills1" found a right triangle with perimeter a square
and diameter of the inscribed circle a cube [or vice versaJ. Let the sides
be (p* d= q*)x, 2-pqx. Then 2p(p + g)x = □ = r2, and the diameter
2q(p - q)x is to be a cube, say r*/s*. From the two values of x we get r
in terms of q7 8.
A. B. Evans162* found a right triangle with integral values for the sides
a, 6, c, diameter d of the inscribed circle and side 8 of the inscribed square
having one angle coincident with the right angle of the triangle and having
*, II, 250, Jetter to Mereennej Jan. 27, 1643.
141 Oeuvrea, II, 252, letter to Mcreenne, Feb. 16, Д643-
1<7 Oeuvn*, II, 265, letter to Carcavi, 1644,
"* L'intermeduure deft math.p 20, 1913, 141-4.
w Recreations Math., 1, 1723, 1735, 52-5,
"* The Gentleman's Math. Companion, London, 5, No. 24, 1821, 59-60,
"i /У, 5, No. 27, 1824, 312-6.
1M /Aid., 5, No. 25, 1822, 157-9-
*« Matt. Quest, Educ, Timen, 19, 1873, 82.
™* Ibid., 21f 1874, 103-4.

188 History of the Theory of Numbers» [Chap. IV
the opposite vertex on the hypotenuse c, and such that d + s is a square.
Take the sides to be the products of the numbers A) by uv. Then 8 =
abcjiab + с2) equals uv times a fractional function of m} n, whose denomi-
denominator is taken as u. Since d — a + b ^ с = 2n{m — n)uvt the condition
d -\- s = D is of the form Av = Q and holds if v = A.
S. Tebay154 noted the existence of an infinitude of pairs of right triangles
with the same hypotenuse such that the differences between the hypotenuse
and the legs are a square and double a square.
G. de LongchampsWe stated and Svechnikoff proved that x2 = #* -|- z2 has
an infinitude of solutions for which x + у is a biquadrate.
Several155 found right triangles with the base 105, and two right triangles
with the same base which is a mean proportion between the two perpen-
perpendiculars.
To find any number of dissimilar rational triangles of equal perimeter,
It. W. D. Christie166 multiplied the sides of special triangles by suitable
common factors, while A. Cunningham employed A) and solved
m(m -)- n) = const.
A. Gerardin1*7 noted that, to find two right triangles having the same
sum of squares of the hypotenuse and one leg, we have to solve
and gave a solution in which x> yf a, & are functions of the seventh degree
of two parameters.
R. Janculescu1*8 noted that the problem to find a right triangle with
integral values for the sides and perpendicular from the right^angle leads
to 1/s2 + 1/y* = ljz\ Thus я* + y2 = P. Let d be the g.c.d. of x = da,
у = dfi} t = dy. Then z — ± dct&ly, so that d must be a multiple of y.
E. Turriere159 discussed right triangles each of whose sides is a sum of
two squares, as 9 = 39, 40 ~ 22 + 62, 41 = 4* 4- 52.
E. Bahier,w pp. 122-148, investigated right triangles with a given
perimeter.
Right triangle with a rationai* augle^bisector.
Diophantus, VI, 18, found a rational right triangle with the bisector
of one acute angle rational. Let the bisector be 5N, altitude 47V, so that
one segment of the base is 3N. The other segment is taken to be 3 — ZN.
Then (by proportion) the hypotenuse is 4 — ±N> Equating its square to
DЛ02 + Z\ we get N = 7/32. Multiply ah1 our numbers by 32. Then
the sides are 28, 96, 100, and the bisector is 35.
С G. Bachet74a in his commentary on the preceding noted that no rational
right triangle has a rational bisector of the right angle.
«* Math, Quert. Educ. Times, 55,1891, 99-101.
"*• Jour- de math. £16нц 1892, 2$2.
115 Ашег. Math. Monthly» 5,189S, 514, 277-9.
>" Math. Quest. Educ. Timee, B), U, 1908,19-21.
Xi» Sphinx-Oedipe, 5, 19Ю, 187.
'"Matheeie, D^3, 1913,119-20.
"• L'eneeignement math., 19, 19l7, 247-252.

Chap, iv] Tables of Right Triangles. 189
J. Kerseyw (p. 143) took the right triangle with the rational sides
AC - p(p* + b2}, ЛВ = p(p* - V), ВС = рBЬр).
The bisector AD of angle A divides the base into two segments
CD = b(p2 + F), BD = b(j>2 - V)
proportional to AC and AB. Since AB : BD = p : by we have
if hf p, Ь are sides of any rational right triangle.
Several1*0 writers found a right triangle with a rational bisector of one
acute angle.
E. Turriere161 found a rational right triangle with rational interior and
exterior bisectors of an acute angle.
Tables of bight triangles with integral sides.
The tables are usually arranged according to the magnitude of the
hypotenuse h or the area A.
An Arab manuscript9 of 972 gave a brief table (see Ch. XVI)*
J. Kersey, Elements of Algebra, Books 3, 4, 1674, 8, h ^ 265.
J. C. Schulze, Sammlung Log., Trig. - . - Tafeln, Berlin, II, 1778, 308,
gave the decimal values of tan w/2 = mjn for 200 pairs of relatively prime
integers m, n each ^ 25, m < n; also right triangles with an angle o>>
A. Aida17 A747-1817) listed the 292 primitive triangles with h < 2000.
Le pere Saorgio, Mem. Acad. Sc. Turin, 6, annees 1792-1800, 1801,
239-252, quoted a table of primitive right triangles from Schulze.
С A. Bretschneider, Archiv Math. Phys., 1, 1841, 96, h Ш 1201.
Du Hays, Jour, de Math., 7, 1842, 331-4, gave four tables each with
32 entries to illustrate the systematic tabulation of primitive right triangles,
using A) with щ n relatively prime, m > л. First, give to m the values
2, 3, — - and to n the values < m and prime to w, such that one of m, n is
even. Second, take 1, 3, 5, ■ • • as the odd side and factor each into two
factors m ± n. Third, begin with the even side 2mn. Fourth, take a sum
of two squares as the hypotenuse.
A. Wiegand, Sammlung Trig. Aufgaben, Leipzig, 1852, 131 triangles
and their angles.
D. W. Hoyt, Math. Monthly (ed., Kunkle), Cambridge, Mass., 2,
I860, 264-5, h < 100.
E. Sang, Trans. Roy. Soc. Edinburgh, 23, III, 1864, 757, h ^ 1105.
S. Tebay, Elements of Mensuration, London and Cambridge, 1868,
Ш-2, gave an incomplete table arranged according to area At the largest
Ay 863550, being an error for 934800. Reprinted by G. B. Halsted, Metrical
Geometry, 1S81, 147-9.
H. Bath, Archiv Math. Phys., 56, 1874, 188-224, used formulas [due to
de Lagny18] to form a double-entry table, and noted an error by Berkhan.28
"• Amer. Math. Monthly, 7P1900,83-5." " ~ ~~~
" 16,40^8.

l&O HlBTORT ОТ THE ТНВОЕТ ОТ NtTMBKRfi.
W. A. Whitworth, Proc. lit. PhiL Soc. Liverpool, 29t 1875* 237, h < 2500.
Whitworth and G. H. Hopkins, Matii. Quest- Educ. Times, 31, 1879,
67-70; D. 3. Hart, Math. Visitor, 1, 1880, 99, forty triangles with
h = 5 13-17 29.
N. YiU, Math. Magazine, 1,4884, 163, primitive with k < 500.
G. B. Airy, Nature, 33, 1886, 532, A < 100.
A. Tiebe, Zeiisehr. Math. Naturw. Unterricht, 18, 1887, 178, 420,
solved a* + i* = A2 by setting h = x + y, whence 2x - al/y — У, 80 that у
is to be choeen as a divisor of a1 (e > 2) such that the difference is even-
Whence he constructed a table with h < 100. Cf. T. Meyer, ibuL, 36,
1905t 339.
H. Ueber and F. von Luhmann, Trig^ Aufgaben, ed. 3, Berlin, 1889,
287-9, gave the 131 primitive triangles with h < 999.
P. G. Egidi, Atti Accad. Pont. Nuovi Lmeei, 50, 1897, 126-7, h Ш 320.
J. Sachs, Tafeln zum Math. Unterticht, Wise. Beilage zum Jahres-
bericht Gym. Baden-Baden, 1905, k < 2000; 2000 < h < 5000, A a pnxluet
of primes 4n -J- 1; one side < 500.
J. Gediting « h < 1000.
A. Martin, Math. Mag., 2, 1910, 301-324 (preface, 2, 1904, 297-300),
tabulated the values of p2 ± g*, 2pq and area Л = pq(p* — q*) for p ^ 65,
4 < Pt 4 prime to p, q even if p is odd. Omitting the entries p = 33,
q = 22, and p = 35, q = 14, we have 862 triangles of which 443 have
A ^ 934800 (the largest A of the 178 triangles in Tebay's table). There
is a table of the sides p* d= g2, 2p<7 of triangles for which p = g + 1 < 157
and those with p s 312, q = 1, whence Л exceeds a leg by 1 or 2 respectively.
P. Barbarin, l'mtermediaire des math.^ 18, 1911, 117-120, gave the
35 pairs of primitive triangles with the same Л < 1000. A. Martin, ibid.,
19, 1912, 41, 134, noted the omission of one pair and stated that there are
41 pairs with 1000 < k < 2000.
A. Martin, Proe. Fifth Internal Congress Math., 2, 1912, 40-58, gave
the primitive triangles with k < 3000, noting two omissions by Sang.
He listed many sets of й (к Ш 15) triangles whose A's are consecutive
integers; also sets of three triangles whose h's are sides of a right or scalene
triangle- A product of n distinct primes Am -\- 1 is the hypotenuse of
C* — l)/2 different right triangles, only 2*-* of which are primitive.
W. Konnemann, Rationale Losungen Aufgaben, Berlin, 1915, h < 1000
(adverse review, Zeitschrift Math. Naturw. Unterricht, 46, 1915, 390).
E. Bahier,K pp. 255-9, tabulated the primitive triangles with a leg ш 300.
On systems of equations including я* + у2 = & see papers 76, 77, 80,
46, 84, 89, 139, 140of Ch. XVI; 5 of Ch. XVII; 51, 146 of Ch. XIX;
354, 357, 360, 362, 366, 369^71, 436 of Ch. XXI; 109, 113, 313 of Ch.
XXII; 207ofCh.XXIIL
not available tob
G. M. Pagmni, Collezkmc dJ OpuscoK Sc., ffiense, 3,18OT, 3-24; GbrnaJe di №ca, Chimica о
Stori» N*t.t Ратпц 3, 1810, 193-207. [Series of raiional right triangles.]
fal Di Aafetellung Pythagoreiecher Zahien, Blatter FortbUdong d. Lehrere u. d. Lehrerin,
4 1911, 99&-1000.

CHAPTER V.
TRIANGLES, QUADRILATERALS AND TETRAHEDRA
WITH RATIONAL SIDES.
Rational or Heron Triangles.
Heron of Alexandria gave the well known formula for the area of a
triangle in terms of the sides and noted that when the sides are 13, 14, 15,
the area is 84. A triangle with rational sides and rational area is called a
rational triangle or Heron triangle»
Brahmegupta1 (born 593 АЛЭ.) noted that, if а, Ъ, с are any rational
numbers,
V + V
are sides of an oblique triangle £ whose2 altitudes and area are rational and
which is formed by the juxtaposition of two right triangles with the common
leg e J
3. Curtiiis2* proposed the following question: Three archers At Bt and
С stand at the same distance from a parrot, В being 66 feet from С, В 50
feet from A, and A 104 feet from C; if the parrot rises 156 feet from the
ground, how far must the archers shoot to reach the parrot? He noted
that they stand at the vertices of a triangle the radius of whose circum-
circumscribed circle is 65 feet, while the parrot is 156 feet above its center. Since
652 + Ш2 = 1692, each archer is 169 feet from the parrot. It is stated to
be difficult to explain why the radius turns out to be an integer. Cf.
Gauss.14* [The triangle is rational since its area is 2*35*11 = 1320.]
C. G. Bachet,3 in his comments on Kophantus VT, 18, treated several
problems, the second of which is to find a triangle with
rational sides and a rational altitude (and hence a
Heron triangle). Taking a right triangle ADC with
the sides 10, 8, 6, he found BD =* N such that N*
-f- 82 shall be the square of a rational number (AB)-
Assuming first that angle ВАС is acute, so that DC :
AD < AD : BD, we must have 6ЛГ < 64, whence
N < 32/3. Let N2 + 82 be the square of 8 - xN; then
i Brahme-Sphut'a-Sidd'hAnt*, Ch, 12, See. 4t § 34 AJgebra with Arith. And Mcaanmtion,
from the Saoacrit of Brtihmegupta aad Bh&cara, tranel. by H. T- Colebrooke, London,
1807,306.
• E. E. Kuimner, Jour. /Qr Math., 37, 1848, L
*• Tractdtue geometricue . . , t Amsterdam, 1617» Quoted by A. G. Kfifitaer, Gsechichte der
" Math., Ill, 294.
'Diopbanti Akx. AritbrQeticorum . . . Ommentariie , . . Avctore C. G. B&cheto, 1621,
416, Diophanti Alex. Atithmeticoruto, cum Conntnectarii^ C, G. Bacheti & Obeerv*-
tionibua D, P- de Fermat (ed4 S. Fermat), ТЫояае, 1670, 315.
191

192 History of the Theory of Numbers. £Chap,V
whence x > 2. Taking x = 5, we have
Nz + 82 - (8 - 5N)\ N - ^,
and the sides are 10, Ц, 8|, while the altitude is 8.
If ЛЛС is oblique, N > 32/3. He took
Bachet's second method of solution is of greater importance, since it
consists in juxtaposing two rational right triangles having a common side
AD. Take as the latter any number, as 12. Seek two squares such that
the sum of each and 122 is a square: 35* + 12* = 372, 16* + 12* = 20*.
Hence by juxtaposition, we get a rational triangle with the sides 37, 20,
35 + 16 = 51, and altitude 12. Using the first relation with 9* + 12* « 15*
or 5* + 122 = 132, we get the rational triangle C7, 15, 35 + 9) or C7, 13,
35 + 5).
F. Vieta' started with a given right triangle with legs Bt D and hypote-
hypotenuse Z, and formed {Diophantus7 of Ch. IV) a second right triangle from
F + D and B, having therefore the altitude A = 23(F + 2>), and multi-
multiplied its sides by 2>, and the sides of the given triangle by A. Juxtaposing
the resulting two triangles with the common altitude AD, we obtain a rational
triangle with the sides AZ, D(F + Df + В*Д D(F + D)* - B*D -f В A,
whose angle at the vertex is acute or obtuse according as F < Z or F > Z.
Ггапй van Schooten* used the juxtaposition of right triangles.
The Japanese manuscript of Matsunago,6 first half of the eighteenth
century, started with any two right triangles with integral sides and multi-
multiplied the sides of each by the hypotenuse of the other and then juxtaposed
the triangles» The sides below 1000 of the resultmg oblique triangles were
tabulated. Removing common factors, he obtained a table of primitive
triangles» From Kurushima (t 1757) he quoted the result that, if
then
nonius H- ndi)> n2{ndv + n^i),
are sides of a triangle with rational area.
Nakane Genkei** in 1722 considered triangles whose sides are consecutive
integers such that the perpendicular upon the longest side from the opposite
vertex shall be rational. Denote the solutions C, 4, 5), A3, 14, 15), E1,
52, 53) and A93, 194, 195) by (ал bJt c,-), j-1, 2, 3, 4. Then
and similarly for the b's and c*s. Whether or not he made the induction
complete does not, however, appear.
* Ad Logisticem Specioeam Nolae Prioree, Ptop. 55, Opera Math., 1646» French tramd. by
F, Ritter, Bull BibL Storia Sc. Mat., I, 1868, 274-5.
* Exercitationum Math., LupL Batav., 1657, 426-432.
* Y. Mikami, Abb Gesclu Math. Wi*., 30,1912, 23(Ы.
M D. Б. Smith and Y. Mikami, A History of Japanese Mathematics, Chicago, 1914,168.

Chap. V] RATIONAL TRIANGLES. 193
L. Euler7 noted that in any triangle with rational sides а, Ъ, с, and
rational area,
A) a :hc- ftw±gr^=Fg>)-P>+g>.rf+ **
pqrs ' pq ' rs '
and that every pair of sides Are in the ratio of two numbers of the form
(a1 + /S*)/aft since
= ~~гГ~ : ~щ~ ' x = -p8±qr7 V = pr^q*,
whence
+ И (р + ^ + О
The portion of Euler's paper containing his derivation of A) is missing.
It is probable that he employed Bachet's method of juxtaposing two right
triangles, using those with the sides
2
f
pq ' pq ' ' rs ' ts J
and obtaining A) with the upper or lower signs according as the component
triangles do not or do overlap.
J. CunlifiV juxtaposed two right triangles with a common side 2r« = 2mn
and hypotenuses r* + &, m2 + л*.
J. Davey' found three triangles with integral sides and areas having
sqnal perimeters and areas in the ratio of a ^ 2, Ъ = 7, с =* 15. Let the
triangles be AFB, BFC, CFD with collinear bases and the common altitude
FE. Take
Then ЛЕ = (r* — l)v/Br), etc. By the equality of the perimeters,
8»- 1 1 P~l _ 1
Then the conditions that the bases be proportional to а, Ь, с reduce to
(a?4 + fe)/r = (a£ 4- b)ft, whence r *= b/(aQ (sheer Ф 0,andtott = c/(ta).
Eliminating r, u7 s between our four relations in r, uf &, t, we get
=0, d^*^^, e -
с
For a = 2t 6 — 7, с = 15, we get the rational root t = 5/3. Taking
£i = 420, we have AF = 541, BF - 525, CF = 476, Z)F = 421, AB = 26,
Ж7 - 91, CD - 195, perim. =* 1002.
To find a triangle ЛВС with integral sides and area such that the dis-
distances from Af В, С to the center О of the inscribed circle shall be integers,
JComra. Arith. Coll., П, 1S49, $4$, poethumotia fragment. Same in Opera poetuma, lv
18^,101»
•The Gentleman^ Math. Companies, London, 3, N6.15, 1812, 398.
^lodks1 Diary, I82It 36-7, Quest. 1364.
14

194
History of the Theory op Nttmbers.
C. Gill10 made a computation which (although not so stated) in effect
consists in finding three right triangles AOF, EOF, СОЕ (see the figure
below) with integral sides such that OF = OE
= OD — r is the radius of the inscribed cir-
circle, but omitted the condition that the sum of
their angles at О shall be two right angles.
If AF = m, BF - n, CE = $, this condition
is mn$ = r*{m + n + $). Thus his solution
fails.
A. Cook11 gave the following solution. Draw
OR perpendicular to АО to meet AB at R.
EThe sides of any rational right triangle are
proportional to r* dh a2, 2r<z.] Hence we may
take
AF~(r*- <*2)/Ba), АО
BO = {t*
By similar triangles,
(r2 + a*)/Ba),
OF
BF
(r2 -
Hence we have RB = BF — FR. Since angles BOR and OCB are equal,
the same lettered triangles are similar. Hence
where d = (r2 - a*)fy* - Ъ*) - 4atn*. Hence
DC = ВС - BD^ 2r*\{r* - а*)Ъ + (r* -
We may assign any values to a, h and any value, exceeding a and b, to r.
For a = 16, b = 18, r = 72, we get AF = 154, АО = 170, BF = 135,
BO = 153, ОС = 120, CD = 96, AB = 289, AC *= 250, Ж? - 231.
Several12 employed Heron's formula
_ g + ^f_ ^ + ^ _|_ S)
for the square of Ше area Л of a triangle with sides 2B, 2S, 2s. T. Baker
wrote x, y, z for the last three factors of Д2. Then A3 = xyz(x -f у + *)•
Let Д = (ox^J. We get л rationally. " A. B. L.u took Л = x - y, S « x,
я = ^ + yj then 3x* — 12#2 = □, whence u2 — 3»* = 1. C. Holt equated
the last three factors of Д2 to 4рУ, (q2 + r2 - p*)*f 4pV; by addition,
jB-h^-h^= (^H-r1-!-p*)\ J> Anderson equated the product of the
four factors to *?x2. Hence
[B* - (& + s2)}2 - ^D^ - s') = ^B5 - y)\
say; hence we get S and then IP,
10Ladiee' Diary, 1824, 43, Quest. 1416.
* L«uW Diaiy, 1825, 34-5»
"The Geotieman'sMath, Сотрыпоп, Londori, 5TNo, 27, lS24t2S&-292. Report in changed
notations ш Math. Mag,, 2,1806, 224-5.

Chap.V] Rational Triangles. 195
C. Gill18 found integral sides xf yf г of a triangle the four diameters of
whose inscribed and escribed circles are integral squares r2 and Rx, R\, R\.
Take z -\- у -\- z = а*, у + z — x *= &*, x -\- z — у = <?, x -\- у — z = <P.
Let the condition a2 = ¥ + c* + <P be satisfied. We get s =- (<i2 - &2)/2,
у, z. It is known that 4Д = rW = Ш* *= R\<? = 2geP.
C. L. A. Kunze14 derived eight rational triangles from the two rational
right triangles {3, 4, 5) and E, 12, 13) by reducing their sides in proper
ratios so that any chosen leg of one shall equal any chosen leg of the other
and then juxtaposing the resulting triangles either with or without over-
overlapping. Schlemilch*6 noted that we may start with any two rational
right triangles.
С F. Gauss,l4a whose attention had been called to Curtius'*1 problem
by Schumacher, stated that the sides of every triangle such that each side
and the radius r of the circumscribed circle are integers are of the form
4abfg(a2 + V)t ± 4дЬ(/ + g) (a*/ - Vg), 4ab(a»/1 + Vf)*
where a, b, f, g are positive integers, while г^Са^ + Ь^ХаУ*-!-^2). We
obtain Curtius' numbers by taking а — g = 1, Ь = 2, / = 10, and deleting
the common factor 8. Many writers14* derived Gauss' formula.
E. W* Grebe18 tabulated for 46 rational triangles the 12 rational values
of the segments of the altitudes and the segments of the sides cut off by the
altitudes.
Grebe1* gave a table of 496 rational triangles, showing also the area,
perimeter, altitudes, and diameter of the circumscribed circle. He began
with 32 rational right triangles D, 3, 5), • - -t A96, 28, 197) with small
ratios of sides, took each pair of these triangles and multiplied their sides
by such factors as produce two triangles whose larger legs are equal. By
juxtaposition he formed a rational acute triangle-
To find a triangle with integral sides whoee area and perimeter are
equal, B. Yates17 took, in accord with A), the sides to be pq(r* + s*)/n,
тв(р2 + <f)fn, (ps + qr)(pr — qs)jn. The latter multiplied by pqrs/n is Ше
area* Equating the area to the perimeter 2pr(ps 4- <F)lnt we get
q*(pr — qs) = 2n.
Integral solutions are found when n = 1, 2, 8, Many solvers used the
segments I, in, n into which the sides a, b, с are divided at the points of
contact of the inscribed circle of radius r. Thus I -\- m = at Z + n — 6,
m -\- n = cw If*is the semi-perimeter, r* =* 2s, whence r = 2. But
rV = sbnn. Hence 4A -\- m -\- n) = Imn. The least side exceeds 2r ^ 4»
Hence we may take I + m = 5, 6, • • • and find integral solutions.
11 The Gentleman's M&th. Companion, London, 6, No. 29, 1826, 509-512.
u Lenrbuch der Geomctrie. Jena, 1842, 205.
"* Briefweehgel xwischen C. F. Gauss and H. С Schumacher (ed4 C. A. F. Peteme), Altoaa,
5» 18ЙЗ, 375; letter of Oct. 21, t847. Quoted in АгсЫу Math. Phye., 44,1S65, 604-6.
we Archiv Math- РЪув., 45,1866, 220-231.
w JElne Gruppe тод Aufgaben fiber dae geradlioige Dreieck, Progr., Marburg, 1&56,
11 Zueammenrtelkmg топ Stttcken rstion&ler ebener Drdecke, Halle, 1864, 248 pp.
17 The Lady's and Gentleman's Diary» London, 1865,49-60» Quest. 2019-

196 History of the Theory or Numbers. [Chap, v
Many" proved that if the sides and area be integers, the area is divisible
by 6. Take the sides to be the products of A) by pqrs. Then the area is
ps + qr)(pr qs).
S. Tebay1* tabulated 237 rational triangles arranged according to the
magnitude of the area, the greatest area being 46410 (cf. Martin**).
J. Wolstenholme10 found a triangle whose sides and area are in arith-
arithmetical progression. Take a — b, a, a + b as the sides, a + 2b as the area.
Then
2Ь= 16 +3a' '
W. Iigowski21 found a triangle whose sides a, b, c, area Ft and radii r
and p of circumscribed and inscribed circles, are all rational. He assumed
that 8 — a ~ pxt $ — Ь = pyt s — с = рг, where $ is the semi-perimeter, and
readily proved that the sides are proportional to
a - х(И -hi), Ь = у(з? -hi), с = (x -h y){xy - 1),
whence
P-sy-1, r = |(x2-hl)(^-hl), F = zy(x + y)(zy-l).
W. Simerka2Ia gave several methods of finding rational triangles and a
table of the 173 having sides ^ 100, showing also the area, tangents of the
half angles, and the coordinates of the vertices (cf. Scherrer*2"). He proved
that the perimeter is always even.
H. Rath22 employed the segments a, ft у of the sides determined by the
points of tangency of the inscribed circle. Then the sides are л -h 0,
« + T, /3 + у and the square of the area is affy(a + 0 + 7). The latter
is a rational square only for a = dj*, 0 = BB, у — ЬС> where В and С are
any two positive relatively prime integers, and likewise for к and j9 while
dfi is the value of the fraction
BC(B + О
к2 - BCj2
when reduced to its lowest terms. Each resulting set of rational numbers
a, 0, у defines a rational triangle, the condition that the sum of any two
sides shall exceed the third being evidently satisfied. His final tables
show relatively prime integral sides, the triangles whose area is a multiple
of some side being listed separate from the others. He gave (p. 213) nine
rational triangles whose sides form an arithmetical progression, the common
difference being here given as a subscript:
C,4,5k A3,14,15),, A5,26,37)lb G5, 86, 97)ш
B5, 88, 51)ia, F1,74,87U A5,28,41)».
« The Ladyb and Geutleman'fl Diary, Lombn, 1866, 61, Quest- 2044.
"Elements of Mensuration, London and Cambridge, 1868, 113-5. Table reprinted by
G. B. Halsted, Metrical Geometry, 1881,167-170.
11 Math. Queet. Educ, Times, 13,1870, 89-90. Same by D. S. Hart, 20, 1874, 56.
* Arehiv Math. Phyg., 46, 1866, 503^.
*• Ibid.t 51, 1870r 19^240.
*Archiv Math. Phya., 56, 1874, 188-224. See the compact exposition by P. Bachmano,
Niedere Zahlentheorie, 2,1910, 440-1. a. Kommerell™ of Ch. XXII,

3 Kational Triangles. 197
D. S. Hart*1 juxtaposed two right triangles with the common leg 2pr
and further legs r(p* — 1), p(r* - 1), and obtained
(p + r)(pr - 1), r(p» + 1), p{r* + 1),
viz., A) for the caae of the upper signs and q = 8 = 1, The last assumption
does not restrict the generality of the result
Hart* noted that the triangle with the sides it — 1, to, w + 1 has a
rational area if Sui* — 12 *= D. He obtained v> = n/d, d = z* - $уг and
took d *= 1, whose general set of solutions is known.
A. B. Evans* found a triangle whose sides а, Ъ, с, radii
* = МИ tan \A){1 -f tan iB)/(l 4- tan JO),
y, z of Malf attics circles, and radius r of the inscribed circle are all rational.
Take cot \A ** m/n, cot %B = p{q, m* -h n1 = П, p1 -h tf = П. Then
tan \A, ete., are rational. A. Martin took cot \C * 3, cot \B = 4; then
the ratios of x, yt г, a, b, с to r are known.
H. S, MonckM showed how to deduce a second from one triangle with
integral sides, two differing by unity.
J. L. McKenzie" found a triangle whose area and Bides are integers,
semi-perimeter is a square, two sides having a given common difference.
D, S. Hart1* discussed rational triajjgles two of whose sides differ by
unity,
R. Hoppe29 discussed triangles with the sides n — r, n, n + r and
rational area A. Thus Д = Jmn, where Sm% = n2 — 4r* Hence n is even,
n = 2p, and m — 2qt whence p1 — 3^ = r1. First, let r = 1. If pk, Qt is
a solution in integers, then is also
pt+i =* 2pk -h %*, ?*+i ^ рк + %Як.
Further, рц. 1 — 4pk -\- p^t = 0 and similarly for the £'s. Hence
8h - p*+i - B + Лв)р* * ^—^g .
The resulting values of n, Д are
n - B + л/3)* + B - л/3)*, Д - -^ {B + ig)» ^ B - л§)«),
for /5 = 0, 1, * ♦•. It is proved that there are no further solutions.
Next, let r be undetermined. Then p : r = 3Xl -f p? : 3X2 — ^, where
X and ft are relatively prime integers. Thus the sides are
3(X"+,*■),
M»th. Quest. Educ. Timee, 23,1875,108.
/Ш., 23,1875, 83-4.
iuirf.»^, 1875, 70-L
iT 24, 1876,35^8.
-, 25,1876, 105-6.
, 28^ 1878, 66-7.
Arehiv Brfath. Phye., И 1879, 441,

198 Hibtoht of the Theobt op Numbers. [Chap, v
W. A, Whitworth10 noted that the triangle with the altitude 12 and sides
13, 14, 15 is the only one in which the altitude and sides are consecutive
integers.
G. Heppel" noted that there are 220 triangles with integral sides = 100
and integral areas, but repeated C9, 41, 50). He listed only 56 rational
scalene triangles with relatively prime sides.
Worpiteky43 gave without proof a formula equivalent to A).
R, Mullet21 considered rational triangles whose sides are consecutive
integers x — 1, x, x + 1- Since the area is to be rational, a? — 4 = 3|f,
whence x = 2u, у = 2v, u1 — 3t? = 1. Hence the triangles are C, 4, 5),
A3,14,15), ete.
A. Martin14 noted that the triangle with the sides 2m2 + 1, 2m5 + 2,
4m5 + 1 has a rational area,
T. Pepin" gave a historical note on rational triangles.
O. Schlomilch3* gave the same method and results bs Hart.M
C* A. Roberts37 noted that, if и is a square and w the double of a
square, и + wf и H- 2u>, 2u + to are the sides of a triangle with rational
area (« + v>) V2uu? and listed many triangles with sides < 500. The triangle
is special since one side equals one-third of the sum of the remaining two.
S. Robins** tabulated rational triangles with a given base and a given
difference between the remaining two sides; also (pp. 262-3) rational
triangles with sides x, x + n, 2x — n for given n's.
H. К BUchfeldt" derived A) by use of Heron's formula for area.
S. Robins40 found rational triangles whose sides are consecutive integers
by taking x — 2 and x + 2 as the segments of the base made by the per-
perpendicular to the base. The altitude is (Зх1 — 3)*, which is made rational
by choice of x by means of convergents to the continued fraction for -V5,
A. Martin*1 juxtaposed two right triangles in various ways to obtain
rational triangles. From Heron's formula for the area A of a triangle with
the sides z, y, z,
}-(Z2 _ з> - ^1 _ fry _ Д2 = (ia!y _ Aq/pyy if Д ^ Ж,
Then
» Matiu Quest. Educ Times, 3ft, 1881, 42.
* 1Ш*, 39,1883, 37-8. Cf. Martin."
»Zeiktehr. Math. Naturw. Untemdrt, 17,1886, 250,
M Archiv Math. Fhja., B), 5> 1887,111-2.
* М&Щ. M&gaxine, % 1890, ft.
* Mem. Aocad. Pont. Nuovi lined, 8, 1892,85.
ш ZertEtchr. MMb Naturw. Untemcht, 24,1893, 401-9.
^ Math. Марте, 2,1893,1Э6.
» Amer. Math, МопШу, 1,1894,13-14, 4Ш-3 (for base fl),
»• Annola of Майц 11,1896-7, 57-60.
«Axoer. M&tJi. Monthly, 5, 1898,150-2.
« MatL Ма^ише, 2,1898, 221-236.

Chap. V] НаТКЖЛЬ TRIANGLES, 199
determines xfy. Taking a; to be the numerator of the resulting fraction»
we have
+ s1) =h
He discussed at length rational triangles two of whose sides differ by a
given integer, making use of a Pell equation gq* — p* «= rb 1.
Т. Н. Safford* juxtaposed the right triangles E, 12, 13), (9, 12, 15) of
areas 30 and 54 to obtain Heron's triangle A3, 14, 15) of area 84, also to
obtain D, 13, 15) of area 54-30. He listed 37 rational right triangles.
D. N. Lehmer41 derived A) by use of the rationality of the sines and
cosines of the three angles, a necessary and sufficient condition for the
rationality of the triangle.
National triangles with consecutive integral sided have been found.41
W. A. Whitwortb and D. Biddie** proved that there are only five
triangles with integral sides whose area equals the perimeter: E, 12, 13),
F, 8, 10), F, 25, 29), G, 15, 20), (9, 10, 17).
A. Martin'* formed rational triangles by the juxtaposition of two
rational right triangles. He tabulated 168 rational triangles of area ^ 46410
not found in Tebay's11 table.
H. Schubert47 considered a Heron triangle with integral sides а, Ъ, с
and area J. If а, д у are the angles, / ^ tan aB and hence also sin a.
and cos a must be rational (such an angle a being called a Heron angle).
Set / *= n/m, where л and m are relatively prime integers. Then
since tan yf2 = cot (a + 0)/2. By a » 2r sin at etc.,
(
Hence
a = mn(p* + (fO Ь - pq(mz + л2), с — (mq + np)(mp — nq), J = mnpqc.
J. Sachs48 gave tables of rational triangles with altitudes < 100; acute
rational triangles with altitudes 100, - - -, 500; rational triangles arranged
according to the least Bide and according to the greatest side. The last
tables axe convenient for the formation by juxtaposition of rational quadri-
quadrilaterals, pentagons, etc.
T. Hannuth4* considered rational triangles with sides a, a + d, a + 2d.
« Тпдаи. Wiecoijein Aa«i Sc, 12,1998^9, Б05-8.
41 АппаЬ Ы Ma!th.t B), 1, 1899-1900, 97-102.
"Ашег. Mfrtb. Monthly, 10,1908,172-3-
Mth, Quoft. Ednc iWa, 5, 1904, £4-бА СЗ-3,
Ме, 2,1904, 275-284.
k in (kr algdsmiecbeii Geometrie, Leq»i^t 1906, 1-16. Festgabe 48
Ы Hb 1905 Rrid Al
fiBk gdsmiec , q^t , g
УавяттЬщ^ d. РЫЫовеа п. SchnhaittDer жа Hamburg, 1905. Reprinted in Aualeee
*ж тешсг Untemcbte- n. YodeeitDe^x^xis, Leipn^ 2,1905, 1-23.
Tafeln жат M*tK Unterricfct, Progr, 794» Baden^ftden, Lopog, 1908.
UmidtU Mh. u. Nfttarw^u, 15,1909,106-6.

200 History of the Theoby of Numbers. [Chap» V
Its area is rational if (a + 3d) (a — d) = 3#*. Hence decompose 3jf in
every way into two factors congruent modulo 4.
E, N. Bariflienw notsd that, if 2p is the perimeter, the area is an integer if
p = (on + i)(fin -hi), p - Ь = \n(yn + 1),
p - a = («n + Щуп + 1), p - с = fin(pn + 1),
and >я = &*- The condition p = S(p — a) is satisfied if 47 -h 0 = 5a,
/3 — 7 = 5. If in p — Ь and p-cwe replace Xn and рп by 5n + 1, the
area is integral and the condition p = 2(p — a) gives for 5n + 1 a value
which is integral if 0 + 7 = 2a; then 5 is a right angle. A. Gerardin noted
that we may set p = (an -h t)(f*n + 0, ete., and take 0 + 7 = 2cc,
P — у ** 2p, t *= (p — 5)ftp
L* Aubry*1 noted that the triangle with the sides x — 1, ж, х -h 1 has an
integral area if (z/2I - Zy* =- 1, i. ev if
a: = 2, 4, 14, . -«, хщ - 4хл^ — х*^.
The area" of any triangle with integral sides and area is a multiple of 6.
B. Hechtu discussed triangles whose sides are integers, also the area
or the four radii of the escribed and inscribed circles.
A. Martin54 proved that in any primitive rational triangle two sides are
odd, the least side is > 2, the difference between the sum of the two smaller
sides and the largest side is not unity, and the area is a multiple of 6.
Every integer > 2 is the least side of an infinitude of primitive rational
triangles.
E. N. Barisien*5 noted that the triangle with the sides 7, 15, 20 has its
area and perimeter each 42. Multiplying the sides by 10, we get a triangle
with integral altitudes.
* H. Botteher58 gave rational triangles with an angle 60° or 120°.
Barisien87 gave complicated formulas for the integral sides of a triangle,
with integral values for the altitudes, area, radius of circumscribed circle,
radii of tritangent circles, segments of the sides made by the altitudes, and
segments of the altitudes made by the orthocenter.
Of several triangles6* with integral sides, area and one altitude, the
least appears to have the sides 4, 13, 14, area 24 and altitude (to side 4) 12.
A. Martin*' added (Ц rational scalene triangles to HeppeTs81 list.
N. Gennimafcas80 proved that any rational triangle is similar to one with
"Spbinx-Oedipe, 5,1Й10, 57-9.
"JKd.,6,1911,188.
» M&tb. Quest. Eduo. Times, 21,1912,17-8. Sec paper 18 above.
« Ueber rationale Dreiecke, Wise, BdL %. Jabresber. Stadt Re&lechule in Konigsberg, 1912,7 pp.
" School Science and Math., 13,1913, 323-6.
"Mfttheeie, D), 3, 1913, 14, 67.
» TJnterriehtebtttter «r Math. u. Naturwies,, 19,1913,13JJ-3.
» Sphinx-Oedipe, 8,1913,182-3; 9,1914,74-5,91,94. Аяяос. fian^. av. ac., 43, 1914,48^57.
Mstheee, D), 4, 1914,114-6 for 7 examples.
и I/iatenn6diaire dee math, 21,1914, 76,143,186-8; 22,1915,119-Ш).
» Math. QueeL Edne. Тпгкя, 25,1914, 76-&
Iint math.y 16, 1914, 43-53.

] Rational Triangles. 201
the sides x1 + y*, A + y*)x, с = A 4- x)(y2 — x). Conversely, if x, yy
y1 — x are positive, these numbers are the sides of a triangle, of area cxy.
E. Turriere*1 noted several methods to find Heron triangles. There is
an infinitude of Heron triangles with sides in arithmetical progression such
that no two are similar. He investigated Heron triangles in which the semi-
perimeter p and p — a, p — b, p — с are all rational squares, and the
analogous problem for inscriptible quadrilaterals. He" found Heron tri-
triangles in which the sum of the squares of two sides is a square.
R R. Schemer*2" made use of the theory of complex integers a + Ы to
obtain the coordinates of the vertices, of the centers of the circumscribed,
inscribed, escribed and Feuerbach circles, of the intersection of the alti-
altitudes, etc., of primitive Heron triangles. Cf. Simerka,21*
M. Rignaux" stated the final formulas of Schubert.47
E. T. Bell stated and W. Hoover* proved incompletely that if u© = 2,
«i-4, - -, w*+i ^ 4u«+! — un, then it* — 1, un, un + 1 are the consecu-
consecutive sides of a triangle with integral area, and all such triangles are given
by this method.
Pairs of Kattonal Triangles.
Frans van Schooten5 found two isosceles rational triangles with equal
perimeters and equal areas. Divide each into halves and let the right
triangles be formed from а, Ь and k, d respectively. By the perimeters,
2(a* + V) + 2{2ab) = 2(k* + <P) + 2BM), a + Ь = к + d.
Set к = a + xt d = b — x. The equality of the areas requires
2s* + 3(a - b)x + a2 - 4ab + b2 - 0, x = }(r + 3b - 3a),
where r2 = a* + b* + 14a6. Set г = a + b + с Thus
a 12b -2c'
The general solution thus involves the parameters Ь, с For b ^ 1, с = 3,
we get a = 5/2, x = 1/2. Multiply the sides by 4. We get the right
triangles B0, 21, 29) and A2, 35, 37). Their doubles have the perimeter
9S and area 420.
J. H, Rahne* devoted 8 pages to this problem, and J. Pell 62 pages.
There is first given the above solution by van Schooten, attributed to
Descartes.
Several" gave straightforward solutions to van Schooten's problem.
J* Cunliffe*7 treated the problem to find two triangles with rational
altitudes and segments of sides and with equal perimeters and equal areas.
11 L'eneeignement math., 18, 1916, 95-110,
MJbui,19, Ш7, 259-261. a. Euler* of Ch. IV.
**Zeitschrift Math. Naturw. Untemcht, 47, 19X6, 513-30.
u Lfatennediain: dee m&tb., 24, 1917, S6.
« Ашег. Matb. Monthly, 24,1917, 295, 471. Cf. Hoppe."
"AJgebr*p Zurich, 1659. EugL tranal. by T. Brancker, augmented by D. P., London, 1668,
13Ы92.
" The Gentleman's Math. Companion, London, 5, No. 26, 1823, 183-5.
и New Series of the Math. Repository (ed,, Th. Leyboura), 2, 1809, II, 54-7,

202 History of the Theoby op Numbers. [Chap, v
He found a pentagon inscribed in a circle with rational sides and areas for
all the triangles into which the pentagon can be divided by diagonals.
Triangles all op whose Sides and Medians abb rational
L. Eulet" denoted the sides by 2a, 2b, 2c, and the medians by/, g, h.
Then
Hence, if 2/, 2gt 2h be taken as Bides of a triangle, its medians are 3a, 36, 3c.
Write <r =* a + b -f с Then
F-cL^+c-a)= Л (« - <0* + <* + о - b) = g*.
Set/ => Ь — c + <rpt g ** a — c + 4 Then
Ь + с - a = 2(b - c)p + ff?2, a+c-b = 2(a-c)q + <rq\
Solving each for с and adding a + bt we have two expressions for <r.
ing these, we get the ratio o! : &' of a : 6. Eider took ay = a and got
Then <r/2 = 1 -h p -|- ^ — Zpqf so that с is known and hence also /, q.
Next,
Л' - (a - 6J 4-a(a-b6-c)= ЛУ + 25^ + C^ + 2Z>^ + ^,
where
A = 1 + 3/>> ^ = - 1 + Up - V - V,
С « - 3A + 2p - 2pl + 6p* - Зр4), Z> ^ 2 - 9p - 3pl + lip1 4- Зр4,
We can obtain rational solutions by setting
D
jq±E Or
Euler examined the simplest cases p=±2(p*=0or±l being excluded).
For p ш - 2, we have A = - 5, 5 = 13, С - 321, Z) = - 32, tf = - 4.
Taking the second expression for Л, we have
Multiplying the resulting values of a, &, • - by 4/3, we get
a « 158, Ь *= 127, с = 131, / ^ 204, g = 261, A ^ 255.
Since f/, §0, |b are sides of a triangle with the medians а, Ь, с as remarked
at the outset, we get the new solution
a = 68, b m 87, с = 85, / ~ 158, g =* 127, h - 131.
Eider's" paper of 1778 deals with triangles in which the distances of the
vertices from the center of gravity are rational and the sides are rational
rt Novi Comm. Acad, Petrop., 18, 1773, 171; Comm. Anth, Coll., 1,1849, 507-15.
" Nova Acta Acad. Feltrop.f 12, 1794,101; Comm. Arith., II, 294r30l.

Chap. V] TRIANGLES WITH RATIONAL MEDIANS. 203
We have f - tf = 3(c* — ¥). Euler took
g _j_ A = Z-pq, g — h = Т9, с + b — pr, с — b — qs.
From
f + tf = 4a« + & + <?, Я = 2c1 + 2b» - a\
we get, on setting p=x-\-y, * = z — yy
^ ^ = ** -h у»
Take a/q ^ л + ^, f/r = x+ uy. Then
у 2(t - M) 2{u-N)'
All conditions are satisfied if we take
N-M x _ (M - N)* - 4
The cases r ^ q and r ~ 3g are excluded since M + N + 0. For
g = lyr « 2, we obtain the solution given above* Forg = 2tr = l,weget
a ^ 404, Ь = 377, с - 619, / = 3 314, g - 3-325, Л = 3-159.
Euler's™ paper of 1779 does not differ materially from the preceding.
Euler's71 paper of 1782 avoided the earlier restrictions on the generality
of the solution. Changing the notations to confonn with his earlier ones,
we may set
From
(h -\
we get
Then/* =
0 -f- С — ft
Then
B)
Take у =
Ял
+
(Ъ + с
Гл/ Л. **
{У т
а
а
4{Р
1
1
+
(A-ff
Ч /
)г + F
1, 0 -
= 7s+
О)(ЗР
)» = 2А» + :
-С)'
/v A
- с — р
6s + a
+ Q)
^^
-а1
P7S,
-0)'
-4,
с;в +
gives
-4,
са
(b
•а5
a eimilar
р г-
L —
= 7S-
Then
-n
4<
n
B)
CQ
■Mo + с
formula
5V ,
*■ ' (
ЧЭДу-
are the
-hP) -
E-
) •
for
■*)f,
/*. Write
дл1 - 5^
squares of
■4.
. Acad. JPetrop., 2,1807-8,10; Comm. Arith., II, 362-5.
*» M&n. Acad. Petrop., 7, X820, 3; Comm. Arith., II, 488-01.

204 Histoky or the Theory of Numbers. [Chaf.V
Set PQ + 1 =* n(P -f Q). We may discard the common factor P -f Q
of у and 5, thus altering a. and /3 in the same ratio, and set 7 = 4,
Ь = P + Q — 4n. The first expression B) is the square of
(P - Q)(P + Q) + 2P(P ~ Q) - 4 - (P + Q)CP - Q - 4i),
which is to be divided by P -f Q. Hence
" = 3P - Q - 4л, | = 3Q - P - 4n.
« p
From the above expressions for P, Q, we readily get n *= — 6/4. Set
С = 16V/52, Л = (9a2 + £•)(** + #), *" * 2(9a4 - p4).
Then 7 = 4, б = l)/Da2j82). Suppressing the common denominator 4a*02
in а, Ь ± с, /, Л ± ^, we get
a = *(D - F)f b + c = a(C +D)t Ъ - с = 0(C - D),
f = 0(D + F)t k + g = 3a(G - D), h - g = fi(fl + D).
Euler71 noted that 2a2 + 2b2 - c2 is a square if
a - (m 4- n)p — (?n — n)^, Ь = (m — «)p -f- {m -f- л)д, с = 2?»p — 2?ig.
It suffices to make the product of the remaining two medians a square»
We obtain a homogeneous quartic in p> q. A special set of values making
it a square is found to be
p = (OTt _j_ ftJ)(9™2 « n*^ 5 = 2wn(9m« + ««).
Euler deduced hisw two solutions and three others:
207, 328, 145; 881, 640, 569; 463, 142, 529,
To make a = 2s2 + %V2 - &> 0 = 2x* + & - y\ у ** 2уг + 2г* - хг
squares, " Atticus " n took x = Ьп — 4m, у = 2m, z — 2m -f- n. Then
a = Gn — 6m)*, and 7 = 48твл — 23n2 = p* determines m. Also,
+ 1649л4 = П
if p = щ whence m = n/2.
J. CunliffeY< treated the problem subject to very special assumptions
and obtained for the halves of the sides 807, 466, 491. Later, he7* gave
another very special treatment and obtained the sides 884, 510, 466 and
medians 208, 659, 683.
N. Fuss74 reproduced the solution in Euler's paper of 1782 with a
replaced by r — ^ p by r -f- s, 7 by pf etc.
J. Cunliffe77 wrote z = AC, у = ВС, z = AB for the sides, and BE,
AFt CD for the medians* Take z = x -f- у - d. Then
44F» ** 2(AB* + AC2) - ВС1 ^ 4s2 + ixy + y1 - 4d(x + y) + 2<P.
n Poethmnou* p^er. Comm, Arith. Coll., 2,1849, p. 649; Opera pcetuma, 1,1862,102-3-
71 ТЪе Ctentleman'e Math. Сошрашод, London, 2, No. 9,1806,17.
71 New Series of the Math. Repository fed., Leybourn), London, 1,1806, П, 44.
n Ibid., 2,1800 II, 31-4.
» Mem. Acad. Sc. St. Petefeburg, 4t 1813, 247-252.
" Tbe Gentleman'e Math, Companion, London, 5, No. 27,1824, 349^53. Extract in I'iAtcr-
mediaire d math., 5, 1898,10-11.

Сяар. V] Triangles with Rational Medians. 205
Equate it to Bx -f- у — т)г and the similar expression for ABE1 to
(x -f- 2y — n)z. Solve the two resulting linear equations for x, у in terms
of d, m} n. Reject the common denominator. Tbus
x = d?Dn - 2m) -f 2d(m2 - n*) — mnBm - n),
у = d*Dm - 2n) - 2d\m* ~ n*) -f mn(m - 2n),
z = 2<F(m -f- n) — 6mnd -f- mn(m -f- n).
Then 4CD1 = 2(x* -f y4) — & becomes a quartic in d which is a square for
_ 3(w + n)(m - nJ{2m* - &mn -f-
- nL - mn(m* -f- »*)
C. Gill7* gave a solution in which the sides are proportional to expres-
expressions in the sines and cosines of two of the angles A, Bt subject to the condi-
condition that tan A/2 equals one of four complicated functions of sin В and
cos B. The numerical example is the same as the first one of Euler's"
paper of 1773.
E. W. Grebe78 thought the problem was a new one. Changing his nota-
notations to conform with Euler's, we see that 2b5 -f- 2c2 — a* = P implies
(b + c + f)(b + c-f) = (a + h- e)(a - Ъ + c).
From this and a similar formula involving gt we get
Ь -f- с -f-/ = m(a -f Ь - с), Ь -f- с - / = — (a - Ь -f- с),
c + a + g= р(Ъ + с-а), с -f- a - g =-{Ь - с -f- a),
where m and p are unknowns. These four relations determine the ratios
of а, Ь, с,/,дяя rational functions of m and p. Then 2a2 + 2b2 — c2 (which
is to equal h2) is made a rational square by choice of p rationally in terms
of m. Then the sides and medians are quintic functions of rn.
C. L» A. Kunze80 gave essentially the solution in Euler's71 paper of 1782.
J. W. Tesch*1 gave Cunlute's" solution.
* E. Haentzschel87 and Schubert88 treated the problem. Cf. papers 101,
106.
The medians of a triangle with rational sides a, 6, с are proportional to
the sides if and only if a2 + c2 = 2b2; such a triangle is called automedian.
Reports of many papers on this equation are given in Ch. XIV.
Triangles with a Rational Median and Rational Sides; Pakallelo-
grahs with Rational Sides and Diagonals.
С G. BachetV fourth problem, added to his comment on Diophantus,
VI, 18, was to find a rational triangle with one rational median. First
" Application of the angular analysis to the solution of indeterminate problems of the second
degree, New York, 1848, 50-2> ReeuJte quoted in rinterm6diaire dea math., 5,1898r 10.
Cf. A. Martin, Math. Quest, Educ. Times, 25, 1870, 96-7; E. Turriere, renseignement
math., 19, 1917, 267-272»
« AKhiv Matb. Phya,, 17,1861, 463-74.
10 Ueber einige Aufg&ben аш dex Dioph. Analyaia, Progr. WeiDoar, 1862t 9.
« L'iniermediairc dee nmtn,, 3, 1896, 237. Repeated, 2OP 1913, 219.
ю Jahreeber. d. Deutechen Math.-Veremiguog, 25, I9l6, 33335L

206 History of the Theory of Numbiibs, [Chap, v
let the angle A from which the median AD is drawn be acute. Let ВС
denote the aide whose mid point is D. Take any number, as 13, which
is a sum of two squares, 2* + 3\ and take DC — 2, AD = 3. Then
AB* + AC1 = 2AD* + 2DC* = 2-13 = 5* + 1*, since the double of a sum
of two squares is a sum of two squares. But 5 and 1 are not values of AB,
AC* Hence we divide 5* + 1* into a sum of two other squares by Dio-
phantus U, 10, vis., E - N)* + A 4- W)*, whence N = 6/5, AB - 34,
AC = 31. Multiplying all by 5, we get AB *= 19, AC = 17, ВС = 20,
AD = 15.
If A is obtuse, take DC =* 3, AD = 2. We get the same values of AB
and AC as before, while ВС = 30, AD = 10 (in place of the misprint 12)*
T. F. de Lagny*1 proved that in any parallelogram the sum of the squares
of the two diagonals equals the sum of the squares of the four sides and
noted the examples 9* -f 131 = 2E* 4- HP), 17* + 31* *= 2A5* -f 20*). To
solve z* +y* = 2(a* -f- b2) in integers, we may, for a = ^takey = 2a — жб/с,
whence я = 4abcf(& -\- c2). Next, a special solution of
is given by s = Ь, у = Ь -f- 2а; to find the general solution, set с = 2а + &,
2 = ciZ>y = b=F 2o7e; then z = (± 2bde =F 2ee?)f(dl -f- e*),
B. A. OouldM found a parallelogram with rational sides a, & and diagonals
x, y< The condition is x* -f- if = 2(a* -hb»)- Seta + b^e7<i—b = ^,
whence x* -f- j/3 = <* + e*. A solution is /я = srf -f- te, fy = ше — uf, if
P = d1 + e\ Wm. Lenhart called the sides a ± a' and diagonals 26, 26',
whence a1 — b2 = b* — a?, which is satisfied if
а, Ь ^ nn' ± mm'; o', a' » nwi' ± wtn'.
J. Maurin*8 gave Gould's solution.
E. Henetw noted that in the triangle with the sides x = pv -f- u,
у = pu — v, г = и + v + p(u — o), where « > t>, p > 1, the median m, is
rational: 2mM *= />(w + v) — t* -f- v. Also m» is rational if
и : v ~ G* - 2)Dp - /i) : G* - 4)B/» -f м), 4 < ^ < 3 + p.
M. A. Gruber87 solved 2(a? -f b2) = c2 + & by setting Ь = a + p,
с = 2а -f- qt whence a follows rationally. W. F. King (pp, 320-2) proceeded
as had Gould.94
H, Schubert" discussed triangles with rational sides o, b, ct and one or
more rational medians, that to side a being designated by ta. Since
B0* - (Ь - сJ = (Ь 4- с)г - а2 «* 4^(s - а), « = $(a -f- Ь -f- c),
the rationality of ^ implies that of x, where
=ь U - i(b - c) =* 8x, ±Ь + i(o - c) = (s - а)/ъ
tt Hint. Acad. Hoy. Sc avec lee Mem., ахшбе 1706^ Parie, 1731, 319-333 (Hist., 83-49).
* Cambridge Miscellany, 1,1943,14.
* L'mtermikUaire dea math.. 3. 1896, 210.
* Ашег. Math. Monthly, 3,180ftr 219-22b
"Auelwe UDterriohte- u. Vori^flungspnxie, Leipzig^ 2, 1905, 68-92; вате in Schubert,^
3S^0

, V] Triangles with a Rational Median. 207
Subtract, replace ex by (s — a)x -f- (a — b)x + (s — c)^ and c-6 by
s - Ъ - (s - c). Thus
з-д g-Ь 9-c_
x
Since s ~ a, etc* shall be positive — 1 < x < 1. Similarly, the rationality
of tb implies the existence of a rational value y, - 1 < у < 1, for which
* — h s - с , s- a
V
The two equations determine the ratios of s — a, • - •. We may set
в — с — (ar — у 4-1)A — x)(l + V) = C*
By addition, 8 ** 3xy + x — у + 1. Hence for any proper fractions x, yy
we have rational values of a, &, c, and of
For ж ^ *, у ^ J, we find a = 17, Ь = 27, с = 16, 2ia « 41, % = 19.
If also ^ is to be rational, we must have a rational solution*, — 1< z < 1,
of
8^- С 8 — fl g — Ь_ ft
Replacing * — e, * — &, ^ — с by their values A, By C7 we obtain a rela-
relation R between x, yt z, quadratic in each* Now the pair of equations
- У) В л (х — у Л- 1W1 — у) 4 С _
have the sum Я and are such that the elimination of z gives
у = G _ 4x - 2x*)/A0* - 5).
He gave eight further pairs of equations with the sum R such that the
elimination of z yields an equation linear in x or yr For the problem of
three rational medians, this method lacks the generality and simplicity
f Elb7
Heron triangles with a rational median; Heron parallelograms,
H. Schubert" defined a Heron parallelogram to be one whose sides,
diagonals and area are rational. Call о, р the angles made by a diagonal
with the concurring sidea a, b; and 0 the angle between the diagonals and
opposite b. Then
a : Ь ^ sin (9 + P) : sin @ — a), a sin or = 6 sin 0,
the second following from the equal areas on each aide of our diagonaL
Hence
2 cot $ = cot a — cot P.
»Aoabte Untemcfate- a. Voriewagsptuiu, Ьэрп& 2, 1905, 36-45, UnterrichtebUUtar
M*Ul il Natiirw., 4 1900, 70-1. Schubert,* 21-%

208 History or the Theory or Numbers. CChap. y
The area being rational, we may set (Schubert**)
tan§* = ^, tan^ = |, tan§6=|,
where mt n are relatively prime integers, etc. Hence
2xy 2mn 2pq '
2(x1 - y*)mnpq = xy(mp + nq)(mq — ftp).
It is concluded erroneously90 that the only integral solutions are
(x, у) = (тгщ} np) or (mp, nq).
Hence there remains in doubt his conclusion that no Heron triangle has
more than one rational median.
R. Giintsche*1 considered a triangle ABC whose sides a, hf c, area I and
median CF are rational. If 8 is the semi-perimeter and p the radius of the
inscribed circle,
cot \A = s(s — a)jly sp ~ I,
so that the cotangents л, ft т of \A, %B, JC must be rational Also,
a + p -f- 7 = сфу. Taking p = (a0 - l)/(<xfl), we have
Let F be the center of A# and v ^ cot l(CFB). From triangles CAF and
we obtain the two values of c/2:
C) e_l + l_,_F_i + ,_>.
To secure symmetry, set ^' = l/£ We obtain
D) 2И^'« - vtf** + ?'л2 - ^ - «) - 2^« = 0,
which is quadratic in each of v, <*t $'. Taking a as a parameter, we may
treat the equation in v, & by Euler's144 method of Ch. XXII. But the
second value of v belonging to /3' is — 1/v, so that the corresponding angle
has been increased by т. То obtain an essentially new solution, introduce
the variable f = vfif in place of p before applying Euler's process. A
similar remark holds for the more general equation
E) px*y + qxy* + rxy + hqx + hpy = 0.
* Other eete of solutions are m =*2,« = l,p *» —2,g — l,z—2py~ lorz - l,y = —2;
ш-»2,п = 1,р = 3,д-1уж=-5,у-4огж=4,у-—3. For x — mqt у - npt
the factor mq - np may be cancelled from the equation in \he tait, giving p(m - 2n)
» qBm — n). Hence 2m — n — Ip, ж — 2n ™ Ij, where I = 1 w 3 (м And n being
relatively prime), Schubert erronaouely exchded I = 3, ав е^*«^ for which ie p ^ 3,
9«1, m-5tn = l; this however does not affect the rebtkm between tan («#) and
tan 01/2).
Н^г. Bertm Math. GeaelL, 4, X905,27-38.

Chap» VJ TRIANGLES WITH RATIONAL ANGLE-BlSKeTOBS, 209
which includes the case treated by Киттег.ш То simplify Eider's process,
set
h Y - k
From the initial pair x = Xq, у = y0, we form
Then Xi9 yt is a new pair of solutions of E)* Similarly, we may start
with io, Ко* For D),
h ш - 1, p = 2«, g - - а, г - 1 - a*, *(tf = - (f + 2)/Bf + !).
Hence from the initial pair **, Й, we get vi => Л*(»оЛ)> tf ж —
From the trivial solution a = pv ль - 1, 0o = ^/P» we 8e*
From these we obtain a new set; etc. We may replace vhy — 1/v, since
C) remains unaltered; we obtain the solution
E. Haentsschel^ repeated Gimtsche's deduction of C), with a, p inter-
interchanged. For symmetry replace the new a by its reciprocal. Hence
2a + 20 r •
The value obtained by solving for r will be rational if
I0ta + ^(^ ^ 1) - a(* + C4^aV - D.
This quartic in & is faeated by use of Weierstraas's elliptic f^ftmction
let. Haentsschel82 of Cb. XYJ There result various particular types of
Heron parallelograms.
Triangles with bational sides and one ob more rational angle-
bisbctobs*
С G* BachetM gave a long construction and discussion leading to the
special acute angled triangle with the sides (reduced 1 :4) 20, 20, 5 and
having 6 as the bisector of either equal angle; also the oblique angled tri-
triangle with the sides 80,125,164 and having 60 as the angle-bisector drawn
to the side 164. [The area of each triangle is irrational.]
J. Kersey91 discussed oblique triangles with rational sides and area and
one rational angle-bisector or median.
mer. Berlin M*th. QeedLr 13, 1913-t, 80-0.
Dioph*nti Aki. A*itbJ ,.., 1621, 410-21, Ed. by 8. Format, 1670,317-0.
* The Henvnts of АДОго, London, Bodka 3 *ad 4, Ш4,144-8,

210
History op the Theory or Numbers*
[Chap.v
N. Fuss" investigated triangles with rational sides a, b, c, rational
angle-bisectors a, p, у and rational area <j% The altitudes are then rational.
Set
Ь -f- с - a = 2ft a + с - Ъ « 2g, а + Ъ - с = 2h.
Then
a + Ъ + с = 2(/ + ? + A), ^ = C/ + ^ + *)ftfc.
He took / = pq7 g *~ qrj k = pr. Then ff is rational if
pq + pr + ?r = *■,
where a is rational. Since a *= g + ^ Ъ = f + А, с = / -f- ^, we get
The quantity under the last radical equals r* -f- $*, which is therefore to be
a square. Similarly, p1 + s2 and q1 + л* are to be squares. Set p = Is,
q = ms, r = т. Then 1 -f- F, etc., are to be squares, while
Im -f- in + mn = 1,
These conditions are satisfied if
Р*-ф Я* -S* 1-lm
l~ 2PQ > Ш " 2RS ' П " Z + m #
For example, let P- Л = 2,Q = S = 1. Then i = m = 3/4, n = 7/24.
Take s = 1 and multiply a, a, etc. by 32. We get
a = 14, 6 = с - 25, a = 24, 0 =
J, Cunliffe* noted that the triangle with the sides
has rational area and angle-bisectors. He97 obtained such a triangle with
the sides 39, 150, 175 by taking three right triangles (m3 + я1, т* - n\
2mn) with a common leg 2mn, where
each right triangle twice.
n = 12, 1; в, 2; 4, 3 and using
» Mem. Acid. 8c. St. Pttfeaboms, 4» 1813, 240-7.
* New Series of the Math. Repository, London, 3, PW 2,1814,13-15,
i*/UdL4Pk2Ul9«l

Chaf.vj Тмашызз with Rational Angle-Bisectoks, 211
Cunliffe'* found a rational triangle ABC with rational values for the
sides, altitudes and angle-bisectors. Circumscribe the circle with the
rational diameter d. Let the perpendicular bisectors mD, nE, pF of the
sides meet the circle at D, Я, F (see right-hand figure on p. 210). Set
a * AD = DB, b = АЖ, с = BF. Then Dm = a*ld, Bn = P/d, Fp « */d
Since the chords а, Ь, с subtend arcs whose sum is the semi-circle, they
serve to form with the diameter an inscribed quadrilateral with sides
a = MP, c = PQ,b = QN,d~ MN. Hence
ДГР2 = d* - a\ MQ2 = d*-b\
and
MP NQ + MNPQ = NPMQf cd ^ V* ~ a*> J¥^V - ab.
Hence if cP — a2 and <F — b2 are rational squares, с as well as a and b are
rational. By the inscribed quadrilateral ACBD,
hence DC is rational. Thus the angle-bisector Dl ** (DBy/DC is rational.
A second solution employs the inscribed circle with radius r and center St
lengths a, b, с of the tangents from A, B, C, and foot T of the perpendicular
from S to AB. Then AS* *= AT* -f Sp = a1 + r*. To satisfy it in
integers, take a = 2mnrf{m* - n2). Similarly, satisfy BS* ^ b2 -f- Л It
is proved that CS* =* с2 + г2 is a rational square by use of
W~ Wright and C. Gill*9 employed an isosceles triangle with the equal
sides С A and CB, altitude CD, and intersection 0 of the angle-bisectora
AP and BQ~ Set a: ^ AD, a =* AC + x = semi-perimeter. Then
CD = Jtf^2ax
will have a rational value ap if я-— |аA — p2). Then
It follows from certain proportions that CP is rational, while AP involves
Vl -fp2. Hence the problem is solved ifl-f-^^D-(l- 0Р)г, say,
which givee p = 2д/(<? — 1). Taking q ^ 3, 4, 5, 7, we get four isosceles
triangles with the same perimeter and having rational sides, areas and
angle-bisectors*
S. Jones108 found я triangle whose sides xt y, z and angle-bisectors are
rational Let nx and ny be the segments of z made by the bisector of the
opposite angle; mx and mz those of y. Hence у■ ^ A + n)mxl{l — mn)t
z t= A + m)nx}(l — mnL The square of the bisector of angle (s, y) is
" H* GentiemaD'a ЬШЬ. Companion, London, 5, No. 27, 1824, 544-9.
" 1Ш., 5f No. 30,1Я27, S8&4.
шТЬе Geatiemaii^ Di^, or Math. Repository, London, 1840, 33-5, Quest. 1400.

212 History or the Theory of Numbers. [Chap.v
xy(l — n1), which is a square if A — n)m(l — win) = a = mV, say,
whence m = A — n)/n. Then у ^ A - п*)х/п\ г = xjn. Then the bi-
bisectors of angles (x7 z) and (y7 z) are rational if 2n2 — n and 2n* + n are
squares. Equating the first to p*nl, we get n* Then 2л1 -f- « = О if
4 - рг = П = {2 - pqY, which determines p. W. Rutherford called the
sides а, Ъ, с; the square of the bisector AD of angle A equals 4Ьсф — а)/
(&H-c)J, where s ~ (a -f Ь -f c)/2. Thus bc*(*-a) = D. Similarly,
ute(s-c) = O,ac$(s-b) = D. Hencee(e - a)(s - 6)(* - с) => П and
the area is rational. Thus the problem is that treated by Cunliffe.9*
J. Davey101 found a triangle ABC in which the sides, the angle-bisector
CD, the median CE> and the segments AE = ЖВ and ED of the base are
all integers. Take
AC = (m + l)p, ВС ~ (m - l)p, AD = (m+l)q7 BD=(m- l)q.
Then АЯ = mqt$D~ q, CD1 = (mJ - l){pl - $")• Take
CD = (m*
whence p »= m*qf(m* — 2). Then
Hence take 5m8 — 4 to be the square of 5{w — l)r/* — 1, thus obtaining
m rationally.
FeldhofT* treated 31 problems on triangles in which certain elements
(area, perimeter, side) are rational, are equal, or are equares. In the tri-
triangle formed by the juxtaposition of two rational right triangles, the angle-
bisectors are rational if two expressions of the form & + 1 are Bquares.1*
Worpitzky1* stated that, if the rational triangle with sides A) has its
angle-bisectors rational, then p = p7 — И, q = 2pv, r ** p? — a1, s = 2p*.
D. Biddiel(* found special oblique triangles having integral values for
the sides, area, altitude from one vertex and bisector of the angle at that
vertex. Use is made of 3 right triangles with a common side.
R. Chartree105 and others found integral values for the sides and the
bisector g of the largest angle such that the perimeter equals mg.
♦ R DolguSin1* gave examples, but no general solution, of the problem
to find all triangles whose area, bisectors, medians, etc. are all rational.
** The Lady's and Gentleman'e Diary, London, 1842, 69. He noted that J. Hoboyd'e
eohitioD, ig4i 57-8, leads only to degenerate triangles whose Ьаде equ&b the difference
of tbe other sides.
i* Einige SaUe uber daa Rationale Dreieck, Prop-., ОяпаЬгйск, I860.
"* For if 2r* ** 2m» is the common aide, ao that the composite triangle hae the aides
b«4> ^ + ^^^b" \h
a + Ъ + с = 2(t* +1^), Ь + с - a -
Hence the quantity under the radical in the expression for the bisector л (Fus**) is a
product of four sums of two sqaaree and hence equals mch & sum. In the ехргеачюп for
the hisectotr 0 оссшя the square root of E - ac(a + Ь -b c)(a + с — b) - 4{т* + «»)
ХС^+л^йС^-п1). RepUdng*by»»«^b«pwegeta-(fe-n^(l+inV^ Hence
E is а тип of two squares. ТЪе product сфу is mtionai aince the area is rational.
^Math. Queet. Educ. Timce, 57,1892, Z2.
"Ibid., 66,1897,102^3.
*" Vest, opytn, Pimiki (Spacimski'a Bote)t Odessa, 1903, No. 355,115-157

Сндр. vj Triangles with Rational Angle-Bisectohs. 213
H. Schubert" (pp. 17-21, or Schubert/* 27-36) considered а Негод
triangle ABC in which the bisector wA of angle A is rational. Since it
divides the triangle into two Heron triangles we need only take A/2 and В
to be Heron angles, i* e*,
. A 2vv A u* — & . n 2pq n p* — (f
Thus sin A and сое Л are rational, so that in his41 formulas for the sides of a
Heron triangle we need only take m = u2 — v*, n *= 2uv> To make wa
and Wb (and hence «?c) rational, take both A/2 and #/2 as Heron angles. He
considered (§ 6) Heron triangles with both a rational bisector and & rational
median.
An anonymous writer107 gave three large integers which are the Bides
of a triangle having integral values for the area, three interior and three
exterior angle-bisectors and the 12 segments cut off by them on the opposite
sides. Also a triangle having integral values for the sides, area, altitude
and two bisectors from the vertex, and the four segments of the base cut
off by the two bisectors. M» Rignaux108 gave a solution in smaller integers
of the last problem.
E. Turnere™ considered a triangle with rational values for the sides
а, Ь7 с and bisector d of the interior angle A. Thus
The rational solutions of this Pell equation are
4
У =
Hence the desired triangle is obtained by assigning any rational values to
b, с and taking a = (be — ?)(& -f- c)fq, d = 2bctfg, q = be + P. In a Heron
triangle, the bisector of angle A is rational if and only if tan {A is rational.
Every Heron triangle whose bisectors are rational is the pedal triangle to
a Heron triangle.
*O. Schulz1*7 (pp. 72-3) treated rational triangles with three rational
angle-bisectors.
Triangles with bational sides and a uneab rei/Ation between
THE ANGLES.
K, Sehwering110 discussed triangles with integral sides one of whose
angles is double another.
J. Heinrichs111 generalized the problem, taking the relation a = тф + у
between the angles. Set В *= 0/2. Then
a : с : Ь ~ cos (n - 1)B : cos (« + 1) В : 2 cos В Vi - cos2 B.
imr L?intenn6diflire dee math». 23.1916, 51-2, 73-
™lbid., 234-7.
^«I/tmeeigiiement mAth., 18, I9l6, 397^07.
"»Gymfl. P*ogrv Coeeftld, 1886.
111 Zeitedv. Matb Nftturwies. Unterricbt, 42,1911, 14Я-153.

214 Histobt or the Theory or Numbers,
Use may be made of the expansion of cos kB in terms of eos В or of
2caskB = (x+ Vrf - 1)* + (* - V^H)*, х = совЯ.
К. Schwering1" took any linear relation between the angles.
Miscellaneous Results on Triangles whose Area need not be
RATlONAb.
A. Girard11^ noted that z=B*+BD+D*, x=2BD+D*, y^2BD+B*
are sides of a triangle in which an angle is 60° [i.e., satisfy г*~х*—ху+у*\,
and the same is true of z, xi=B*—B*f y. Alsoz, x, xx are sides of a triangle
in which an angle is 120° [i.e., z1 «= x* + xxx + *У.
To find integral sides a, b, с of a triangle ЛВС such that, if P is the point
within it from which the sides subtend equal angles, the distances x = AP,
у *= BPt z a* CP are expressed by integers, we have
c* ~ x* + xy + ^ Ь1 «= «* + sz + г1, а* = уг + уг-\-г1.
Many solvers took с = x + y — m> Ь = x-\- г — n and obtained two
values for x, from which we get z *= (hy — mn)/(y — k)f where h and к are
known. Then
determines у rationally. Cf. papers 116 and 123; also 65, 67, 63, 70-73
of Ch. XIX.
Berton stated and J. de Virieu1" proved that the area of a triangle is
not rational if the sum of the sides, without a common factor 2, is odd.
W> S> B* Woolhouse11* proved that, if three numbers ^ n are taken at
random from a list of such triples and if pn is the probability they will be
sides of a possible triangle, then p%f p^.u р„+1 are in arithmetical progression
if n is even. He found the probability that three integers ^ n named by
three dirTerent persona or by the ваше person will be proportional to the
sides of a real triangle.
S. Bills1" found the least integral sides ВС, CAt AB of a triangle for
which x =* OAf у = OB and z — ОС make equal angles and are measured
byintegers.u> First, AB1 *= x* + xy + y* = D,^C2^xs-h^-h^=Dif
p* - 1 g* - 1
Take q « 2. Then ВС1 = & + yz + z* = П if 25p* -f - - ■ ^ D, which
holds lip = 9/4, whence x~44Q,y = 325, z = 264.
H. S. Monek117 gave a very special discussion of the problem to find
the least triangle with sides in arithmetical progression and altitudes in
« АгеЫт Mmtb РЬу*., (З), 21, Iftl3, 125-136.
"*L'AntlL de S. Stevm ... par A. Girard, Leide, 1625, 676; Lee Oemm» Hsth. de 8.
Sterin, per A. Giraid, 1634,169.
^ Tbe I*dyb and Gentleman's Dbry, London, 1844, 60-1, Quoit. 1705.
■"Noov. Amu M*tb, B), 3r 1864,166-170.
" Miith. Quoit. Educ Timee, 9t 1868, 63^5, 91-2.
Ш/ВДЗД1ЭТ4«Ы
4
,1874,106-9.

Chap. V] TRIANGLES WHOSE АйЕА NEfiD NOT BE RATIONAL. 215
barmonical progression. Ite Bides are the halves of the sides of a triangle
whose area is divisible by each side. A. B. Evans1" noted that the altitudes
Pi vary inversely as the sides a, b, c, whence the condition \в а + с = 2b.
Let x = cot }A, у = cot $B. Thus 2y =- x + (x -f y)/(xy^ - 1), which
gives у rationally. Then, if r is the radius of the inscribed circle,
a = r(cot
Evans and A, Martin119 found rational triangles with integral sides and
lines from the vertices to the center О of the inscribed circle, by use of
OA = resell.
M. Weill stated and E. Cesaro1" proved that D, B, 6) is the only triangle
whose sides are consecutive integers and the ratio of two of whose angles
is an integer.
1С Schweringm noted that the ratios of the sines of the three angles
a, ft, у are rational if the sides are rational Assigning values to tan a/2
and tan 0/2, whose ratio is rational, we have tan yj2 and hence the ratios
of a ± Ь ± с and therefore the ratios of а, Ъ, с. Не discussed the problem
to find a point О inside an equilateral triangle with the given rational side a
such that the distances from О to the vertices shall be rational.
Zuge** gave the general solution of s^^ + jf— 2xy cosot, where
cos a is rational. [But the topic is of little interest since we obtain a
triangle with rational sides z, y, z by assigning to them any rational values
such tbat x -f- у > zy etc.J
A. B. Evans** noted that, if ВС = 399, AC = 455, ЛВ = ЬП, СО * 195,
BO =* 264, Л0 = 325, the lines joining О to the vertices of triangle ABC
make equal angles .ш
Several121 gave triangles with integral sides and an angle 60°.
A. Martin1* discussed the last problem.
R. A. Johnson1* gave expressions for the integral sides of any triangle
with a given rational value for the cosine of one angle.
Several gave pairs of triangles with integral sides baving a common
base and equal altitudes.
E. Turriere1** found points whose distances from the three vertices of
a given triangle with rational aides are all rational.
N. Affisfcon gave special triangles with integral sides and points whose
distances from the vertices are integers.
шМд4Ь. Quo*. Educ Time*, 22,1876, 5i.
*'1Ш., 103-3.
"Mattes», 9,1889,142-3. Abo proof by Weill, Nouv. Ann. Math., D), H, 1914,526-7.
*" Geom. Aufgsben mit rationales Lfeungen, Progr. Duren, 1808.
ш АгсЫт Mith. РЬугц B), 17,1900, 354.
ш Matlb Quert. Educ Time*, 72,1900,77.
«ZdMnift Mftth. Natarw. Untemdit, 45, Ш4,184^5.
m Amtf. Math. Monthly, 21,19l4, Ж-Q. &. Neuberg». * of Ch. XUL
Ш/Ь^22^27
QumL Educ. Dines, 27,1915,
t th ^, 1917, 262-7.
Math, QumL and Solqtkme, 5,1918,37,

216 History of the Theory of Numbers.
On the ratios of the sides to the radius of the inscribed circle see
Gecono,1» Ch. XXIII.
ТЪе foDowmc papen were not available for report:
С КЫл^ thier Pyttaerattcbe u. Heronieche Zahlen, ProfT,, Ttuppau, 1906.
Б, Haentascbel, Dae Rationale inder dgebr&iedien (feometrie [ал addreeej, Unterrichteblutter
ЪЫЬ. N*turw., 21,1915,1-5.
Rational Quadrilaterals
A rational quadrilateral is one whose sides, diagonals and area are
expressed by rational numbers.
Brahmegupta1 (§ 38) stated that " the legs of two right triangles multi-
multiplied reciprocally by the hypotenuses give the four sides of a trapezium.'1
Bhascara1" (born 1114) illustrated this construction of a rational quadri-
quadrilateral by starting with the right triangles C,4, 5), E,12,13). Multiplying
the legs of the first by the hypotenuse of the
second, we get two opposite sides of the quad-
quadrilateral; multiplying the legs of the second by
the hypotenuse of the first, we get the two re-
remaining sides of the quadrilateral. One diago-
diagonal is the sum 4-12 + 3 5 = 63 of the products
of the legs of one triangle by the corresponding
legs of the other. The other diagonal is
4 5 + 3 12 = 56.
As the Commentator Ganeea A545 A.D.) indicated (p. 81), the quadri-
quadrilateral is formed by the juxtaposition of four right triangles obtained by
multiplying the sides of each given triangle by the perpendicular and base
of the other. Bhascara noted that if we take the sides of the quadrilateral
in the new sequence 25, 39, 52, 60, one diagonal is still 56, but the other
is now the product 65 of the two hypotenuses. He noted (§§ 179-184, pp.
76-8) that the quadrilateral with the sides 40, 51, 68, 75 and diagonals 77,
85 has the area 3234.
M. Chasles1" made clear the true sense of Brahmegupta's theorem.
Let о, Ъ, с, d, e be integers, such as 3, 4, 5, 12, 13, for which a2 + b2 <=* A
c* -f- <P = A Construct the quadrilateral ABCD with perpendicular diag-
diagonals A€% BD, crossing at / (see figure above), with
AI = ac, CI = bd, BI ^ady DI = he.
Then
AB = at, BC= cd, CD = be, AD = A
Hence the sides are rational and the quadrilateral is inscriptible in a circle,
since AI-CI ** BI-DI; its diameter is cef2. The area is
\{ac + bd)(bc + ad).
| !9Ь2; Colebrooke,' pp. 80-83.
p«5tthirftorique, BraxeUc», 1837, Notel2, p. 440; cd. 2, Paris, 1876; ed. 3, Ruie, I88flf
p. 421. a. O. Terquem, Nauv. Алл. MMh., 5, Ш46, 636; H. G. Zeutbea, Bibliothec»
Mrih C), 5,1904,108.

Chap. V] RATIONAL QUADRILATERALS 217
From one inscnptible quadrilateral we get two others (but not with per-
perpendicular diagonals) by permuting the sides. The area of each of the
three quadrilateralB is the product of the three distinct diagonals divided
by double the area of the circumscribed circle (A. Girard; proof by Grebe,
Manuel de Geom., 1831, 435).
L. N. M. Camot"* noted that the segments of the diagonals of & quad-
quadrilateral are expressible rationally in terms of the sides and diagonals.
E, E. Rummer11* noted that Chasles unriddled the obscurity of Brah-
megupta without perceiving the method used by the latter, and expressed
Brahmegupta's theorem in the following form. If the four side* of a
quadrilateral, inscriptible in a circle, have the values
(a* + Ъ*)(с* — <P), (aj — &*)(с* -f- #), 2cd{d?
where a, ht ci d are rational, then both diagonals (perpendicular to each
other), the segments of them, the area of the quadrilateral and the diameter
of the circumscribed circle are all rationel.
Kummer showed how to obtain all rational
quadrilaterals. Let ABCD have rational sides and
diagonals. Then the segments a, ft у, В of the
diagonals are rational. For, by
& = a7 4- AC1 - 2a AC соэ и,
coeu is rational; likewise cos v and cos (u -f- v).
Hence sin и sin v is rational; also sin2 «and there- A«
fore siii u/stn v. But a
a_aint£> d sinw 0_asintt
/S einti' 3 sin r' 5 5 sint?'
Hence &!&> 1 -f- Э/5 *= ^^/^, ^ und 0 are rational. Similarly, a and у are
rational. Next, с =» сое to is rational, in view of
A) a1 = a1 + j8* - 2afc
Set с ^ m/n, where яг, n are relatively prime. Without loss of generality,
we may assume that a, a, 0 are integers with no common factor. To treat
one of two analogous cases leading to like results, let n be odd. Then n
must divide сф. Thus a = rai, 0 = $filt n = r»,
B) a1 = г**? + s*tf -
Now at, fi\ are relatively prime, since a common factor would divide
We may take ffi odd. The product of B) by r* may be given the form
= (nJ - m1)^, Fl~ar + r**l- «ft, Л = or - A, + «ft.
If Fi and Fj were both divisible by a prime factor p of ft, then 2r*^! and
hence mi would be divisible by p, likewise a by B), whereas a, a, 0 do not
u» G4om£trie de position, Fbris, 1803,391-3.
w Jour, far Math,, 37,1848,1-20.

218 HieroET op тик Theobt of Numbers.
have the common factor p. Hence
Ft = /Л Л = 9*> V* = $u Л = n* -
tilth
Aft z у
Divide the latter equation by n and set { ~ fyf(nz). Thus
The rationality of { is thua a neoessaiy condition for the rationality of the
ratios of tha sides of triangle ЛЕВ. It is a sufficient condition, since
a ?-<* + !
by A). There are similar formulas for the remaining three triangles whose
angles at В are v> and r — id. Taking 0 as the unit of length, we have
5 (y
D _
W ос 2х ' y~ 2y
where f, 4, x, у are rational. By multiplication, we obtain two values for $.
Hence we have the condition
r« ft + g)' ~ ! fr - c)' - 1 (T - cY - 1 (y + cY - 1
^O; 2J 2x 2V ' 2y '
Hence for any set of rational solutions of E), euch that [ с \ < I, we obtain
a quadrilateral with rational diagonals and rational rides
BA
where I « 1 — c*, while a, 7 are given by C).
Let also the area Ua& + РУ + 7^ + *«) sin w of the quadrilateral be
rational, and hence also sin w. The rational solutions of sin1 to -fc* = lare
2X \* - 1
Hence to obtain all rational quadrilaterals we have only to seek the rational
solution» c> £, n, x, у of F) for which с is of the fonn (XJ - 1)/(X* + 1).
Now E) is a quadratic equation in у whose discriminant must be a square:
G) {ox1 - 2*(* + y)x - at)*
Hence we may obtain all rational quadrilaterals as follows; Give arbitrary
rational values to £, *, X and set

Chap. V} RATIONAL QtTADRlLATfcUAl^. 219
Determine fill rational solutions** xt г of G). Then E) determines two
rational values of y, and C), D), F) give the segments of the diagonals and
the sides as rational numbers.
W. Ugowski1* and J. CunHjfe1** gave special rational inscribed quadri-
quadrilaterals»
D. S. Hart141 desired an mscriptible quadrilateral with integral Bides
a, h, c, d and diagooafe x> y. Thus xy = ac -\- bd,x :y = be + ad : ab -\- cd,
so that the product of the three sums is to be a square, say the square of
abe -f- d(a* + Ы* -f- c*)/2, which determines d, A, B. Evans took the sides
AB *= x7 ВС *= mx, CD ^ nx, AD ^ px. As known.
AC1 =* (mp + ft)aa* B& = (mp
w -f- pn
The last gives p rationally. Let
a = a* fi = e», ffip -h n ^ {g + ту/(а*д* - I)}1,
which gives trc. Hart1*7 found a trapesoid with integral values for the sides,
diagonals, area and perpendicular between the parallel sides»
G. Darboux1" based a geometrical theory of quadrilaterals upon two
equations
where a, b, c, d are the sides, and tf = е*~\ u>f being the angle between
the jth side and any line in the plane. Regarding the fs as homogeneous
coordinates, we have & plane eubie curve.
O-Scbldmilch,1» started with two right triangles ^ = A-^, 2«, l-l-o2)
and Tfif reduced their sides proportionally to obtain a common leg,
and juxtaposed them to obtain в triangle with the sides A + a*)/?*
(<x -f 0)A — ap), A -f- /3*)a. Treating two such oblique triangles similarly,
we obtain a quadrilateral with the sides A + ja?Hf A -f- 0*)a, A
A -f- F)yu, where
(
The sides^ diagonals and area are rational if a, ■ ■ •. fi are*
S» Rohms140 listed rational trapeziums whose area equals the square
root of the product of the four sides, found by use of convergents to Va* -f- f.
H. Schubert11 (pp. 49-54) considered quadrilaterals inscribed in a circle
of radius r. Let 2ah • • -, 2o* be the arcs subtended by the sides* Then
"ТЬот^пф!* flotations of G), Kemocrobtoined neweoiutionft by tbe method of
U СЬ. ХХП »nd thttft deduced т*гкш» rales for forming m(iozk&l qmidril^tenle.
ш Arebiv Math. Fbj**, <7f 1867,113-6.
"•• New Series of MaUl jBepostory (e^ Т. ЬгтЬоиго), 2,1809,1, 74-5, 225-6.
ш M»th. Qpert. Educ. ТЫ», 20,1874, 94^5l
ит/М^ 80-81- For Ьшкогу of шваг^рШе qaadriWcnde with given швм, 2t, 1874,29^35.
"»B«dLScM*ib, Artr^ B), a, I, l«79t 10»-128; Comptee Benditf Pads, 88t 1879,1188,1252.
"• ZtttMOir. Mafch. N«tvrw. Unteniefat, 34,18U3, 401-9.
ш Am*. Math. Monthly, 6,1896,181-2,

220 HiffroBT of the Theory of Numbers. [Chap, v
the sides are 2r sin «,-, the diagonals are
e = 2r sin (<*! -f- «0, /= 2r sin (oc* -f- «0-
The area ie §tf sin («i + a»). In the very special case in which the tangents
of \<*u la*, 1«* a*6 rational, as well as one side or r, the four sides, diagonals
and area will be rational
A. Gerardm Juxtaposed two right triangles with a common hypotenuse
to obtain a quadrilateral whose sides have the values quoted from Brah-
megupta by Kummer; also a second quadrilateral.
E. N. Barisien142 noted the insoriptible quadrilateral with the sides
AB - 75, ВС = 68, CD = 40, DA - 51, segments of diagonals (at right
angles) AI = 45, BI ** 60, CI = 32, DI = 24, and diameter 85 of circum-
circumscribed circle.
F, Neiss"* treated rational triangles and rational quadrilaterals.
I. Newton14" treated the problem to find the diameter x = DA of a
circle having an inscribed quadrilateral ABCD, three of whose consecutive
sides a = AB, Ь «= ВС, с - CD are given, while the fourth side is the diam-
diameter. We have x1 - (a* -f b* + c*)x - 2abc *= 0. E. Haentssschel and
E> Lampe"*7 found rational quadrilaterals of this type by the method of
Кшшпег.ш
E. Haenteschel14» treated rational quadrilaterals with perpendicular
diagonals by setting с = 0, t ** 1, in Summer's work. Condition G) Is
now а*(х* — 1)* -f- 4tV = z1; methods of finding rational solutions are
developed. An evident special solution is obtained by taking £ *= ij; then
у = xt AB = BCt CD = AD, and the quadrilateral is given by the juxta-
juxtaposition of two congruent rational triangles. Next, taking x = t\ = c(d%
у = £ = a/6, we get Brahmegupta's quadrilateral as quoted by Kummer.
More general solutions are found by use of Weierstraes' f-function.
Haentzschel1" noted that the determination of a quadrilateral with
rational sides, diagonals, area, and radii of the inscribed and circumscribed
circles, depends on the rational solution of
(^+1)<И + 1){(^ + 1)(>*-М)+4^} = D.
By use of Weierstrass' f-function, he found two infinite sets of rational
solutions, including the special solutions by O. Scbulz1*7 (pp. 38-103).
Ankum'e method to deduce a rational tetrahedron from a rational quadri-
quadrilateral Is applied to the quadrilaterals found here.
E. N. Barifiien1* noted that in the quadrilateral with the sides
AB «* 1625, ВС - 2535, CD ** 3900, DA = 3380,
w SphinxwOedipe, ft, Ш1,187.
"Ш1Ьш, D), 3, ШЗ, 363u &*notod (p.U) the quadribtenl with rooea»ve rid« IS»
20, 24, 7, <Цопл!в 20, 25, And area 234.
"*• RAtionak Dreiecke, Vlerwi» . . ., Dis.^ Leipxig, 19\L
ie* Arithinetica munsalie, Amsterdam, 1,1761, TVt Ch, X, 14O-L5O-
»* Zdtecbrift Math- Natnrw. Tjnteindit, 4», 1915, HO-t; 4», 1918» 139-144,144^5.
ш Sttiungsber. Berlin U%th. CeadL, 14,1015,23-81-
»*7Wrf., 14, 1915,85^M.
«» L'int«m6<liAir« <Ы rnfttbp 23,191ft, 196-0.

. VJ RATIONAL PYRAMIDS; RATIONAL TRIHEDRAL ANGLES. 221
with diagonals crossing at right angles at /, and having E, F, G> ^^J8 Jyf
projections of / on the sides and K} Lt M, N as its projections on Et t FG,
GH> HB, there are integral values for the distances from I to these 12 points,
for the 8 segments on the sides, and for the 8 segments on EF, FGt OH, HE*
E. Тштгёге1" gave known results on inscriptible quadrilaterals with
rational sides and diagonals.
W. F. Beard stated and G. N. Watson1" proved that if two circles with
centers О and 0', and radii R and R\ are such that quadrilaterals can be
inscribed in the first and circumscribed about the second circle, then the
least integral values for H, R\ с = 00* are 35, 24, 5, while general solutions
follow from
(Rz ^с*)г= {Rr(R + c)ji + [R>(R _ c)(,e
For rational quadrilaterals, see Turriere,^ Euler148 and Schwermg1»;
also, Euler82 of Ch. XV. Cf. Berton** of Ch. XXIIL
Rational Inscribed Polygons.
L. Euler14* gave a construction to find a polygon with any number w
of sides, inscribed in s circle with center О and radius unity, eucn that the
sides, all diagonals, and the area are rational. Employ n — I arbitrary
angles 2A, 2Bt ->, and take as the nth angle one whoee sine and cosine
equal the sine and negative of the cosine of the sum of those л ~" * angles.
Take arc AB = 2LA, arc ВС = 2B, arc CD = 2C, etc. Hence side AB\й
2 sin A, side ВС is 2 sin B, - -, diagonal AC is 2 sin (A 4- B)> ' "'■ lo
make all the sines and cosines rational, take sin.4 ^ 2ob/(^ "^J ete"
Since triangle AOB equals sin Л cos A, the area is rational- "e ви^6
complicated expressions which serve as rational sides and diagonals of
an inscribed quadrilateral, but do not make the area rational.
H. Schubert4* (pp. 28-38, or Schubert," pp. 55-67) considered an in-
inscribed polygon with the sides ai, ■ ■ • f a». Let 2щ be the arc subtended by
ui. Let n — X of the <x's (and hence all) be Heron angles.47 Lt
tan К = qjpi, 4r = П (p) 4- $!),
so tiiat r ie the radius of the circumscribed circle. Then the siaes
at = 2r sin at are rational, also all diagonals since the ratio of а°У one ?Jf
is the sine of a sum of certain a's. The area (sin 2аг + - ■ ■ + &n 2ал)тя/2
is rational.
J. Cunliffe*7 found rational inscribed pentagons.
Rational Ptbamids; Rationai Твшеоваь
A rational pyramid ie one whose edges and volume V are rstionaL
R, Hoppe141 considered a rational trihedral angle (one having rational
sines aad cosines of the face and dihedral angles). Let o. 6. с be tbf tangents
t math., 15, 19l6,406-410.
v Matb, QneeL and flohitioiM, 4,19X7,31-2.
"*ОретжродЫта, 1, 1862, 229 (abort 1781).
*• Arehiv Math- a, fbje., 61,1877, 6

222 Ндаговт or the Theory or Numbers. [Сна*. V
of the half face angles. Then the cosine of the dihedral angle (b, c) is
гу + <л - o»(i + frV)]/f2bc(l + a4)). Adding and subtracting 1, we otb
tain as factors of the numerators
D = а + Ъ + с — аЪс, A = - а+ Ъ-\-с + abc,
В = а-Ъ + с + аЬс, С = а + Ъ-с + аЬс.
Hence г = sin (&, с) = ^АВСОЦ2ЬсA + а1)}. If /, p, A are the tangents
of the half dihedral angles, then
s = у/а + я, (i + wp + *«>/» - а + оа + л/*
etc. If the latter equation has rational solutions, we obtain 32 distinct
rational trihedrals, since we may replace Ь by its reciprocal, eto.
To obtain a rational tetrahedron, we may take two rational trihedrals
having a common dihedral angle and subject to the condition that the
edges converge (in the earlier notation, W < 1, «/ < 1, / = /0- While
the tetrahedron now has a rational volume, it remains to make the sixth
edge rational. The condition is that Ь\ + Ь\ + cf + cj — 2 — 2Ь1Ъ1схсг — 2m
be a square, where
V)(« + «0A + Л "
К. Schwering140 discussed rational tetrahedra by use of the formula
A - cos2a)(l -cos^p) - (cos 7 - cos a cosjS)',
where /, g, h are the edges from the vertex D, and a, ft у are the face
angles at D, while athfc are the sides of the base of the tetrahedron.
The first problem is to choose rational values of the cosines such that F
shall be the square of a rational number. The first term of F must be the
sum of two squares. Give 1 — cos2 a the form of a fraction whose de-
denominator is a perfect square. Then its numerator is a divisor of a sum
of the squares of two integers and hence is itself the sum of two squares.
Thus 1 — cos2 a. equals the sum of two rational squares. Hence cos2 a is
one of three rational squares whose sum is unity; likewise for cos2 j3. Con-
Consider the integral squares equal to the sums of the squares of three inte-
integers; for instance
If
G^lP + iV + i* Of-JiJ+iVj + n
we take
M л М t MM,- NP, +
«** = -, CO8/J = ^, COST- Ш
and find that F ie the sqoare of (NNt +
"Jdur.fifrMitb., 115,1*95,301 7.

Сна*. V3 Rational Pyramids; Rational Trihedral Angles. 223
The next problem is to find rational solutions of
ft c*
where the cosines are given rational numbers. Set
a
Then
g(l - Xs) = ЭДХ + сова), /(I - M*) = 2AG* + cosjS),
Hence g/f паз the value
cosy)
M + e^ -Й *
If cos a ^ cos/? *= cosy ^ 0, we have a rectangular tetrahedron and
the problem reduces to that treated by Euler1 of Ch. XIX to find three
squares such that their sums by pairs are squares. Ibis process of Euler
leads in the general problem to
_ X «= РЧ* + <**t) -f 2p(cos *-f-сое
For example, let AT ~ ЛГ ~ 0, P ~ Q *= 3, Ad = Pt ^ 2, ATj 1,
Qi ^ 3. Then
Thus /, ^ A are proportional to (вр + 4)& — 2p- 4)Eр« -f 2p — 2),
б(вр + 4)(p» - l)(p» + 2p + 2), 3(p» - l)(p* - 4^ - 2)(p» + Sp 4- 6),
For p =» 0, we remove the factor 4, and get / = 8, g ^ - 12, A « ft,
a =t 15, Ь ^ — 7, с = 12, V = 96, in which the signs may be taken positive.
Forp 2 we get/= 112, * = 72, Л = 135, a = 153,6 = 103, с = 152,
V = 120960.
To obtain a rational quadrilateral, set p -\- у = a or 2т — a- For
example, for cos а = cos 0 = cos у = — J, we have
f={7p*-» 4)(p* - 4)Bp - 1), g = S&- l)(p> + 2)Bp - 1),
Л = p(p* - l)(p 4- 4)(p* - 12p + 8).
Thus, for p ^ — ^, we have the rational quadrilateral ABCD, in which
ЛВ = 138, ВС ~ 192, ОТ - 168, />Л = 127, AC => 283, ДВ ^ 120.
We obtain a simpler solution by taking X = д. Thus, for cos л ^ — 3/7,
cos 0 = 0, cos 7 = 2/7, f = — 2, we have X= - 3 and/ * в, ? = 7, Л ~ 8,
a = 9, Ь = 10, с = 11, V = 48. For cos а = cos у = J, сов 0 => — J,
* «= 2, we get the rational quadrilateral AB = 48, ЯС = 57, CD - 73,
DA ~80,AC~ вЗ, ЯХ> = 112.
H. Schubert47 (pp. 50-7, or Schubert,88 92-104) employed a rational
polygon inscribed in a eircle of radius r and center C. Draw a perpendicular
to its plane at С and of length k such that in the right triangle of legs h
and r the angle opposite Л is a Heron angle*7 д. Thus we have a rational

224 Histobt or the Theory of Numbers, [Chap, v
pyramid. For example, if the Bides of the triangular base are 13, 14, 15,
take cos ц *= 65/97, sin fx ** 72/97; then the altitude is h = 9, lateral edge
97/8, and volume 252.
Schubert1" discussed rational spherical triangles, i e., having rational
values for the tangents of half of each side and angle.
R. Guntsche1*2 made use of F. BesselTs1" relations between the face and
trihedral angles and reduced the problem of the rational tetrahedron to
& diophantine equation quadratic in q and quadratic in г with coefficients
involving an arbitrary parameter p. EulerV44 process of Ch. XXII is
used to find solutions q, r rational in p, so that the six edges, the surface
areas and volume are expressed rationally in p.
Gunteche1" considered tetrahedra whose edges, surface areas and
volume are all rational and having all faces congruent* He reduced the
problem to the solution of
ftyB + Ф + 6- 1)(*0 - * - $ - I) = h\
but did not solve it in general. But seven particular sets of solutions in-
involving an arbitrary parameter are found.186 The tetrahedra of Hoppe14*
are all of the type here considered.
E. Haenteschel"* wrote Giintsche'e cubic function in the form
#«<* - B) - 4^* - *{& - 6)
and reduced it to Weierstrass' normal form 4П{* — е,) by the substitution
- В
obtaining^ « - &!Z\ebe% = =F 6*/4 + &/S ^ S/1. By use of Weierstraas'
F-function, he solved 4П(в — e$ = **. The case в — 7/3 is treated in detail.
• 0. Sehulz treated rational tetrahedra.
For special tetrahedra, see papers 30-31 of Ch. XIX.
Unterrichto- imd Vorleeuu^prejie, 3,1906,202-250.
r. Berlin Math. GeeeU., 9,1907, 2-16.
» Arohiv M^tb Phy5.> 65,1880, 363-^372, on epberical trianglee Tfith rational та1вдв ft* the
sinee &nd coanes of the angles and «idea. Cf. M. ВашЬов, (З), 2671918,195-6.
"SitzungBber. Berlin Math. Geeell.T 6,1907, 38-53.
»HegavetwoaucbeetemArchivMatb»Phy8M C), 11,1907, 37l.
"•SiUungaber» Berlin Math, Geeell., 12, lfilS, 101-8. Continued, 17 1618, 37-9.
ш TJeber Tetraeder mit rationalen Maes^ahlen der KantenlfingeD und dee Vohwnen*,
1914,262 pp. Cf. HaentMcheL"*

CHAPTER VI.
SUM OF TWO SQUARES.
Diophantus, II, 10, divided a given number 13 = 2* + 32, which is a
sum of two squares, into two other squares, {г + 2J + (mz — 3)*, by taking
m = 2, whence z = 8/5. In III, 22, Diophantus required four numbers Xi
such that each of the eight expressions E = Bхд* =Ь Xi shall be a square.
In any right triangle (p, b, h), k* d= 2рЬ -П. [If К* = р\ + # (» = 1,
■ -, 4), take x%r = 2pjb#*, Zxi =* hx; then E = x*(k* ± 2р&) « П.]
Hence we seek four right triangles with equal hypotenuses. We must
therefore find a square which can be expressed as a sum of two squares in
four ways. Take the right triangles C, 4, 5) and E, 12, 13); multiply the
sides of each by the hypotenuse of the other. We obtain the triangles
{39, 52, 65) and B5, 60, 65) with equal hypotenuses. The number 65 can
be expressed as a sum of two squares in two ways: 65 = 4* + 7* = Is + 8f,
since 65 is the product of 13 and 5, each a sum of two squares. Now form*
the right triangle C3, 56, 65) from 7, 4 and A6, 63, 65) from 8,1. We now
have four right triangles with equal hypotenuses. |~If we carry out the
corresponding process on the right triangles (a1 — 6s, 2ab, a% + Ь*), (с* — <P,
2cd, c* + <F), we obtain by multiplication two triangles with the hypotenueef
A) (a2 + ^(c2 + tf) = (ac ± bdf + (ad =f he)*.
The right triangles formed from ac±bd and ad^Pbc give two new tri-
triangles with the same hypotenuse, provided cjd is distinct from afb, b/a,
]
()/()]
Diophantus, V, 12, treated the division of unity into two parts such
that, if a given number a is added to each part, the sums are (rational)
squares. The problem is equivalent to the representation of 2a + 1 аз а
sum of two squares. It is stated that a must not be odd [so that no number
4n — 1 is a sum of two squares} Unfortunately the text of the second
part of the necessary condition is very obscure. C. G. J. Jacobi1 emended
it to read that 2a + 1 must have no factor of the form 4л — 1; P. Tannery
and T. L. Heath, in their editions of Diophantus, read prime factor; but
neither correction makes the criterion exact.
Diophantus, VI, 15, stated that 15 is not a sum of two (rational) squares.
Mohammed Ben Alhocain,2 an Arab of the tenth century, gave a table
of numbers equal to a sum of two squares, formed by adding each square to
itself and to the larger squares. It is stated falsely that if an even number
is a sum of two squares, one of them is unify.
• See Cli. IV, Diopbantus.'
f For & Hke composition of factors a1 - йэ see Eulei* of Ch. ХП.
1Berichte Akad, Wim. Berlin, 1847, 2G5-278; Werke, 7, 1891, 332-344 (report below).
Same by H. Н&пЫ, Zur Gcecbtchte der Math., 1874,100.
1 Cf. F. Woepeke, Atti Accad. Pbnt. Nuovi lined, 14,1860-1, 306^0.
225

226 History of the Theory of Numbers. [Ckap, VI
Leonardo Pisano,3 in his Liber Quadratorum of 1225, proved A) and
used it to solve x2 + уг = a2 -\- h2t given a solution of c* + d? = e7:
x = (ac + bd)!e, у = (ad - bc)/e.
This solution was reproduced without proof by Lucas Paciuolo and Cardan
in their arithmetics (full titles on p. 6 and p. 8 of Vol. I).
F. Vieta4 noted that X2 = F* + G2, Z* - £2 + D* imply
(Г) {XZ)* = (BG ± DFJ -f {BF =F /XT)*.
If В and В are the hypotenuses, M, N and MD/Bt ND/B the pairs of
legs of two similar right triangles, a third right triangle with the legs
(BM ± DN)IB and (BN =F DM)jB has the hypotenuse V#* + D2. Гп
the special case F = B>G = Z>, A') becomes (X2J = BBDY + {^ - D*)*;
the right triangle with the sides 2BD, D* - £*, X2 is called the triangle of
double angle. Using the latter and the given triangle (B, D, X), and
applying the same rule, we obtain the triangle CBD2 - B*> D2 - ЗВ*П, X*)
of triple angle, etc. ^equivalent to De Moivre's formulas for cos na> sin na
in terms of cos a, sin a].
Vieta,5 to express Z* = B2 + IP as the sum of two new squares, employed
a second right triangle (F, G, X) to obtain A'), whence £cf. L. Pisano3}
F^DG\
X J '
He noted that the method of Diophantus II, 10 consists in denoting the
sides of the required squares by A + B, SAjR — D. Thus
Hence from (B; Dt Z) and the triangle BSR, S2 - К*, S* + i?2), formed
from S, Й, construct a third triangle by (Г) and reduce the sides in the
ratio R1 + Bz.
G. Xylander,6 in his comment on Diophantus V, 12, stated incorrectly
that a must be the double of a prime,
C. G. Bachet7 remarked that 10 is the double of a prime, while
2-10 + 1 = 21 is neither a square nor the sum of two integral squares,
and expressed his belief that 21 is not the sum of two rational squares»
While Diophantus seemed to infer that the double of the even number a,
increased by unity, should be a prime, this would exclude 22, 58, 62, for
which 2-22 + 1 - 45 = 36 + 9, etc., whereas 45, 117 are not primes,
"~ "Tra Scritti inediti, 1854, 66-70, 74-5; Scritti L. Piaano, 2, 1862, 256. Review by O.
Tcrquem, Annali Sc, Mat, Fi^ 7r 1866, 138; Nouv. Ann. Math,, 15,1856, Bull, BibL
Hist., 61, a, Woepcke, Jour, de Matb,p 20, 1855, 57; A, Genocchi, Annali Be, Mat,
F{b,p 6y 1855, 241-4; M» Chaelee, Jour, de Mftth., 2, 1837, 42-9, who ^ve a geometrical
proof*
<Ad Logiaticem Specioeam Notae Prioree, Prope, 46-48; Opera Math., 1646, 34. French
trauel. by F. Ritter, BuU. BibL Storm Sc. Mat., 1, 1868, 267-9.
'Zetetica, 1591, IV, 2, 3; Opera Math., 1646y 62-3.
1 Diophanti Alex. Rerum Arith. libri вех, Basel, l5757129, L 9.
7 Diophanti Alex. Arith,, 1621, 301-4,

Chap. VI] SUM OP TWO SqTXABES. 227
He treated the generalization to divide any number (as 2) into two parts
such that, if a given number (as 4) is added to each part, the sums are
squares,—whence 10 is to be expressed as a sum of two squares each > 4,
FermatV comment was: "The true condition (namely, that which is
general and which excludes all the numbers which are inadmissible) is that
the given number a most not be odd and that 2a -f 1, when divided by the
largest square entering it as a factor, must not be divisible by a prime
4n - L»
A. Girard* (t Dec. 9, 1632) hed already made a determination of the
numbers expressible as a sum of two integral squares: every square, every
prime 4n + 1, a product formed of such numbers, and the double of one
of the foregoing.
Bachet7 (p. 173) in his comment on Diophantus III, 22 found that 5525
is the sum of the squares of 55 and 50, 62 and 41, 70 and 25, 71 and 22,
73 and 14, 74 and 7. Also 1073 = 322 + 7* - 28* + 17* is a sum of two
squares in four ways. Thus 5525 -1073 is a sum of two squares in 24 ways,
all being given. He stated and proved A) ш his Porisms, IIIr 7.
Fermat1* mede, apropos of Bachet's preceding comments, the remarks:
(A) Every prime of the form An + 1 is the hypotenuse of a right
triangle in a single way, its square in two ways, its cube in three, its bi-
quadrats in four, and so on indefinitely»
(B) The same prime [4n -f 1J and its square are the sums of two
squares in a single way, its cube and biquadrate in two ways, its fifth and
sixth powers in three ways, and so on indefinitely.
(C) If a prime which is the sum of two squares be multiplied by an-
another prime also the sum of two squares, the product will be the sum of two
squares in two distinct ways; if the first prime be multiplied by the square
of the second prime, the product will be the sum of two squares in three
distinct ways; if the first prime be multiplied by the cube of the second,
the produet will be the sum of two squares in four distinct ways, and so on
indefinitely»
(D) It is now easy to determine in how many ways w a given number
can be the hypotenuse of a right triangle» For the number pa<frcs, where
p, g, r are primes of the form An + 1, while s is a square having no such
prime factor,
w = 2cBab -\-a + b) +2a& + a + b + c.
Here, and in (E), Fermat used numerical values.
(E) To find a number which is an hypotenuse in an assigned number
w of ways, take the prime factors of 2w + 1, subtract 1 from each and
• Oeuyne, III, 266,
• L'arith. de Simon Stevux ■ < ■ annotations par A. Giraid, Leide, 1625, 622; Oeuvree Math.
de Simon Stevin par Albert Giranl, 1634, p. 156, ool. l, note on DiopLantue V, 12. Cf.
G.Vacca, BibKotheca Math,, C),2,1901, 358-9. Cf, G. Maupin, Opinions etCurioeit*e
touchant la Math^mfttique, Paris, 2, 1902, 158-325.
"Oeuvree, 1,293; Шг 243-6. Diophanti Alex. Arith,, ed., S. Fermat, 1670, 127.

228 History of the Theory of Numbers. [Сна*, VI
take half of Hie remainder as the exponent of any prime 4n + 1. [Since
2» + 1 - Ba + l)Bfr + i){2c + 1)- • ■,
by Z>0 For w = 7, 15 = B + 1)B >2 + 1), and ptf answers the question.
(F) To find a number which shall be the sum of two squares in any
assigned number w of ways» For w = 10, set 2w = 2 2 5. Subtracting
1 from each prime factor, we get 1, 1, 4. Take three primes of the form
4n + 1; for example, 3, 13, 17. The number sought is the product of two
of these by the fourth power of the third.
(G) Conversely, to find in how many ways a given number, say 325,
is the sum of two squares, consider its prime factors of the form An + 1.
Since 325 = 5M3, we take \\2Л +2 + 1 + 1} =3. Then 325 is the
sum of two squares in three ways. For three exponents а, Ь, с, the number
of ways is k/2 if к = (о + 1}(Ь + l)(c + 1) is even, but is {к ~ l)/2 if к
is odd.
(H) To find an integer which is the hypotenuse of any assigned number
tv of right triangles, and which if increased by a given number a becomes
a square. The question is difficult. If w = a = 2, 2023 and 3362 satisfy
the conditions, as do also an infinitude of numbers.
That no number 4n — 1 is a square or a sum of two rational squares
was communicated to Descartes March 22, 163S, as having been proved
by Fermat Descartes11 proved this for integral squares by observing
that a square is of the form 4k or 8k + 1.
Fermat1* stated that he had proved that a number is neither a square nor
the sum of two squares, integral or fractional, if its quotient by the largest
square dividing it contains a prime factor 4n — 1; and that x2 + y1 is
divisible by no prime 4n — 1 if x and у are relatively prime»
Fennat (Oeuvres, II, 213) stated the contents of A, B, Z>, E in a letter
to Mersenne, Dec. 25, 1640. Frenicle, in a letter to Fermat {ibid., 241),
Sept. 6, 1641, proposed the problem to find the least number in F. T.
Pepin18 noted that this problem and Z> are answered by the theory of
quadratic forms.
Fermat14 called the theorem that every prime 4n + 1 is a Bum of two
squares [cited henceforth as Girard's* theorem^ the fundamental theorem
on right triangles. He" stated that he possessed an irrefutable proof.
Elsewhere he18 stated that his proof was by the method of indefinite descent:
" If a prime 4» + 1 is not a sum of two squares, there exists a smaller
prime of the same nature, then a third still smaller, etc., until the number
5 is reached/' thus leading to a contradiction. He found it much more
difficult to apply the method to such an affirmative question than to
negative theorems (cf. Fermat,* eta, Ch. XXII); for lie former, "the
method had to be supplemented by some new principles."
u Oeuvn* de Deecartee, II, 92; letter to Мегэеспе, March 31, 1638. Cf. p'l$5.
» Oeimee, П, 203-4; letter to Roberval, Aug., 1640.
" Меиюгте Acced, Pont. Nuovi lined, 8P 1892, 84-108; Oeuvree de FencAt, 4P 1912, 205-7.
» Oeuvrefl, II, 221; letter to Frenide, Jime 15P1641.
* Оеитае», И, 313,403; Ш, 315; lettera to Faecal, Sept. 25r1654, and to Digby, June 19,1668.
"O П, 432; letter to Ctravi, communicated to Huygens, Aug. 14,1659.

Chap. VI] STJM OP TWO SQUARES. 229
FreniclelT concluded from numerical tables that, it pi, Pt, ■ ■ ■ are
distinct primes, each the hypotenuse of a light triangle (a necessary and
sufficient condition being that the prime is of the form 4k + 1), a number
tf = p*1 ... pi* is the hypotenuse of exactly 2* primitive right triangles
(i. e., with relatively prime legs). He recognized that the problem reduces
to the question of the number of ways in which the proposed number N
can be expressed as the product of two relatively prime factors. The non-
primitive triangles are obtained from the primitive triangles whose hypot-
hypotenuses are the factors of N. Fennat's rule D is given. Problem G is dis-
discussed (pp. 34-нШ).
John Kersey,18 to treat & + tf = d* + Ъ* of Diophantus II, 10, set
$ = ra + b,y = sa-d. Thuso = 2{sd - rb)f(9* + r*), so that the values
of x7 у follow. He also treated the problem [Bachet,7 304] with the restric-
restriction that x or у shall fall within given limits,
Claude Jaquemet,1* in a letter Jan. 26, 1690, proved that an integer not
a square, which divides no sum of two squares without dividing each
square, is not a sum of two squares, integral or fractional. A manuscript
by Jaquemet or N. Malebranche proved also that a number which
divides a sum of two relatively prime squares is itself a sum of two squares;
but the later proof by Euler24 is far simpler. Cf. Bhfocara,» § 88, of Ch. XII.
The Japanese Matsunago,20 the first half of the 18th century, would
solve 2? + у* = к by setting hfl — r5 + R, where r5 is the greatest square
contained in k, and forming the equations
ax =* 2r — 1, a2 = ax — 2, a, =* a2 - 2, ■ ■ -,
bx = 2r + 1, Ъ* = Ъг + 2, h = b* + 2,
From 2B subtract successively Ь^ Ъ*у - - -. When a difference is negative,
add the corresponding a.. If the remainder zero is reached, and a', 6'
are the values last employed, a solution is
It was stated that a set of solutions of x? + у2 = г8 is given by
x = (m* — Зп*)т, у = Cm* — n*}n, z = m* + n\
L. Euler21 proved that, if neither a nor b is divisible by the prime
p = 4n — 1, then a2 + ft2 is not divisible by p. For, a411 — fr1* is
divisible by p and hence a<lf~~* + b*" is not; thus the factor aJ + &* of
the latter is not divisible by p.
Euler22 stated that if 4m + 1 is composite it is either not a sum Ш of
two squares or is so in more than one way; if ab and a are B1, b is a BL
" Mem. Acftd. Hoy. So., 5t 1666-99, 6d. Pans, 1729, 22-34,156-163.
" The Element* of Algehia, London, Book 3,1674, 9-17, 20-23.
» Bull. КЫ. Storia Sc. Mat.« F^a., 12,1879, 800-*, 644; 13,1880, Ш.
» Y. Mibmi, Abb Gescbicbte Math. Witt., 30,1912,233.
CdMiPh(dP) 1, 1843, U7; letter to GoMbach, Maroh 6,1742,
20 C Aith I 53 § 16 Fh L
py(,), , , ; , ,
Novi Comm. Aead. Ptetrop, 1, 1747-8, 20; Cornm. Aiith., I, 53, § 16. French tranaL
in Noov. Ann. Math., 12,1853, 46.
), 1,1W3,134, letter to Gddbacn, June 30,

230 HlSTOfcY OF THE ТНЕОЕТ OF NUMBEKS. CChap.VI
He stated he had a rigorous proof. He stated Feb. 16, 1745 (p. 312) that
it has not yet been proved that the gum of the squares of two relatively
prime integers has no divisor other than a 13, nor that every prime 4n -f 1
is a 13, uniquely.
Chr. Goldbach2* proved Fermat's statement that a prime 4k — 1 cannot
divide the sum of two relatively prime squares. Let o1 be the minimum
square of the form Dn — \)m — 1. Set v — 4n — 1. Then
v{m - 2a + v) - 1 = (a - »)*,
so that a* ^ (a — v)*, whence v Ш 2a. Similarly,
{4{л - a + m) - l\m - 1 = (a - 2mO, a* ^ (a - 2m)*, m^a.
Thus a* + 1 = vm ^ 2am ^ 2a?, a = 0 or 1, values leading to contradic-
contradictions.
Euler24 proved the Lemma: Every divisor of the sum of two relatively
prime squares is itself Hie sum of two squares.
It is first shown that, ifp = c2 + d2isa prime and pq — a* + &*, then
q is а Ш. Since c*(a* -f &*) — «4c2 + #) ^ divisible by p, one of the factors
be ± ad is of the form mp. Set h = me + x, a --= ±md + y. Then
ex db dy ?= 0, But с is prime to d. Thus* x — ?w?, у = =F no. Hence
It now follows from A) that> if the primes pu • ^ pk and the product
Pi ■ ■ p*q are all 13, then g is a 13. Hence if pq, but not g, is а И, р has a
prime factor not а Ш.
Let p divide a1 + Ъг, where a, IS> are relatively prime, while p is not
a S]. Set a = mp rfc c, 6 = up =h rf, 0 ^ с ^ Jp, 0 ^ d ^ |p. Then
с2 Ч- d2 ^ P? = $P*' Hence g has a prime factor r ^ Jp, not a 12. As
before, the divisor r of c2 + d2 divides a sum e2 -Ь/2 ^ Jr2, and e*4-/2
has a prime factor ^ ^r not a EL Proceeding in this manner we ulti-
ultimately reach a contradiction with the fact that the sum of two sufficiently
small squares has all its prime factors sums of two squares.
Euler gave a " tentative proof " of Girard's theorem that every prime
p = 4n + lisa[E. If neither a nor Ь is divisible by pf a** — b*n is divisible
by p. If p divides Hie factor a2n + &**, a 23, then p is a EL It remains
to showf that a?n — ^n is not divisible by p for some pair of values of л, о
Q>roved later by Euler2*].
Since p — а2 -Ь b2 impnes 2p = (a + ЬJ + (л ^ ЬJ, and conversely
2p = a2 + b2 implies p = a2 + ^8*r where a ^ {a + &)/2, ^ = (a — &)/2 are
integers, there are as many representations of p as of 2p as a sum of two
squares (including the case in which one square is zero).
«Соггшр Math.Pby«, {edM Fuse), 1,1S43, 255P letter to Euler, Sept. 28,1743, Enler,p.
258, ezprewed surprise at the eimpHcrty of the proof.
»Jitf.p 416-9; letter to Goldbach, May 6, 1747. Novi Comm( Acad» PWrop-, 4, 1752-3
{1749), 3-40; Сош*. Aritb4 1,155-173.
♦ Ь the letter, it м concluded from bc±ad = m (<?+&) that md4= a ia diviable by e;
Thus ^ а = c* - dmpb ~ cm -\- d*.
t bi Uje letter, U is «toted Uk&t there are innumerable awes in which ^ - Ь*- и not
by 411 + 1.

Chap. VI] SUM OF TWO SQUARES. 231
From Girard's theorem and A) it was concluded that any number is
a B1 if it has the form 2*a2b, where each prime factor of & is of the form
Euler* later succeeded in establishing the point which he could not
prove in bis preceding paper.24 If the differences (a + 1Jя — aZn of the
first order of 1, 2P\ 32я, • ■ >, Dn)*m were all divisible by p, the differences of
order 2n would be divisible by p, whereas they equal Bn)l This point
can also be proved by means of Euler's^ criterion for quadratic residues;
however, Euler proved this criterion by the method of differences» In fche
former* paper (§ 70), Euler noted that the negative of a residue of a square
when divided by a prime 4n — 1 is not the residue of a square, whence
a2 + ft2 is not divisible by 4n — 1 if a and b are not» Since a product of
primes of the form 4k + 1 is of that form, it follows (§ 73) that 4n - 1,
whether prime or composite, is not a divisor of a sum of two relatively prime
squares.
Lagrange9 of Ch. VIII proved that if a B1 divides a CS the quotient is a [2.
Euler27 proved A) by multiplying (a + bi)(c + di) by its conjugate.
Euler22 gave a more elegant proof of the Lemma.24 Let N divide
P2 + Q*, where P and Q are relatively prime. Set
P - jN =fc v> Q^gN ±q, 0 ^p ^ $N, 0^q^*N.
Then p2 + $* = Nn, where n ^ fiV. Set p = <m + a, q = fin -f- &, where
a and Ь are numerically ^ %n. Set A = aa + b$. Then
Hence a2 + &* = яя', л' < Jn. Thus ^ = «(a5 + /53) + 2Л + n'. By A),
nn'fa5 + ff) - (af + Ь*)(л* + 0*) - Л2 + B\ В ^а& - Ъа.
Hence ЛГл' = (»' Н- Л)? + В2. Just as this was derived from Nn — p2 + ^2,
so from it we get Nn" = B1, n" ^ in', etc., finally AM ~ B1.
C. G. J. Jacobi1 (p. S41) repeated this proof and stated that, while it
contained nothing not known to Diophantus, there is no ground for the
assumption that the latter actually possessed the proof.
Euler2* gave a second proof of Girard's theorem. Since — 1 is a
quadratic residue of every prime p = 4» -f- 1» there exists a square b2
with the residue — 1, so that p divides 1 + b2. Hence, by the Lemma,
p is a B.
In a posthumous manuscript, Euler80 proved the first step in the above
Lemma. Let P — p2 + g3 be divisible by A =? a2 + fr2, where a is prime
* Correep. Math, Fhys. (ed., Foea), 1,1843, 493; letter to Goldbach, April 12P 1749. JToti
Comjn» Ac«!. Petrop., 5,175^-5 A751), 3; Comm. Arith.p I, 210.
* Novi Oomm. Acad, Pfttrop., 7,1768^-9 A755), 49, eeq., § 78; Comm. Arith., I, 273.
17 Algeb», St. Peterabmg, 2, 1770, §5 168-172. French traaaL, Lyonp 2, 1774, pp. 201-8.
Opc№ OmniA, A)^ I, 417-420.
"Acta Emditonim Iipe., 1773, 193; АОл Acwi. Petrop., I, 2, 1780 A772), 48; Comm.
Aritb», 1,540. Proof reproduced by Weber-WelLrtein, Encyklopadie der Elem. Math., 1
(Alg. und Ала1увп), I9G3, 244-250.
» Оривс ш!., 1,1783 A772), p. 64 eeq., § 36; Comm. Arith,, I 483.
••Trftctatuedenumeionim, 1E64—7; Camm. Arith., II, 572. Same in Open. Poetuma, 1,
1862,7X

232 Histoey of the Theory of Numbers. [Chap, vi
to b. Since A is prime to a and 6, we may set p = mA d= /a, q = nA ± gb.
Thus /*a* + дЧР is divisible by A. The error in the conclusion that g = f
was pointed out in a marginal nots by means of the case p = 17, q — 6,
a = 7, Ь = 4- However, (p* — /^b2 and hence p* — /* is divisible by A.
If we assume that A is a prime, we see that g ± / is divisible by A7 so that
q ~ vA zhfb. Hence
PjA = (f ± ma db ЛI + (dz w* =F ?nu)*.
Thus Euler's proof of the first stsp in the Lemma is valid if A is a prime.
He gave (p. 570) another proof by setting p = ma — rib, q = na -f~ mb -\- s.
Then P = А (яг1 + n7) + sfc, fc = 2(na + mb) -f- s. Since A is a prime,
either s ^= tA or k — — (A. In either case,
PjA - (m + 60* + (n + a*J.
J. L. Lagrange11 deduced from Wilson's theorem the fact that the
prime 4д + 1 divides A-2 ■ * ■ 2л)8 + 1. He82 proved the Lemma in con-
connection with the general problem to find the form of the divisors of numbers
represented by B& + Ctu + Du2. He81 deduced Girard's theorem from
the fact that a prime p of the form 4n + 1 divides x*n + 1 for 2n integral
values of x numerically < \p (it being a factor of x*~l — 1).
Beguelin7* of Ch. I failed in his attempt to prove Girard's theorem.
P. S. Laplace" remarked that every prime 4n + 1 mil be a [2 if proved
to divide a G2, in view of Lagrange." But4n + 1 divides (а2я + 1)(а2я — 1)
and not the last factor for every a, since
Bл)! = {Bn + l)**-l} -2»(Bn)«-- 1} + ■••,
by the formula for the 2nth order of differences of xln — 1 for x = 1
J. Leslie84* solved z* + у* = а2 -|- Ь8 by setting
x -b a ^ F — v)m, з — a — F 4- 2/)/m.
C. F^ Kausler2* gave tentative numerical methods of expressing a given
number A as a sum of 2, 3 or 4 squares.
Let A = 4C + 1 = BP)' + BQ + IJ. Then С - P2 + Q(Q 4-1).
If C = 2Z> + 1, then Р = 2Г+1 and D - iQ(Q + I) = 2Г(Г + 1).
Hence we subtract from D in turn the halves of the pronic numbers
Q(Q + 1), given by a table (extending to Q = 225), and note if any re-
remainder is double a pronic number. If С = 2D, then P = 2T and we
use D - iQ(Q + 1) = 2Г2.
A number A = 43 + 2 can only be the sum of two odd squares, whence
Thus В = 20. Set P = Q + tf. Solving the quadratic for Q, we see
» Nouv. Mem. Ac*L Berita, annfe 1771 A773), 125; Oeuyree, III, 431.
a Ihid^ ахшбе 1773, 275; Oeuvrae, Ш, 707»
^.^^11166 1775,351; Oeuvree, Ш,789-790.
м ТЪбцпв abregee dee monftbree premiera, 1776, p. 24.
«»Ti«m. Boy. Soc Edinbuisb^ 2, 1790, 193,
m Nova, Acta Adad. PfcttopH U, «d annum 1793 A798), ШвЬою, 125-156.

Chap. VI] SUM OF TWO SftUABBS- 233
that 4C -\- 1 — R2 must be a square. The problem thus reduces to finding
two squares with the sum 4C* + 1, treated in the first case.
The methods employed to express A as a SI or Й1 are no better than the
similar one of subtracting from A in turn squares, or sums of two squares,
and ascertaining if the remainder is a G2.
Kausler56 extended his table of pronic numbers to Q = 1000, and gave
their halves and quarters, and applied them as in the former paper. Given
A — a2 + fr2, to solve x* 4- уг — A, set x — a + 2wa, у = 2na — Ъ* Then
a =• (rib — ma)l(mz -f~ n*) is to be integral. Let wiT n be relatively prime»
Tben b = an + /??и, where 0 = (am + a)/nis an integer. The latter gives
n = -pa + |ia, #г ~ <?a + /*ft where p/g is a convergent to л//?. Then the
former arfves a relation between a, ft p, g, >i which is not solved.
C. F. Gaussar applied the theory of binary quadratic forms to prove that
every prime 4tt + lisaG3ina single way. In a foot-note he considered
M *= 2*Sa*b* *"», where a, fc, • • • are distinct primes of the form 4те + 1>
and S is the product of all the prime factors 4n + 3 of ЛГ, If £ is not a
square, M is not a \2\. It is stated that, if S is a square, there are
decompositions of ilf into a sum of two squares, when one of the exponents
a, 0, ■ • ■ is odd; but к + J if a, 0, • ■ • are all even» Here the squares and
not their roots are counted»
A. M. Legendre38 had already given the last result-
Legendre89 developed Vp into a continued fraction with the
quotients a a & ■-■ д у, ■ • • $ <x 2a - * *,
la rttq m /o /
convergent^ -- »— — — • ■ ■ — - ■ • •,
0 1 Пц п g^ g
where f* — pg* = — 1. Then by use of the convergent^ corresponding
/ т(п/щ)
g п(п!щ) -f щ>
Substituting these values into/2 — pg° = — (тщ — VhnJ, we get
то* — pn* — — {ml — pnS)»
But if (Vp + /o)/A and (Vp -|- /)/!> are the complete quotients corre-
corresponding to mo/no, m/tt, then
m2 — pn* = (тпяо — mon)D, ml — pn3 — — (тщ —
Hence A = D, so that DZ>0 -h 73 = p ^ves p = D* + 73,
Nova Acta Acad» Petrop., 14, ad annoa 1797-8 A805), 232-267.
Disquisitional Aiith., 1801, Art. 182; Werke, 1,1863, 159-163.
Tb&me dea nombree, 1798, p, 293; ed. 3P 1830,1, 314 {tiaud. by Maeer, I, 309),
Theorie dee nombreer ed. 2> 1808, 59-60? cd» 3,1830,1, 70-1. (Maeer, Ir 71-73). a.
Diriohlet,*» \ 83r long Footuote, Cf, Euler™ (end), of Ch. ХП.

234 History of the Theory or Numbers- ЦСнар. vi
Legendre" stated that every divisor of a sum of two relatively prime
squares is a sum of two relatively prime squares» P. Volpicelli41 noted
that the latter need not be relatively prime since d = 2197 - 392 + 26*
is a divisor of Ш = 119* 4- 1202 [but d also equals 9* 4- 46*}
P. Barlow42 stated that a number 4n + 1 is a prime if a El in one way
only. [He should have said relatively prime squares; 45 - 36 4- 9 is а GD
in a single way. For Euler's proofs of the correct theorem see Ch. XIV
of VoL I].
A. Cauchy43 obtained A) by taking the norm of the product of two
complex numbers.
С. К Gauss44 stated that, if a prime p = 4Л 4- 1 ie expressed in the form
e2 4- Pi e odd, / even, then =h e and d= / equal the minimum residues (l eo
between — p/2 and 4- p/2) modulo p of \rl{k\) and Jr2, respectively, where
The residue of ± e is positive or negative according as the positive value
of e is of the form 4ти + 1 or 4m + 3, But there is given no general rule
as to the sign of d= / (cf, Goldscheider130).
Gauss46 noted that the number of sets of integers x, у for which
& 4- V2 ^ A is
= 1 4-
where q = [44/2} r - g 4- [^]^ an^ CO denotes the greatest integer ^ I.
Denote by f(A) the number of representations of A by x2 4- 03, which ia
8 if A is a prime 4n 4- 1, while for A = 2и8алЪ* • • • (as in Gauss17)
/(A) ^
or 0, according as S is a square or not. The mean of f(A) is т. Set
4-/C?n); the mean of j\m) is 4тг/3. Set
/"И
the mean of /'Ч^) *8 16т/15. Proceeding, we approach the mean 4 and
find that
4 = TT'4't'f'H-*(to infinity),
the denominators being the successive odd primes pt and the numerators
being p ± I.
«Th6oriedesnombreap 179SP 190; ed, 2, 1808, 175; cd. 3, 18Э0,1, 203 (Maeer, 1, 204).
u Ann&li di Sc. Mat- Fis., 4, 1853, 296.
41 Theory of Numbers, London, 1811, p. 205.
u Coure d'analyee de J'eoole polyt., 1, l82l, 181.
" Gott. «elehrte Anarj 1, 1825; Comm. 8oc. 8C. Gott, recent,» 6, 1828; Werke, IIr 18вЗ, 168,
90-1. Cf^ Bachmann,*1 KreiHteilung, Ch. X.
« Forth. MS., Werke, II, 1863, 269-275, 292; Gau»-Maeer, HOhere Aritb., 1889, 656-WL
Cf. EJaenatein,11 Hennite»1". w

Ц Stjm or Two Squabes. 235
C. G. J. Jaeobi* stated in a letter to Legendre, Sept. 9, 1828, that the
theorems relative to numbers represented as a 23 follow from
A. Genocchi^ noted the conclusion that, if x2 4- y? = n has JV\ @ or 2)
sets of solutions with xory zero, and ЛГ2 other sets, iVi 4- 2tf2 is double the
excess of the number of divisors 4m 4- 1 of n over the number of divisors
4m 4- 3 of n.
Jaeobi47 gave the formulae
where ш, n range over all odd integers such that all prime factors of m
are = 3 (mod 4), all of n are = 1 (mod 4), while \f/(n) is the number of
factors of n and hence is the excess of the number of divisors 4k 4- 1 of
over the number of divisors 4k 4- 3 of
A comparison of the square of the latter series with the former shows that
the number of representations of 2mbi as a sum of two odd squares is the
excess of the number of divisors 4k 4- 1 of mbi over the divisors 4& 4- 3.
Jaeobi48 proved that
where A(x) is the excess of the number of divisors Am 4- 1 of x over the
number of divisors 4m 4- 3» Although not explicitly stated by Jaeobi,
it follows that the number of representations of x as a El is 4AU} Qcf.
Dirichlet52]. Evident corollaries relating to the evenness or oddness of
the two squares were noted by J. W. L. Glaisher."
Jacobiw gave an arithmetical proof of his47 first theorem: If p is odd,
the number of sets of positive integral solutions of y1 4- z* = 2p is the
excess E of the number of factors 4m -\- 1 of p over the number of factors
4m 4- 3. Let
p^aA ...PV'..V,
where a, " -, /> are primes 4m + I, and a', • • -, a' are primes 4w 4- 3.
The factors of p are the terms of the product
(l+a+ ... + ,/).■. (l + p+ ... +
«JourfUrMatb,, 80, 1875,341; Werke,I»434.
« Fundament* Nova Func. ЕШр.г 1829, 106 C1), 107, 103E), 184G)? Werke, I, 162Cl),
163,159E), 235G). a. Jaeobi» of Gh. IIL
«Fund. Nova Func» ЕШр., 107,184F); Werke, 1,162-3, 235F).
** Quar» Jour. Math., 38, 1907, 7.
» Jour. fQr MatiL, 12,1834,167-«; Werke, VI, 245-7.

236 History of the Theory of Numbers. [Chap, vi
Set a = - • - p = 1, *' =* •••=*'=*— 1. Then a factor 4m + 1 is
replaced by 4- 1, a factor 4m + 3 by — 1. Henee the product is replaced
by E. Thus
Hence E = 0 unless A', - - -, S' are all even. If the latter are all even, E
is the number of factors of n = aA • • • рл, while p = nQ*, where every prime
factor of Q is of the form 4m 4- 3» Now 2p is not а Й unless p is of this
form nQ2. Also 2л<2* *= y2 4- ** requires that # and z be divisible by Q>
while 2n ~ w2 4- л2 has as many sets of positive solutions as n has factors
(all the factors of n being ofythe form 4m 4- 1).
A. D. Wheeler51 gave trivial or known results on 12).
G. L. Dirichlet*2 obtained, as a special case of a general thoerem on
quadratic forms, JacobiV8 result that, if n is odd and positive, the number
of sets of solutions of x2 4- У2 ~ « is tbe quadruple of the excess of the
number of divisors 4k -\- 1 of n over the number of divisors 4k 4- 3.
A. Cauchy43 proved Gauss* " result that, Кр = хг + у*,
Cauchy54 proved identities of the type
A4- 2t 4- 2t* 4- «• 4- - 0f = A 4- 2£ 4- »• + - - ■)'
G, Eisensteinw gave the values of А, В in p = An 4- 1 = A* 4- B* and
p = 3n 4- 1 = A* - AB 4- £*, where p is a prime. He" stated that the
number of lattice points inside and on the circumference of a circle of radius
Vw and center at the origin is
>{«-[?№]-И—}•
С. G. J. Jacobi" gave the representation as a B) of each prime
4n + 1 = 11981.
Jacobi1 noted in 1847 that an insignificant change in the text of
Dioph&ntus V, 12 gives the result that, if a number without a square
factor is a O, neither itself nor a factor of it has the form 4л — 1, and
expressed his belief that Diophantus had a proof, though he gave none,
since all that is essential to a proof was in the Greek mathematics and is
11 Amer. Jour. Sc. and Art* <e<L, B. Silliman), 25, 1834, 87. ^~ ~~
и Jour, fur Math., 21, 1840, 3; Werkc, I, 463» Zahientheorie, $ 91.
a Мёш. Ac. Sc. Pane, 17,1840, 726; Oeuvn», A), 3, 1911, 414.
« Comptee Rendus Paris, 17, 1843, 523, 567; Oeuvree, A), VIII, 50, 54T
* Jour, fur Math., 27,1844, 274.
«Jtof., 28, 1844, 248. СГ Gau»," 8иШ* and Сау]еу.л Proved *Jeo by H. АЫЬогп,
Udber BerecbnuDg von Summon von grfeeten Gei«en auf geometriflchem Wege, Progr.
Hamburg, 1881, 18.
" Jour, fur Math., 30,1846,174^6; Weifaj, VI, 2ft6-7. Jurat*, Мею. Math., 34,1904, 132.

Chap. VTl SUM OP Two SQUARES. 237
in the spirit of their method» From this point of view, Jacobi proved that,
if a given odd number N is the вшп of the squares of two integers b and с
having no common factor 4*1 — 1, every prime factor p of N is of the form
4k 4- 1» When Ь, с are relatively prime, the proof shows that p and hence
every divisor of N is a sum of two rational squares. The fact that every
divisor is a sum of two integral squares is established by an argument
perhaps not known to Diophantus and not necessary for his assertion»
F. Arndt," using continued fractions as had Legendre" for the ease of a
prime, proved that the hth power of a prime 4n 4- 1 is a El in 2* ways»
J. B» KuhV9 gave the representation as a El of each prime ^ 10529.
V. A» Lebesgue* noted that x* + у* = z* + & becomes pq = r$ if we set
2x=p + q + r-s, 2tf = p + g — r + s,
2z ^ p — g + r + s, 2t ^ p — q — r — s.
C. Hermite*1 developed a/p iuto a continued fraction, where a1 = — 1
(mod p), and employed two consecutive convergents m/n, m'/n\ such that
n < Vp, n' > Vp. Then
Since (na — трУ + n2 is a multiple of p and is < 2p, it equals p.
J. A. Serret62 employed q* = — 1 (mod p), 5 < p, and developed p/^
into a continued fraction so that the number of quotients is even (replacing
if necessary the last quotient Q by Q — 1 4- 1). In the series of quotients
the terms equidistant from the extremes are shown to be equal. Let mjn
be the convergent which includes the quotients of the first half of the series,
and *Яо/по the preceding convergent» Then the continued fraction whose
quotients are those of the second half of the series has the value т/1щ.
If w is the common middle quotient, the convergent following mjn equals
nw 4-
Replacing и by m/m*, we get the entire continued fraction. Thus
q mn + '
L. WantzeP stated that the use of complex integers affords the simplest
proof that every prime divisor of a El is a El. He proved that no complex
prime a 4- Ы divides a product without dividing one factor Qdue to Gauss].
и Jour, fur Math., 31, 184вг 343-358; extract of Di»., Sundiae, 1845» Arndt,1* Ch. XII. "
" Tafeln der Quadrat- und Kubik-Zahlen aller Z&hten bie Hundert Tausend » » » , Li
1848, Table 2.
"Nouv. Ann» Math., 7,1848, 37»
« Jour, de Math., 13,1848,15^ О&лъы, I, 2$4; Nottv» Ann. Math., 12, 1853, 45;
philoroalique de Pane, 1845,13-14.
■ Alg«>reSap^r., ed. 1,1849,331; J<wu\ de math., A), 13,1848,12-14; Nouv. Ann. Math., \2,
1853] 12: Society phUoomtique de Fane, 1848,12-13,
« Sodetl philomatique de Parie, 1848,19-22.

238 History of the Theory of Numbers, [Chap. VI
P. Volpicelli" noted that, if z = a) + b) (j = 1, —, m), A) shows that
z2 is a sum of two squares in m{m — 1) ways, not necessarily distinct. If
z ^ m2 + пг = р* -\- q2, then
z = (a\ 4- &J)(aJ + Ъ\), axa^ — —-—, bj>z — - ,
t n 4- q , n — q
To show that a number having a prime factor p = An 4- Z is not a sum of
two relatively prime squares, raise a* = pg — b3 to the power 2л 4- 1,
whence 5 - a^1 4- b^1 is a multiple of />, whereas * = 2 (mod p) by
Fermat's theorem» In attempting to prove that every prime p = 4n 4- 1
is a El, he employed relatively prime integers z, y, not divisible by p and
one even. By Fermat's theorem, x1* — y*n — pQ. Since every odd num-
number can be expressed as a difference of two squares, he claimed that we can
satisfy xin — угп = Q, whence p = (x*J 4- (y*J. By use of A), a product
of к distinct primes of the form 4« 4- 1 is a sum of two squares in 2*
ways, and only in that many ways» Several examples illustrate the method
to express A as a El by use of the continued fraction for VX The nth
power of a El is a El in n/2 or (n 4- l)/2 ways, according as n is even or
odd.
Volpicelli*5 considered the number у of ways of expressing z as a \2\,
when each prime factor of z is a El. When z is a product of к distinct
primes, v = 2*^1. When just two of these к primes have exponents m
and m\ his three formulas can be combined into the single one у ^ 2*"^"м+м/?
where /x = m/2 or (m -\- l)/2 according as m is even or odd, and similarly
for ц\ When the roots of the two squares are given double signs, the
number is 4?»
Volpicelliw considered Gauss'S7 theorem on the number у of the ways
of expressing P = aab? • ■ ■ as a El, when «, 6, ■ • • are distinct primes of
the form 4n 4- 1- bet N = (a + l)(/3 4-1) • • • be the number of divisors
of P. Let N' be the number of ways of expressing P as a product of two
factors A, B. Then JV' = (JV 4- l)/2 or Nj2 according as or, ft - • are all
even or not all even. If P is a product of two distinct factors > 1 each
expressible as а И, the product theorem A) yields two expressions for
P as El, and conversely- Thus if P is not a square, v = Nr = Nj2.
If P is a square, у — 1 = Nf — 2, v = (N — l)/2, whereas Gauss gave
p = (N 4- l)/2. pt is merely a question as to the inclusion or exclusion
of P = P 4- 0, cf. Genocchi.74^ The special cases in which P is a power
of a prime or a product of distinct primes are treated (pp. 71-81). He67
insisted until7* 1854 that there is a misprint in Gauss' formula»
* Raooolla di Lettore . . . Fis. ed Mat. (Palomb»), Rom», 5, 1849, 263» 313» 392, 402.
* Giomale Arcadioo di Sc., Let, ed Arti, Roma» 119, 1849-50, 20-26; Annali di Sc. Mat. e
Fis.p lp 1850, 156.
" Atti Accad. Pont. Nuovi Шсец 4P 1850-1, 23-31. Same by Volpicelli«
♦7 Nouv. Ann. Math,, 9, 1S5O, 305-8; AnoaU di Sc. Mat. e Fie., 1, 1850, 527-531; 2, 1851,
61-1.

Chap. VI] ЗшС OF TWO SQUARES» 239
V. A. bebesgue*8 proved that y2+1^2?*i$y^Ot m> 1, by use of
complex numbers.
G. Bellavitis** stated that every solution of x2 + y2 = 5 • 13 • 17 is given
by
X 4- yi = B ± i)(Z d= 2i)D rfc г).
If each Ci is a prime 4k + 1, x* -\- y2 = c?c? ♦ - - has ft=K»ii+l)Gttj+l)* * *
or A: — | essentially different sets of solutions, according as у = 0 gives
во solution or a solution.
E. Prouhet™ proved Gauss' *7 formula.
D. Chelini*2 gave an " elegant proof " of Gauss' formula by noting that
every solution of x2 + У2 = (a2 4- Ь2I"^ 4- й*) - • is given by the de-
development of
z + yi = (a-\- ЫУ(а - Щ^\ах + МЧ»! - V")" ■
where ?i=0, 1, ■ ♦ ♦, m; щ = 0, 1, — >, wii; etc.
A. Genocchi*2 noted that CheUni71 did not prove that the solutions
obtained are all different, nor that no other solutions exist.
V. Bouniakowsky73 proved that every prime 8k 4- 5 is a El by use of
his formula A0), Ch. X, Vol. I, involving sums of divisors.
H. Suhle73a noted that Jacobi's*8 theorem implies the generalization that
the number of positive solutions xt у of $* 4- у2 = р is the excess of the
number of divisors 4/я 4- 1 of p over the number of divisors Am 4- 3. He
proved Eisenstein's56 result.
C. Hermite74 noted that, to express as a 121 a number A for which
л2 « — 1 (mod A) is solvable, it suffices to consider the form
Ax2 4- 2axy 4- A-'ia2 4- l)y\
which is reducible to X1 4- Y2.
A. Genocchi76 considered the number of representations of n by u2 4- v2.
By the remark of Euler** (end), it suffices to take n odd» Let t be the g.c.d.
of w, v. If n has a prime factor p — 4w 4- 3, set n — prn't t — jft*>
where n' and t' are prime to p. Since p cannot divide a 12, тг = 2pt so
that the product of all the prime divisors 4m 4- 3 of n is a square which
divides u2 and «*. It thus suffices to treat the case in which every prime
factor of n Is of the form 4ттг 4- 1- For such an nt set n =* prn\ p being a
prime not dividing n\ Then
(u 4- iv)(u — iv) — (q -t &)*($ — ^Yn\ q2 4- т2 = p.
Now у ±гг are complex primes, and decomposition into such primes is
unique.- Thus
и 4- iv — i*{q 4- гт)л(^ — ir)k{uf 4- iV),
*»Nouv. Aun. Math., 9,1850, 178-181.
" AmiAli di Sc. Mat. e Fis., 1, 1850, 4^-5.
11 Comptee Rendoa Ратщ, 33, 1851, 225-6.
^ Annali di Sc. Mat. e Kfl-P 3, 1852r 126-9,
n Nouv. Ann, Math., 12, 1853, 235-6,
лМёт, Ac. Sc, St. P&eisbouiE, (fyt 5, 1853, 303.
** De quonmdam theorize пшпегогши theoremattun appiicatione, Berlin, 1863, 18, 26.
H Joor. fur Mftth.r 47, J854r 345; Oeuvn», I, 237.
n Nour. Ann» Math., 13,1854,158-170,

240 History of the Theoby of Numbers. [Chap, yi
where the final factor divides n\ Multiplying by the conjugate, we get
л = p*+* (*'2 + »'•). Hence Л -f к = x, n' = u'2 +|Л The multiplica-
multiplication of и + iv by i-1 at most interchanges u* and o5. Hence the effective
solutions u, v are given by
и + ^ = (q + гг)л(д - *)'-*(«' + г*') (Л - 0, I, - - -, w),
where u\ v' range over the N' solutions of u* + v* ^ л'. If we change
the sign of t/ and replace Л by т — h, we get и — iv. If т is even, and n'
is a square u'*f the representation n = (p'VJ is excluded. Hence the
number of representations of n as a El is \(* + 1)JV', unless т is even and n
is a square, and then is \ \ (* 4- !)#' — 1 (. The number of representations
of n' is §ЛГ' or ^(ЛГ' — 1) according as N' is even or odd. Hence by induc-
induction we obtain Gauss' " result that if a, b, ■ - are distinct primes Am 4- 1,
the number of representations of n ~ a*b* - • - as a El is \N or J(^V — 1),
according as n is or is not a square, where N = (a + !)(& 4-1) ■ • -. The
second would be $(N 4- 1) if we count also the case of n 4- 0. Hence the
" correction " by Volpicelli67 is unnecessary.
P. Volpicelli76 retracted his67 claim of an error on the part of Gauss*7
and Legendre,38 but gave к — £ as the number of representations of M
as a El when fi and a, 0, - * are all even, i. e., when M itself is a square»
Concerning Euler's remark, quoted by Genocchi,** that an integer And its
double have the same number of representations as a El, Volpicelli (p. 185)
stated that p = 4225 has only four [omitting p = 652 4- Oj, while 2p has
five, representations.
A. Genocchi77 answered the latter objection by noting that zero is to be
counted as an integer. He remarked (p. 495) that the " new " case noted
by Volpicelli (that of M a square) had been treated by Fermat, who dis-
discussed the number of ways a number is the hypotenuse of a rational right
triangle.
A. Cayley7* noted that a formula of Dirichlet's** becomes, for D = — 1,
A + V + 2$16 + 2$« 4- .. .)fe 4- q* + q* + ■ ■ 0
1 - q* 1 - fl* r 1 - 9» 1 - qu ^
H. J. S. Smith,7» in accord with Gauss24 of Ch. II, denoted by [gt ■••
the numerator of the common fraction equal to the continued fraction
,11 1
and employed Euler's75 relations (Ch» XII)
B)
C) bi ■ • ■ gJ
Fie., 5t 1854, 176-186; Joar. for Math., 49, 1855, 119-122.
" Annali di Sc M&t. e Fia.? 5, 1854, 491-8-
« Cambridee imd Dublin Math. Jour., 9,1854, 163-5.
™ Jour, fur Math., 50,1855, 91-2; Coll. Р&ретв, I, 33-4, Reproduced by Borel and Drab,
Introifaction к Ь th6orie dee nombn», 1895,109-12; СЬгувЫ, Algebra, ed. 1, II, lSS9t
471; «L % 11,499.

СнаргУ1] Sum of Two Squares. 241
For pa given integer, let/ii, - • •, /*, denote the integers prime to p and < %p.
In the continued fraction for p/tnt Ctf* " * 3«] is UOW P* *** v*^ °f B)>
E^» *" ' tfi] arises from some p/p*< Let p be a prime 4X 4- 1, so that
с — 2X. Hence there is some ftk Ф 1 which coincides Tvith p* and thus
there is a set of quotients gu - - -, g« symmetrical from the ends. If n
were odd, n = 2i — 1 ^ 3, p — [$i - -• £;-i7igi-i - * • q J has the factor
[?! • • • g<_i] by C). Hence л = 2г and
P =* bi - • № • • • ffi] = Cffi ' • • ffj + bi"- ^i-i?.
C. G. Reuschie80 expressed as a sum of two squares each prime 4tt + 1
up to 12377, and to 24917 for those primes for which 10 is a quadratic
Tociduc*
A. Cayley81 wrote E'(n/k) = 1 or 0 according as /i/t is iiii integer or not
and proved that the number of ways the integer n is a El is
v = B'(n) - E\nlZ) + E'(n/5) - E'(n/7) + .. -,
if n = л2 4- f? is counted twice when а Ф fr Hence v is the number of
lattice points on the quadrant of the circle with radius Vw and center at
the origin. Eisenstein's* formula follows readily»
X liouville*2 stated the formula
summed for * = I, 3, 5, — - and for в = 0, 1, 2, ♦ - >, E^l and imph'ed
that it is connected with sums of two squares» It was proved geometrically
by L. Goldschmidt,w who showed that the right member is the number of
lattice points in a quadrant of the circle т* 4- в* = n.
F. Unferdinger8* proved, by use of norms of complex numbers, that a
product of n sums of two squares can be expressed as a (Z) in 2** ways,
distinct in general.
S. Kaminsky85 proved that a? 4- yz *=? ps? is impossible in integers if p
is a prime 4n 4- 3.
F. Woepcke* proved by induction from p} pn} pn+1 to pn** that any
power of a prime 4m 4- 1 can be expressed in one and but one way as a
sum of two relatively prime squares. The proof shows that the number
of all decompositions (primitive or not) of jf as a El is (X 4- l)/2 if p is
odd, X/2 if p ^ 2, Hence follows Gauss' *7 formula. Also the number
of primitive decompositions of pV • • ■ pi' is 2*, if each p^ is of the form
Am 4- 1.
J« Plana*7 used Jacobi's4* formula to prove Gauss'ir result on the number
of ways of expressing N = 2*8*рлр^ ■ •• as a1 4- b*, where p, p\ •• • are
" Matiu Abb., Neue Z&hlenth. IbboUen, Progr. Stuttgart, 1856. Errata by Cunningham,
Мею. Math,, 34,1904-5,133-5.
л Quar. Jour, Math., 1, 1857, 186-191»
«■ Jour» de Math., <2), 6, I860, 287-8.
u Beitrfige xur Ibeorie dcr quad. Formen, Dias. Gdttmgen, SonderBhau»eaT 1881.
"Archiv Math. Phya., 34, I860, 83-100.
■ Noav. Апл. Math., A), 20, 1861, 97-9.
» Atti Accad. Font. Nuovi lincca, 14» 1860-1, 311-6.
* Mem. Accad. Turin, B), 20,1863,123-6.

242 History of the Theory op Kumbbbs. [Chap, vi
primes 4k 4- 1» To find a, b without trial, express p, pT as El by continued
fractions and apply A) and
G. L. Dirichlet** used the theory of binary quadratic forms to prove
that, if m is a product of powers of ^ primes 4A -f- 1, the number of sets of
relatively prime solutions x, j/of^2-|-y2 = ?nis 2*+a. The number (S 91)
of all sets of solutions is the quadruple of the excess of the number of its
divisors 4Л + 1 over the number of its divisors 4h + 3.
A. Vermehren,8* to express ^ as a sum of two squares, put z ~ и + v;
then г2 = u*(u 4- Sv) 4- v*(Zu + v). He took и 4- 3v = 4n\ du 4- v = 4m\
F. Unferdingerw noted that the product of the expansions of (a ± Ы)ш
gives (a2'+ b2I™ = A2 4- ^, where Л, В are known polynomials. HeM
had shown that a product P of n sums of two squares can be expressed as a
El in 2*'1 ways distinct in general. The same result therefore holds for P"\
G. C. Gerono90* proved that every divisor of a sum of two relatively
prime squares is a sum of two relatively prime squares»
V, Eugenio" proved the Lemma" as follows. Let M divide P* 4" Q2,
where P is prime to Qt and call PfjQf the next to the last convergent of the
continued fraction for P/Q. Then PQ' - P'Q = ± 1. By A), M divides
(PP' 4- Off)* 4- 1. Thus M divides N* 4- 1, where ^V is an integer < M.
Express M/N as a continued fraction with an even number of quotients:
where n = 2s. Let Mi/Nlf -., M*/N* = MjN be the successive con-
vergents. Then
D) Jfm = MAt 4- ilf^i, ^V^i - Х<а< 4- JV*.i, MJV^ - JVJIf^ - (- l)%
,„ M 1 11
E) *+
Kow N* + 1 - MN'. Thus by D,), M{Nf - N^) --
Thus M divides N - Мл^ < M. Hence M„_x = N. Thus E) equals
f, and a = а„„1, etc» Hence
. . 1 11 1 1 itf.
But Jf^t ^ JVt. Thus MjN ^ (Af! + NDt'MjV., M = M\ + N\.
"Zahlentbeorie, 568P 1863; ed. 2,1871; ed. 3,1879; ed. 4T1804.
« Die Pythagoriiechen 2»Ыбп, Projp-. Domechule, GusUow, 1863.
"Archiv Math. РЬув.р 49,1869, 116^-7.
*• Nouv, Ann- Math4 B)r 871809, 454-6, 559-
» GiornAic di Mat., 8, Ш0г Г" "

Chap. VI] SUM OF Two SqUAKES. 243
P. Seeling* proved that if A is & prime Am + 1 the period of the con*
tinned fraction for V? has an odd number of terms. Hence A is a El,
J. Peteraen" reproduced EuIerV* proof that every divisor of a sum of
two relatively prime squares is a El» Then by Wilson's theorem, every
prime 4n 4- 1 is a EL He proved Gauss'Jr result on the number of solu-
solutions of a* -f- j/1 = Л.
L. Lorenz" proved that
whence m* + n2 = N has 4(а^ — bN) solutions if aN is the number of
divisors of the form 4m 4- 1 of N, and fr^ the number of divisors of the
form 4m + 3.
P. Bachmann" employed the theory of roots of unity to prove that
every prime p = 4n 4- 1 is a sum of two squares, to compute the squares,
and to prove Gauss' * result.
J. W. L. Glaisher* would strike out of the list of numbers
12 3 4 5 6»-»
_3 _6 -9 -12 -15 -18 ...
5 10 15 20 25 30 ■ -
- 7 - 14 - 21 - 28 - 35 - 42 .».
9 18 27 36 45 54 .»»
every one whose negative occurs in the list» Each remaining positive
number 1, 2, 4, 5, 8, 9, 10, - ■ - is a El and every El occurs in the final set.
The proof is by JacobiV formula. He gave a like scheme to obtain the
numbers expressible as a sum of two odd squares»
R. Hoppe" proved that every prime p = 4tt4-lisaEh The values
of r = a? for x *= 1, • —, 2n are incongruent modulo p. But r1* = 1 has
only 2n roots and — r is a root» Hence to each x corresponds an integer у
such that y2 = — r. Thus a* + y2 = pq> If pj is a factor of qt we get
x\ + y\ = piqu Since the g's decrease, we finally get a qk ~ 1, whence
zj + yl - ph. The remaining factors of g*_, are El, whence q^x is a El.
Then ?«>_! ^ Elfa*-! = El, etc. Finally, p is a El.
F. L. F. Chavarmes" considered an integer N whose prime factors are
distinct and each of the form 4e -\- land hence a El. ThusA^ = П(а* 4- /S2),
Set JVt - (a> + 0*)G2 + *2), ^2 = Ni(* 4- Г), • ■ •, whence Nt = x\ 4- ^
for xt = ay ± ^5, yi - 07 =F a5. Similarly, each pair xu yt yields two
Archiv Math. Phye4 62, 1871,
Tldaekrift for Math., C), 1,1871, 80-4.
» Die Lehre топ der Kreirtlieilung, 1872,122-137, 235.
* Math. Queet. Edtic. Tfcnea, 20, 1873, 87; Britiah Адес. Report, 4в, 1873r 10-12 (Trmne.
Sect.).
w Aichiv Math. Phye., 66,1874, 223.
M Bull. Soc. Vaudoiee des Sc. NatuieUee, Laiu&zme, 13» 1874-5, 477-509.

244
History of the Theory of №шсвекв.
[TJha*. vi
sete of solutions x%t y% of N2 - x\ + y\ = (x) + yj)(«* + f1). Then
tfj = Z3 + J^ has 8 sets, etc. It la proved (pp. 503-6) that if p and p'
are primes 4e — 1, no one of p, pf or pp' is ft GO.
V. SchfegelM stated that the numbers {8X + 7L* are the only ones
not a sum of fewer than four squares; the numbeis DX + 3J* and the
products of two relatively prime numbers of that form are the only numbers
not a sum of fewer than three squares. The numbers representable ад a GO
are s-2*, where s — 4(X2 + И + p) + 1. The numbers representable in л
ways as a S3 are 2Г times the product of n factors a,
T. MuirIW noted that by Lagrange's theorem any integer A is of the form
x2 H- y* if in the continued fraction for VA the period of the partial de-
denominators has an odd number of terms, Muir1<u gave formulas for x
and y* For, the general expression for such an integer is A = R* + S,
where ata%
example,
Then A = x2
2x
У =
а„а„
+ iC(a2 • • ■ atI,
is the period, while К is a continuant. For
- 1
0
0
1
<h
- 1
0
0
1
a*
а^г)\М +iC(tfi
K(a1 "*a^
When Af =^ iC(at • • - <hO Л — x2 + y1 is also the sum of 3 squares.
E, Lucas104 gave the complete solution of ti* + 1? =« y* and stated that
the same process applies to u* + & ^ y*.
S. КоЬеИв"" derived all the decompositions into the sum. of two squares
of an odd positive integer D, containing no square factor, and such that
P — Dm1 — — 1 is solvable in integers, by developing into a continued
fraction *4NjMt where M and N are complementary factors of D and
M <*fn. For D odd, we take M < <4Щ.
G: bL Halphen1" considered the sum s(x) of the positive divisors d ot ь
positive integer x such that x/d is odd. Then
frfr) - 9(x - 1) - 8{x - 4) + ш(х - 9) ± «(* - n^) + -..,
H Zwtechrift Math, Phye., 21,1876, 79-80,
1M Proo. London Math. Soc., 8,1S76-7, 215-9. The Expraekm of 9 Quadratic Sard ая а
Continued Fraction, Glasgow, 1874, $ 51. Euler" of Ou XII wrote (a, b) for X(a, b).
M Proc. Hoy. Soc. Edinb., 1S73-4, 234.
»" Bull. Bihl. Stotia Sc. Mat. Fie.t 10t 1877, 243» Cf. J. Berirand, Ы« вёт.
I^ria, 1850,244; 1851,224. Cf. Lucas* of Ch. XXII.
im Proc. London Math. Soc.r 9,1877^8, 187-106.
^ BuIL Soc. Math» Гпшсе, б, 1ЭТ7-8, llfr-120,17&-X80.

Chat. VT\ SUM ОГ Two SQUARES, 245
the series being continued as long as x — ra1 is positive; if x is a square,
s@) is replaced by x/2. The proof is by use of the series for
Hence if a; is not a square and no x — raJ is a square, a(x) is a multiple of 4,
Thus *(x) is a multiple of 4 when x is not a square or a GO. If also x is a
prime, л is of the form 4m — 1, since a(x) = x + 1. Hence every prime
not a B1 is of the form 4m — 1, so that every prime 4m + 1 is a GO.
Я Realis1* proved that every prime An + 1 is the quotient of x2 + y*
by the common factor of x2 and y1, where
X - a» + f? - У, У - (т - «)* + G - Й1 " 7J-
For the latter values and
и - а» + (а - 7I " (« - fl\ ■ « * + № - У)г - @ - «)",
we have x2 + y* = u* + t£, identically, and they furnish all the solutions.
E. Lucas"* proved that every prime 4k -\- 1 is a GO by use of " satins "
Пв formed of the points (xt y) with x = 0,1, •• *f n such that у is the residue
of ax modulo n where a is prime to n and a <n. Since each parallel to
the p-axis contains one and but one point of the satin, ox = 1 (mod n)
has a unique solution. If/1 + 1 = 0 is solvable, у ^ fx gives/y=/*u^ — з,
and the satin n/ is unaltered by a rotation through a right angle and is a
square satin. If n is a prime p = 4fc + 1, we can separate 2, 3, • - •, p — 2
into (p — 5)/4 sets of four numbers like а, л, p — a, p — cry where aa = 1
(mod p), and one set p,p — f7 such that/(p — /) = X, whence/2 + 1 ^ 0
is solvable. Thus p divides a sum of two squares. Since the satin is
formed of squares having p as a side, p is a sum of two squares.
T. Hammth107 proved that every prime p = 4n + 1 divides a sum of
two relatively prime squares. Let g be an odd primitive root of p and
set ^ ^ 2 (mod p). Then g24 + 2* = 0 (mod p), e * X + (p - l)/4.
S. Gunther10* proved A) by use of lattice (gitter) points. ]No three
lattice points are vertices of a regular triangle. The geometrical proof by
Lucas shows that
x2 -|- y* = w2 + ь1 = 2(t*x + i#)
have no rational solutions. Ifa2isa[2J,aisaEL
For the knight's path problem in chess, we have (pp, 14-16) the system
of equations
(Xi - xi+1)* + (y< - у^У - 5 <i - I, 2, • • -f n* - 1),
and, if the path is closed, also
(z* - x,y + (у- - yt)» - 5.
M Nouv. Ann. Math., B)r 18,1879, 5004.
*" Ь'Ьцрздшге avile, Turin» 1880' French tnniL, Aeec>c» fiuujj,, 40, 1911, 72^87- Cf»
A. Aubry, renBeign«»ment math., 13, 1911, 200; Sphinx-Oedipe, num^ro врёсЫ, Jan.,
1912, 10-13.
lfl* Awhiv Math, Fhy*, 66.1881, 327-8,
"*Z*it0cbrift Math. Natorw. Uniemcht, 13, 1882, 94-08,102.

246 Histoby of the Theory of Numbers. QChap. vi
If the path is symmetrical, there are further conditions. He gave ft history
of the subject»
№. V. BougaieP9 applied elliptic functions to the decomposition of
numbers into squares (with relation to JacobiV7 Fnndamenta Nova).
E. Fauquembergue1* noted that a cube Ф 1 is never a sum of squares
of two consecutive integers.
E. Cesaro111 considered the function ф(п) ^ 2^(a), where a ranges over
all the positive integers for which n — a* is a square. Then
For/(s) = l^fnjistbenumberofpositiveintegralsolutionsofx2 + tf* = n;
then 2^(j) equals лтг/4 asymptotically, whence the number of ways of
decomposing a number into a sum of two squares is in mean */4.
T. J. Stieltjes11* states that if f(n) is the number of solutions of
x* -)- y2 _ щ зл^ jf lfi jg ^e largest odd integer ^ л(п, then
/(I) +/(9) +/A7) +•••+/(»)
-•S«-»'
/E)+ /A3)+ /B1)
where, in the last, к =* Ц(\^Г+4 - 1)]. If ф{х) is the sum of the odd
divisors of x,
4A) + *E) + - - • + фD* + 1), фA) + *C) + .. • + *B» - 1),
0A) + 0B) +...+*(»)
are expressed as sums of greatest integers.
T\ Pepin111 proved that, if m is an odd number not a square,
{2 + (- 1)—}Eи2 - flt)Z(m - *■),
where X(^) is the sum of the odd divisors of к and <r{k) is the sum of all
the divisors of k. Let mbea prime Al + 1. Hence
1 = 2B0/** - mMm - V) (mod 2).
Thus among the differences m — 4p* occur an odd number of squares, so
that m is a GO.
1M Math. Soc. Moscow, II, I883r 200^12, 415-456, 616-602; 12,1886,1-21.
"• Nouv, Ann. Math.r C), 2,1883, 430.
^ M£ra. Soc. Roy. Sc. <Je Iiige, B), J0? 1883, No, 6, pp. 192-4, 224,
ш Comptea Rendue Fane, 97, 1883, 839-891.
« Atti Accad. Pont. Nuovi Lmcei, 37, 1883^, «.

Chap. VI] Sum OF Two SquaBES. 247
E, Catalan1" expressed з =* rf1*** + y4**2 as the sum of the squares of
two polynomials, and & as such a sum in two ways (p. 51). By use (p. 63)
of (x d= iy)(x* zk iy*) • • • (x2*1 ± iy*n~l) = -P H- tQ, we get 2*~x decomposi-
decompositions of (x2 -h y2)^ + V4) ■ • • (x2* + y2*) as * 121.
Catalan1" noted that, if a + 5 ^ GO, and n = 2»,
C, Hermite111 stated that if /(л) is the number of solutions of хг + y1 = n}
where J?i(x) =[« + Ц- И =* C^J - 2[>] ^ Ле function used by
Gauss.
Hermite11* proved by use of expansions of elliptic functions
t -/A) +/B) + • - +/<Q - «(- 1)^йСС/а1
^ = № + /A0) + - - • + /(8C + 2) =■ 4Z( - 1)-TBC + c)/Bc - 1I
summed fora = 1, 3, 5, ...; с ^ 1, 2, 3, - . He stated that
+ 1)/41 Also, for n - [(V4C + 1 + 1)/21
He proved Gauss*tt result for s; also, J. Iiouville's1** result
L, Gegenbauer114 concluded from a general theorem on quadratic forms
that the number of ways any number r which is odd or the double of an odd
number can be represented as a sum of two squares is the quadruple
of the number of decompositions into two relatively prime factors of
those divisors of r which have only prime factors of the form 4s + 1 and
a square as complementary factor. The number of representations by
x2 + y* of those divisors of r whose complementary divisor is a product of
«• Attt Accad. Pont. Nuovi Lincei, 37,1883-i, 80.
"'Matheew,*, 1884,70.
i" Ашег. Jour, Math.r 6, 1884, 173-4.
»' Bui]. Ac. Sc, St. Peterebours, 29, 1884, 343-7 (Oeuvres, IV, 159-163); reprinted, Acta
Math., 5,1884-5, 320.
i» Jour, de Math., <2>, 5, I860, 287-8.
™ Sibr. Akad. Wifie, Wien (MatbO, 90, Cp 1884, 438.

248 Histoby of the Theory of Numbers. [Chap. VI
an even number of primes exceeds the number of representattons of the
remaining divisors by the excess of the number of those divisors, with
complementary square divisor, of the form 4s + 1 over the number of such
divisors of the form 4s — 1.
T* Рерщ120 quoted Dirichlet's*2 theorem that the number of representa-
representations of an odd number n by хг + у2 is 4p, where
.'- Г
p —
is a sum of Legendre-Jacobi symbols. It follows readily that the number
of representations of 2n is 4p and the number of decompositions is p. Since
p is the excess of the number of divisors 41 + 1 over the number of divisors
4J _|_ 3f we have JacobiJsw theorem that the number of decompositions of 2n
is that excess. likewise, 2"n ~ x2 + y1 has 4/> solutions,
3. H6alism noted that if p is a prime or a product of primes of the
form 4q + 1, all integral solutions of x2 + у* = p are found from the
identity
by giving to a and b such integral values that the second member takes the
value p. Thus the problem reduces to that of expressing q as a sum of two
triangular numbers. If p is odd or the double of an odd number and if
p = x2 H- у2, where x and у are relatively prime, then
J. W. Bock1** employed the n{2n — 1) pairs formed by two of 1*, 2*, • ч
Bn)\ From any pair x\} y*t whose sum is not divisible by the prime
p = 4n + 1, we obtain 2n ineongruent sums v*x\ + м*у\, v = 1, • • •, 2n.
If &j ~h ^j ^s not congruent to one of these sums, nor to zero, it leads to 2n
newsumsp*x! H-y^j; etc. But 2n does not dividenBn — 1). Hencethere
exists a sum & « A2 + B2 divisible by p, 0 < Л < Jp, 0 < В < Jp. In
the attsmpt to prove that, if s is divisible by a prime q = a1 + bJ, the
quotient is a sum of two squares, the quotient is taken to be c2 + <P, с and
d not being assumed integral» By A), q(c* + d?) is of the form x2 + уг*
From г = xl -|- y2, it is concluded erroneously that A = x отуу В = у or i.
R. lipfichitz123 noted that all real substitutions of determinant unity
for which z\ + x\ = y\ + ^J (i. e., automorphs) are given by multiplying
(X* + iXi2)(zt + гх2) - {Ло — гХц)(^1 + гу2)
by Xq — гХи and equating the real terms and the imaginary terms, and
d. Nuovi Liiicei, 38,1884r-6,166.
Nouv. Amu. Math», C)t 4r 1885, Z6Z-Q; Oeuvree de Fermat, IV, 218-220-
Mitt» Math, GvelL Hamburg, 1,1885,101^4.
Untenuchungen Uber die Бшшпеп von Quadraten, Bonn, 1886^ 147 pp. French
by J. Molk, Jour, de M&th., D), 2, 1886, 373-139. Summary in Bull, dee So. Math.
Aatr., B), 10,1,1886,103-183.

Chap, VII SUM OF Two ^QUAKES, 249
conversely. In particular, all rational automorphs of x] -\- x\ are derived
by taking Xo and Л13 to be relatively prime integers» To show (p. 384)
that every prime p=4r + lasaE], use a solution of w2 -f 1 = 0 (mod p)
and set d = u&, where fcis any integer not divisible by p. We can choose
relatively prime integers /*,, p2i such that тр9 and rp?i are numerically
< p/2 and congruent modulo p to ^ and £i respectively. Take p& — — (Ни
Then r*(pl + p%) is < hv* and is divisible by p. Hence pj + pjt ** pi, where
f < p/2. Determine fa and 0lt numerically < 1/2 and congruent modulo
tio pn and pJ3 respectively. Then ф1 + <£?, =* U\ where J' ^ */2. Then
(Фо — гфа)(ра -h ipit) = т'1(р'о -f *>!s),
where pi, p:7 are relatively prime. Hence p02 + p\\ =* pk, к = р/тг* ^ t/2.
Repeating this process, we finally get Xo = p(;\ Л12 — p% such that
Xj + *?t = V> and
F) Wi - AiA = 0, Xj^x + X<^2 = 0 (mod p).
Similarly we can find a complex integer with relatively prune coordinates
Xo, \u, whose norm is any power py of p and which satisfies F) modulo py.
If то ^ pyq* . •., where p, $, - - are pnmes = 1 (mod 4), or if m is the
double of such a product, apply the preceding discussion for each py and
take the product of the resulting complex integers. By using all sets of
solutions of £? + £з ^ 0 (mod pr)t we get every proper representation of
m as a GO and each once and but once.
C. Hermite124 proved by use of elliptic functions that, if M = ±n + 1,
summed for m = I, 3, 5, * - , where /(») is the number of representations
of n as a GO. The asymptotic value of S is |3/т»
A. Berger12* gave an elementary proof of the theorem that, if n is a
positive odd integer, the number of all sets of solutions of z* + y1 — n
is 42(— l)^1^, where 5 ranges over all positive divisors of n. While
Dirichlet's proof was by transcendental analysis, Berger uses only the
known nuinber (Dirichlet88) of relatively prime seta of solutions.
Berger1* proved that if л is a positive integer the number of sets of
integers x, у for which x2 + y1 = n is 42 sin Ятг/2 {Berger126).
C. Hermite127 proved Gauss'« formula for the nuinber of sets of integers
x, у for which x* + у* ^ A.
E. Catalan"8 noted that, if x2 + y2 + «* is a square,
If Л» - AC = - rf, (Ca - AcJ - 4{Bc - Cb)(Ab - Ba) - GO.
"*Joui\fQrM*th.,9&, 1886,324-8; Ocuvree, IVy 209-214. Cf,
л Adta Math., 9, 18S6-7, Э01-7.
» Ofvemgt af Kongl. Vetenakape-Akad. F0rhaDdlM 44,18S7, 153-8.
w Amer. Jour. Math., », 1887» 381-в; Oeuvrea, IV, 241-250.
»«M»thc«i, 7t 1887, 120,144,

260 History of the Theory of Numbers. [Chap, vi
J. W. L. Glaisher1" wrote 4G(n) for the excess of the number of repre-
representations of n in the form (GrI + Fe + II over the number of those in
the form @r 4- 2I + F* 4- 3)*, provided n = 1 (mod 12), whence the
representations of n as a GS are of one of those two types. If p, q are rela-
relatively prime numbers 12k + 1, (?(pr) = G(p)G(r). He evaluated (?(a*), a
being & prime. The number of representations of n as a GO is 4E(n),
where E(n) is the excess of the number of divisors 4k + 1 of n over the
number of divisors 4k + 3. There are noted simple relations between
E(n) and G{n), It is shown (p. 195) by elliptic functions that the number
of representations of 4n + 1 as a sum of an even and an odd square is
4EDn 4- 1); the number of representations of Sn + 2 as a sum of two odd
squares is 4J£Dn + 1). Hence if n = 1 (mod 4), n and 2n have the same
number of representations as SL Next, EC&n + 9) = EDn + 1). The
number of compositions of a number as a sum of two squares, both of the
form A2n + IJ or both of the form {\2n + 5J, or one of each form, is
expressed in terms of functions E and G, Similarly for representations by
the forms at the beginning of this summary. Let (pp. 211-3) m be odd,
a even, Ъ odd and not divisible by 3, с = 1, d = 5 (mod 12); then the
number of representations by За2 + Ъ*, За2 + c2, 3a2 + <P, 3m% + c2 or
Зто* + d* is expressed in terms of G and the excess Я(п) of the number of
divisors = 1 (mod 3) of n over the number of divisors = 2 (mod 3).
F. Goldscheider"° discussed the sign of /, not determined by
Gauss/4
L. Gegenbauerm noted that Hermite's124 formula ia one of a set which
follows from a general formula for the sum of the values taken by an
arbitrary function f{y) wher у ranges over all those divisors ^ Vjfc of
к = 4л + 1 or 4n + 3.
E* Lucas132 gave two proofs by use of continued fractions that every
divisor of a sum of two relatively prime squares is a GO.
JL Th. Vahlen183 deduced from the theory of partitions the fact that
every odd integer is a GO in В ways, if дг + иг and (— g)* + u2 are regarded
as different ways, while E is the excess of the number of factors 4m + 1
over the number of factors 4m + 3, He noted that this fact is equivalent
to the theorem of Jaeobi" in view of a remark by Euler44 (end). Since
every integer N is the product of an even power of 2 by an odd integer or
by the double of an odd integer, the number of sets of solutions ^ 0 of
x2 + y* =5 AT is E. He gave a summation formula for the number of primi-
primitive representations as a GO.
From a representation a2 + ¥ + c* + d* of an odd prime p we obtain
a multiple of 32 representations by permuting a, •• -, d or changing their
signs, except when two are zero, the factor being then 12 4. But there are
) representations of p. Thus if p = a* + h? has N sets of solutions
i» Proc. London Math. Soc., 21, 1889-90,162-215.
ш Dafi Reaprozit&tageeeU der achtea Potenzreete, Progr. Berlin, 1889, 26-29.
™ Sit«unB*er. Akftd. Wise, Wien (Math.), 99, Ш, 180О, 387-M3.
v Tbtorie deu nombrea, 1591, 454-6.
w* Jour. fQr Math., 112,1893, 2S-32.

т, VI] STJM OP TWO SqUABES. 251
b > a > 0, then S*(j>) = 4SN (mod 32). For p - 4n + 1,
*(p) ** 2Bn + 1)
and N is odd»
A. Matrot114 noted that, ifp=2A + lisa prime, and a is not divisible
by p, a* ^ ± 1 (mod p) by Fennat's theorem. If the upper sign held for
every a,
8h =* 1* + • • • + (p - 1)* ^ p - 1 (mod p),
whereas, for q < p — 1, «« = 0 (mod p), as shown by induction. Hence
tbere exists an a for which a* = — 1. Let Л = 2fc. Thus j> divides a GO.
That p is а И follows as in his 1891: paper on 3L
E, Catalan1*5 repeated the proof by Eugenio."
H. Weber1* proved that every prime n — 4f -h 1 is a GO by use of the
four periods each of/ terms of nth roots of unity.
C. Stunner117 proved that l-h^ + 2^nif|x|> landnhasanodd
divisor > I.
Several1" treated x2 + {x + IJ = y1, whence ** - 2w« = - 1 if
t = 2x + 1, и = y\
Stdrmer1" applied a theorem on Pell's equation (Stdnner250 of Ch. XII)
to find the complete solution of 1 + x2 — kA*' * * * Л;# in positive integers,
where к, Аъ •. ^ Л» are given positive integers. In particular, there is a
new proof that 1 + x2 *= ym or 2j/* is impossible if a; > 1, у > 1, n being an
odd prime.
M* A* Gruber140 gave a table and identities for 4л + 1 =* GO.
Several writers1*1 discussed ж2 + p1 = tf for p a prime.
3. de Longchampe142 noted that AT4 is a GO or Ш if ЛГ/Х — 1 is a square
or C, since
ЛГ* т 16Х{ЛГ - X)(JV - 2X)S + (ЛГ2 ~ 8ХЛГ
R, Hoppe148 used Girard's theorem to prove that a number ia a GO or
not according as it has no prime factor of the form An — 1 to an odd power
or at least one such prime power factor.
J. H. McDonald1" gave a direct proof of Jacobi'e" result on the number
of representations of an odd positive number as a GO.
С A. Laisant1* noted that (a4**2 + l)/{a* + 1) is always a GO.
»< Jour, de math, №ma, D), 2,18ЙЗ, 73.
»■ Mem. AauL Roy, Bdlgique, 52,18ЮЧ, 17.
w Lehrbuch der Algebra, 1,1895, 583-5; cd. 2,1,1898, 632-4.
i"L'interai6diau€desmatb.p3,1896,171; 5,1898, 94 for n ~ 2Г.
Mlbid.7 4, 1897, 212-5,
ш VidenBkAbe-SelekabeLe Skrifter, Chnstumui, 1897, No. 2.
»• Amer. Math. Monthly, 5,1898, 240-3.
«* L'intenn&li&ire de» roath,, 5,1898,157-9; 16,1«», 177,
i*1 Ш&, 7,1900, 66. MUprint of 2ЛГ - X for N - 2K
ш Archiv M«th, Phye., B), 17,1«Ю, 128, 333.
"* Ргос and Тппв. Roy- Soc, Canada, <2), 6,1900, Sec. Ш, 77-8,
»»Nouv, Ann. Math., D), 1,1901, 239-240.

252 Bistort of the TWoby of Nttm«f,rs, [Chap. VI
H. Schubert noted that, if in x2 + y* = u2 -f z* the unknowns have
no common factor, either all four are odd or in each member one number
is odd and one even. In the first case,
whence we must factor an arbitrary number g in two ways with always
one factor even and the other odd» In the second case, g must be a product
of two even factors and also a product of an even and an odd factor.
R. E. MoritzHfla proved that every rational number not a square can
be expressed in an infinitude of ways as a quotient of two sums or two
differences of two squares, and gave one such expression for each such
number < 100.
A. Palmstrom147 noted that 2? = y1 + t2 implies л: =* a2 + b2, whence
у = a8 -\- ab* or a1 — 3ab2 [^provided у and z are relatively prime]. P. К
Teilhet"8 obtained all the solutions.
A. Thue14* proved that a prime divisor of a 23 is a \2\.
Several14** found three consecutive integers each а И, including
B«J + B»)ff 8tf + 1, Bn - IJ + Bn -r IJ, provided the second be a GO,
i. e,7 n be triangular, n = (лъ1 + w)/2.
L. E» Diekson160 proved that all factors of a sum of two relatively prime
squares are sums of two squares by use of the theorem that if a and Ъ are
relatively prime every prime divisor of a2 + Ъ2 is 01 the form 4n + 1 and
the theorem that every prime 4n + 1 is a sum of squares of two relatively
prime integers.
G. Fontene proved Gauss'37 theorem by showing that, if A, B}
are primes 4A + 1, there is a A, 1) correspondence between the decomposi-
decompositions of АШБ? * — as a product of two factors and its decomposition into a
sum of two squares, provided we fix the order of the two squares whose sum
is A, or B, etc.
A. Cunningham162 expressed each prime 4n + 1 < 100000 as a EK
P. Pasternak1" proved that all solutions of z* + y1 = v* + w* are
x = ma + np, v ~ mo? — np, у = поз — mp, ш = п« + тр,
whence
x2 + y2 - (m2 + n')(o>2 + p*).
Thus every integer which can be expressed as a \2\ in more than one way is
itself a product of two sums of two squares. From known theorems it is
said to now follow that no prime 4n + 1 is а ЕЯ in more than one way.
^NiedereAnaL, 1,1902,167-171; ed, 2, 1908,
'«■Ueber Continuanten . . ,t Dias. Strafflbur^, Gottingen, 1902» Cf, MoriU'0 of Ch. IX.
117 L'intermedi&ire dea math., 8, 1901, 302.
M lbi*Lt 10,1903,210-1.
Ml Oveiidgl D» VSden, Selet. Fdrb.t KriHUania, 1902, No. 7.
**• Math. Quest. Educ. Timeer B)t 3, 1903, 41-3.
«*Amer, Math, Monthly, 10r 1903, 23.
ш Nouv. Anu. Math., D)r 3, 1903, 108-tl5.
«Quadratic Partition*, London, 1904. Errata, Meea- Math., 34,1904-5,132.
wZithr. Math. Naturw, Unterricht, ^7, 1906, 33-35.

Chap. VI] StTM ОТ TwO SQUARES. 253
A. G^rardin154 discussed the solution of
(lOx + m)* + (Wy + py = 100a, a - h1 + <P, m < 10, p < 10>
Since ?n* 4- p* = 20A, we have in = 2, p *= 4 or 6; m = 4 or 6, p = 8.
These cases are treated in turn. To solve (pp. 8&-90) x2 + y1 ^ a2 + fr2,
set z = a + mk, Ь = у -\- h, m(x + a) => b + y. Then
h = 2(y - am)l{m* - 1),
and the general solution is said to be
(am1 - 2my + a)s + y*(m* - IJ = tf(™2 - I)« + (ym2 - 2am + y)*.
W. Sierpmski1^ gave a long proof that, if А (я) is the number of pairs of
integers u, v for which u* -\- v1 ^ x, A(x) =* тх + 0(zl!*)f for О denned as
in Landau,179 while тг is the usual constant,
E. Jacobsthal14* proved that, if p is a prime = 1 (mod 4), p = a2 + b2,
where, in terms of Legendre's symbols,
where г is any quadratic residue (as — 1) of p, and n any non-residue» Also,
a s (p — 3)/2 (mod 8)» Proof is given of formulas, equivalent to Gauss1,41
for the residues of а, Ъ modulo p.
Identities157 solving a* + 6* = 2c11 have been given»
W. Sierpinski168 evaluated sums like
where т{п) and rs(«) denote the number of decompositions of n into 2
and 8 squares.
* E. ]N\ Barisien159 expressed 2* as a ratio of two GO»
J. Sommer1* applied ideals to show that every prime in + 1 is a Eh
L. Aubry cited known results.
G. Bisconcini1K proved that n is а Ю if and only if n contains no odd
power of a prime 4ft — 1, and deduced all decompositions of pr as a GO,
given that of the prime p = 4ft + 1» HeIto proved that, if pi is a prime
4ft + 1, ptl - • • p^r has 2та~1 рюрег decompositions into S3; also Gauss7"
theorem. He treated (pp. 68-80) the decomposition of fractions into
one of the forms x2 db y2-
e, 1906-7,112-9.
»* Praoe mat.^fiz., Waraaw, 17, 1906, 77-118 (Polish). See papers 179, 180, 189, 198-203,
ш Anwendufigen einer Formel aue der Theorie der qaadratiecben Rcete, Bias, Berlin, 1906,13;
Jour, fur Matli., 132, 1907, 238-245.
UT Uintermediaire dee math., 13, 1906, 62, 184; 14, 1907, 72»
m Prace mat.-fiz., Warsaw, \St 1907,1-60 (Polieh). Reviewed in Jahrb» Forxechritte Math.,
38t 319-21; ВиД» dee Sc. Math», {2)r 37, 11,1913, 30-31.
u» Bull, Sc. Math» Eton., 12,1907, 262-6.
M0 Vorlesungen uber Zahlentheorie, 1907, 112, 123-4. French tr&neL (of ravieed text) by A.
L6vy, 1911, 105, 117-^9.
to I^eneeiEnement math., 9,1907, 421,
m Periodico di Mat., 22, 19O7r 270-285.
™lbid>t 23,1908, 9-23.

254 History of the Theory op Numbers. LChap, VI
F, Ferrari1*1 found the known solution of x* -f y* = z* by use of
z ~ r -f si.
H, Brocard1* noted that n7 -f (n -f IJ = mk has solutions for к = 2,
but not for к = 3.
E. Landau111 considered the number B{x) of positive integers ш x
which are 121 and gave * long proof that
where г ranges over all primes of the form 4m -f 3.
E* Landau167 applied binary quadratic forms to show that a number is
a 121 if and only if it has no prime factor 4m -f 3 to an odd power.
E. X. Barisien118 used the epicycloid to derive the identity
(8^ - 6£ - 6* + 3I + 4A - **)A -f 3* - 4t*)* = 13 - 12t,
whence 12- 13* is a EJ if I = A - 0*)/(l + ^).
M. Kaba and L. E. Pickson16* deduced, by use of special theta functions,
Hence there is no representation as a 121 of a number having a prime factor
4m + 3 with an odd exponent, and no proper representation when such a
factor has an even exponent. If P = pf* • • • pl% where pu ■ -, p* are all
the distinct primes of the form 4m + 3 which divide e, and if in, — *, irt
are all even, there are as many improper representations of e as there are
representations of ejP; every representation of e is of the type (P1^J
-h (PlilyJ. Hence the problem reduces to the case in which every prime
factor of e is of the form 4m + 1. Then the number of representatioDs of e
asaGSis^ + 1) -.-(ir. + l).
P. Bachmann170 gave an exposition of the work of Lagrange32 and
Vahlen.1»
Welschm stated that the general solution of«2-h^ = y2 + 22is
2x = ab + cd, 2y =ac+bd, 2z = oh - cd, 2u « ac - M,
where a, d are even, or Ъ> с are even, or all four are odd.
L. Aubry1" proved that x2 4- (x + IJ + mk if к is not a power of 2.
A. Deltour171 applied continuants (Muir"x) to prove that a prime
4A + 1 is a 121 in one and but one way.
'« Pteriodico di Mat,, 25,1009-10, 59-66; Supplem. al Period, di Mat., 12, 1908^9,
i* L'inteimediAire dee matb., L5] I908r 1&-19.
"• Archiv Matb. PhyeLr C), 13,1908, 305-12.
m Handbueh . . . Vertenung der PrimzaUeD, 1,1909, 549-550.
"* Abboc. frang. ev. вс, 38, 1909,101-7.
"•Amer. Math. Monthly, 16,1909, 85-7.
^ №edere Zahleotheorie, 2, 1910,304-319 D77),
in LintenD&liain: dea matb-, l7r 1910, 96,118, 205.
">1Ш., 18P 1911, 8-9; errata, 113; Sphinx-Oedipe, пшпбго арёсЫ, March, 1914, 15-16;
errata, 39.
™ Nouv. Авп. Math., D), 11, 1911, 116,

Chap. У1Д SUM ОТ Two SQUARES. 255
Marchandm presented the known application of complex integers
a + bi to find all decompositions of a product of primes 4n -f- 1 as a I2L
Paulmier1*1 gave solutions of x* + tf = A% for five special values of A*
Several writers17* found x such that x -h 1 and я* -h 2 are sums of two
squares.
J. K. Heydonm noted that, if a, b, • • • are distinct primes,
ap>-i&*-i . _ ^ и ш 2^H—* or 0 ways.
P. Lambert178 applied complex integers a -\- Ы. He gave two proofs
that a divisor of я 121 is a 121.
E. Landau178 proved that, if A(x) is the number of pairs of integer*
u, v for which иг + & ^ x} then A(x) = тх -f Ofir1***), for every « > 0.
Kere/(x) = 0(g(x)) means a function such that there exist two numbers £
and A for which | f(x) \ < Agfa) when x ^ $. Although the result is
not quite as sharp as that by Sierpinski,IW the proof is much shorter.
Landau1*0 gave a new proof of the theorem due to SierpinskU*
R. Bricardm gave an elementary proof that every prime p = 4n + I
is в. 121. By Wilson's theorem, m2 + 1 = 0 (mod p) for m — [(p — 1)/2]1.
Write ж^ for the minimiini residue of mi modulo p. Consider the p — 1
points Af< = (x^, г). The square of the distance AMf, between any two
of these points is divisible by p. It is shown that the least of these squares
is < 2p if p > 32 and hence equals p. A like proof shows that every prime
8g ± 1 is of the form x7 - 2y*.
F. Ferrari182 noted that the least number decomposable щ 2* distinct
ways as a sum of two relatively prime squares ф 0 is the product, found by
A), of the first n -{- 1 primes of the form 4& + 1. For this least x = p\ -h gj
(г ** 1, " *, 2*), set yi ^ p) — q*if Zi — 2p^i\ then x* = y] -h zj is the least
square decomposable in 2* ways as a I2L To find the least 0? -f- l)th
^wer decomposable щ 2' ways as a 121, use P ^*Ъ\ + c\ (i ~ 1, •• •, 2P),
whence ЩЪ* -h cj) = F*^x has 2P decompositions.
A. Aubrym noted that A) can be derived from Brahmegupta's (Ch. V)
inscribed quadrilateral ABCD whose diagonals meet at right angles at 0,
by evaluating the perpendiculars BE and OJ to DC.
E. Haentzschel1*4 noted that his *" method in Ch. XXI to deduce a new
solution of ax? -h — * + d — y* from one solution may be applied to
x2 + y2 = z2 in two ways according as x or у is taken as the variable. He
m I/intermedi»ire dee math,, 18,19ll, 228-232.
m/Wd.r 19,1912,151.
"*/btf., 55-7, 257»
»" Math» Quest. Educ. Timce, B), 21,1912, 98-«.
"• Nouv. Ало. Math., D), 12,1912, 408-421.
1W Gfittingen Nachnchten, 1912, 691-2, Gionaale di Mat., 51, 1913, 73-81.
iw AniiaU di Mat., C), 20, 1913, 1-2S; SiUuneBber. Akad- Wim. Wien (Math.), Ш, 1912,
Ila, 229S-2328.
ш Nouy. Ann. Math., D)r 13,1913, 558-562.
*" Perlodioo di Mat., 28,1913, 71-8.
ш Sphinx-Oedipe, numero нрёсЫ, Jutxe, 1913, 23-24.
ш Sitsungaber. Berlin Math. Geeell., 13,1914, 92-6.

256 Histoht of the Theory or Numbers» [Chap, vi
quoted from A, Fleck18* the solution
(a'c + 2abd- Ъ*сУ + (Vd + 2abc- aW = (a* + V)*, a* + V = c2 + #,
which includes the primitive solution (a* - Зоб*J + (За2*? - Ь5}2 « (a2 + Ь*У
by Euler* of Ch. XX.
* Hesse186 gave the general solution of x* + y* - z\
Several writers187 found solutions of хг + у2 = z4.
* J. G. van der Corput1*8 treated sums of two squares.
G, H* Hardy38* wrote r(n) and R(n) for the number of integral solutions
of p* _|_ y* = n and of m2 + v1 ^ л, respectively, and set й(х) = xx -Ь ^*(:г)-
He proved the existence of a positive constant К such that each of
P{z) > Kx4\ P(x) < - Kx1*
is satisfied by values of x surpassing all limit. Hence in Sierpinski'sws result
P(x) = O(zlt*), with 0 defined as by Landaur17y the exponent ^ cannofc
be replaced by a number < i- He gave an explicit analytic expression
for P{x) in terms of Beesel's functions.
Hardy190 proved that, for every positive <, P(x) is on the average
), L e.,
G* Bonfantinix9X proved that, if a number n not a prime is a И, it equals
either a product of several factors each a SI or such a product multiplied
by a Bquare which is a common factor of the given Bquares whose sum is n.
Conversely, if m is a product of several sums of two squares and if m is
not an even power of % m is a И.
G. Koenigs and Lu Bastien1*2 discussed the number of decompositions
of (a1 + b*N ** & Ш.
A. Gerardin198 noted that P - 2ku* = 1 implies
By means of the fact that every prime of the form 4n + 1 is a factor of
a number J* -h 1, R. D» Carmichael1We proved by Fermat's method of infinite
descent that such a prime is a SI.
* A. L. Bartelds1*4 gave an elementary proof of Girard's theorem.
T. Hayashilw proved that y»'+ 1 ф ^ if у ф 0>
i» Vouwche Zeitung zu B^rliD, June 1,1013.
w Untcirichteblatter fQr Math. u. NftturwiB8.r 20r 1914, 15. Щеп1*веЪе!, р. 55,
Heeae'a paper,
^ Amer» Math. Montblyr 21,1914,199-201.
1Ю Nieuw Archief voojt Wiakunde, 11,1914^5, 6L
'M Quar. Jour. Math., 46, Ш5, 263-283j Proc. London Math. Soc., B), 15, 1916, 15-16.
** Proc. London Math. Soc,, B), 15, 1916, 192-213.
111 SuppL al Periodico d\ Mat,, IS, 1915, 81-6, By use of Boufantini1* of СЬ. ХШ.
"» Lintermediaire dea math.. 22T 1915, 253-4; 23, 1916, 345
1Я/Ш.Г 22,1915,57,
"•* Diophantine Analysis, 1915, 39-40.
1M ^TiHkandig Tijdechriftr 12, 1915-6, 159-166»
ш Nouv. Ann» Mftth.r D), 16, 1916,150.

. ylj Sum of Two Squakes. 257
M. WeilF9* noted that the product of p sums of two squares is a sum of
two equares m 2P~^1 distinct ways.
IVL Chalaux1*7 proved Girard'e theorem by induction using the fact that
if a prime is a SI and divides a sum of two relatively prime squares, the
quotient is a sum of two relatively prime squares.
E. Landau1*8 proved bis179 former theorem by means of a new simplifi-
simplification of Pfeiffer's method (cf.pp. 305, 322 of VoL 1 of this History).
Неш next considered the lower limit a of the constants for which A{x) »
tx -h O(x*)f and proved that a = £. Later he*** proved a theorem on the
number of lattice points in certain regions which is a generalisation of the
main theorem applied in bis17* above papers.
* K. Szilysen*01 stated empirically an asymptotic formula for the num-
number of pairs of integers for which x* + у1 ^ Nt a formula already proved
by Lipschita.
M. Bignaux308 announced a table in manuscript of the decompositions
as a B of the 3908 decomposable numbers < 10000»
G. H. Hardy** deduced Landau's179 theorem very simply by two dif-
different methods from the theorems in Hardy's190former paper. IP04*»!,- • - -
a* are primes of the form 4fc + 1, there are 4(л + 1)ж sets of solutions of
x* + y* = (&i <k ш " <**)n, ш 21*** of which x and у are relatively prime.
On the cumber of solutions of x* 4- Dy)J *= nt see Nasimoff58 of Ch.
XIII. 0nf + ^(m4 n*) z*y see papers 142-5 of CIl ХШ and the
cross-references given there. On I + x2 = 2y*, see Euler7 of Ch» XIV and
Cunmngham™ of Ch. XX. In Ch. XVII are given reports on papers on
a number and its square both sums of two consecutive squares; cf. Meyl"
of Ch. IV. On x2 + n* + y1, see Pepin10 and Hayashi" of Ch. XX. On
x+y= П, x1 + tf1 =* «*f see papers 40, 48, 50,52, 54^56, 63 of Ch. XXTL
On systems of equations including x7 -h y1 = z2, see papers 353, 363, 368 of
Ch. XXL Equal sums of two squares occur on p. 37, p. 206 \ in paper
107a of Ch. VII; 18 of Ch. XIIIj papers 21, 35, 45, 62, 80 of Ch. XV; 7,
9, 18, 20 of Ch. XVHI; 4,13,15, 33,37, 42,46-50, 75, 102,133,149 of Ch.
XIX; 177 of Ch. XXII; 45 of Ch. XXIV. In VoL I were cited the papers
by Euler4'7 and Gauss,w pp. 381-2, containing tables of primes and factors
of numbers x7 -{-y'; by Lucas" and Catalan,81 pp. 402-3r on special num-
numbers which are GS; by Liouville28, p. 286; and various papers, pp. 360-1,
on factoring numbers which are Й in two ways.
'*Nouv. Ann. Math., D), 16,191в, ЗП-1,
"' Ibid.t D), 17,1917, 305-8.
**G6ttingen Nachrichlen, 1915,148-160.
*»/**., 161-171.
"/WA, 20&-244; 1917,96-101. Cf. Revue aem«etrielk, 27,1, 1918,16,18.
™ Math. Ы tennee. erteeiW (Hungarian Acad. Be.), 35,1917, 54-6.
*»L'intermedmre des math. t 25, 1918,143; 36,1919,54-55.
*»Proc Lcmdon Math. Soc., B), 18, 1919> 201-4.
«-Amer. Math. Monthly, 26, 1919, 367^8,

CHAPTER VIL
SUM OF THREE SQUARES,
Diophantus V, 14 relates to the division of unity into three parts such
that if the same given number a be added to each part the sums will be
squares* This problem is equivalent to the determination of three squares,
each > af whose sum is 3a -h L Diophantus stated that a must not be
of the form HI + 2.
C. G. Bachet1 stated that this condition is not sufficient and gave as a
sufficient condition that a must not be of the form 8& + 2 or 32fc + 9,
stating that he had tested the numbers a < 325. He also divided 5 into
three parts such that each increased by 3 is a square; since
3-3 + 5 = 1 + 2' + 32,
he took the sides of the squares to be 1 -f 7N, 2 + Nt 3 — 5N, whence
N = 4/25.
Fermat2 remarked that Bachet's condition fails to exclude a = 37, 149,
etc., and himself gave the correct sufficient condition that a must not be of
one of the forms
8*+ 2, 4-8*+ 24 + 1, 42>8* + 2'42+4 + 1,
4*-8ft + 2-43+4*+4 + l,
[Thus a must not equal
. 4*-8fc + 2-4" + D" - l)/3 = [B4* + 7L* -
so that 3a + 1 must not be of the form B4& + 7L* and hence not
(8w + 7L*, since m is a multiple of 3 if 3a + 1 is of the latter form»]
Regiomontanus* (Johannes Mtiller, 1436-1476) proposed in a letter the
problem of solving the pair of equations
я + У + z = H6, x2 + у* + * = 4624 = 682.
Fermat* stated that no integer 8k -f- 7 is the sum of three rational
squares. Descartes5 proved this for integral squares by noting that a
square is of one of the forms 4fr or 8& 4- 1.
Fermatf treated the problem to find two numbers each of which, as
well as their sum, is composed of three squares only [not composed of one
or two squares]. He took any such number, as 11, and multiplied it by
two squares whose sum is a square, for example, 9 and 16» The problem
was proposed by Sainte-Croix to Descartes in April, 1638, with the illustra-
* Diophanti Abx, Arith., 1621, 310-3. """ "
* Oeuvres, I, 314-5; French transl.r III, 257-8.
1C. T. de Murr, Memorabilia BibL, I, 1786, 1±5.
«Oeuvraa, II, 66; III, 2S7; letter to M«rseflner Sept. or Oct., 1636. The latter communi-
communicated it to Descartes
* Oeuvres, II, 92; letter from Deecartea to Mereenne, March 31, 163S. See also p. 195.
* Oeuvreer II, 29, 57" letters to Мепюаве, July 16 and Sept. 2,1636.
259

260 History of the Theoey of Numbers. [Cka*. vii
tion 3, U. In his reply, Descartes7 gave оЧ2г b2 + 2 (о and Ъ odd);
he8 took the interpretation that each required number and their sum shall
be the sum of three squares in one and but one way, and gave nine examples
including
22 =0+9 + 4, 35 = 25+9 + 1, 57 ^49 + 4 + 4.
But Sainte-Croix desired that each be the sum of 3, but not of 4, squares.
Fennatg asserted that the double of any prime 8n — 1 is the sum of
three squares; he desired that Brouncker and Wallis seek a proof. Refer-
Reference will be made under the subject of binary quadratic forms to the
assertion of Fennat and proof by Lagrange that any prime Sh + 1 or 8& + 3
is expressible in one and but one way as the sum of a square and double of a
square.
The Japanese Matsunago™ in the first half of the eighteenth century
solved x7 + y* + z2 = u2 by taking x and у at pleasure, expressing x* + y1
as a product of two factors and equating the latter to и — z and и + z.
He noted that x* + у* + z* = и4 has the solutions
и = m2 + n\
L. Euler21 noted that if Fermat's theorem that every number a? is a sum
of three triangular numbers (a2 + a)/2 is truer then every number Sx + 3
is a sum of three squares Ba + IJ.
Euler12 noted that, to prove that a prime 8wt + 3 is of the form 2a1 + b\
one needs the theorems (of which he had no proofs): If the integer n is not
a sum of two integral squares, then no integer np2 is a sum of two integral
squares; if n is not a sum of three integral squares, it is not a sum of three
fractional squares.
May 6, 1747 (p. 414), Euler wrote that he had verified for small integers
m that there always exists a triangular number Л *» (x* + x)j2 such that
4G» — A) + 1 is a prime. If this be true, set n ~ m — A; then 4n + 1 is
a ffl and 2D?i + 1) is а И. Set a ^ 2x + 1. Then n » m — Д gives
8m + 1 = 8n + a2. Hence 81» + 3 « 2Dn + 1) + л1 ia a HL On pp.
442-5, Euler and Chr. Goldbacb discussed without result the problem to
express 8?n + 3 as a Sk June 25, 1748 (pp. 45&-460), Euler expressed his
belief that any number 4n + 1 or 4д + 2 is a 13. The latter would gjve
4n + 2 - BaJ + B6 +1)* + Bc + IJ, 2n +1 - 2a2 + Be)'+BeS+l)',
for Ъ ^ d + e, с — d — e, whence any odd number is of the form
2X2 + & + z\
March 24,1750 (p. 512), Goldbach gave the identity
/р + ^ + ра-р-тУ-ри- ft' + Bb - yy + (g - g - 7)8.
rOeavree, II, 167; letter to Mereenne, Jane 3,1638' Cbuvree de Fennat. 4,1912, 57.
'Oeuvree de Descartes, П, 180-2.
9 Cteavrwr II, 405; Ш, 318; letter to К Digby, June, 1658.
» Y. Mik&mi, Abh. Gcachjchte M&tb. Wise., 30,1912, 233.
u Cortesp. Math. Phys. (cd., Puss), 1, ШЗ, 45; letter to Gddbach, Oct. 17,1730.
»1Ш.У 263; Oct.15, 1745/

Chap. VII] Sum OF THREE SQUABEe. 261
June 9, 1750 (p. 515), Euler expressed this аз the first of the following:
a* + & + & = Bm - aJ + Bm - &)* + Bm - c)f, if a + b + с = 3m;
a* + fr1 + c1 = (*» - a)J + (m - Ъ)* + Bm - cJ, if a + Ь + 2e =* 3m;
a2 + P + c» = Bm - aI + Dm - bJ + Dm ~ cJ, if a + 2b + 2c « 9m;
and gave five more euch formulae and «i-milar ones for SL
Euler11 verified that if m ^ 187 and n is of the form 8iV + 3, then m
is the sum of an odd square and the double of a prime 4n + 1- Since
4n + 1 = a1 + b\ 2Dn + 1) = (a + W1 + (« - bJ, a»d the m'e щ ques-
question are BL
J. L. Lagrange1* remarked that a prime 8n — 1 is of the form 24n — 1
or 24n + 7^ Since he had proved that any prime 24n -h 7 is of the form
VT + &?, ita double equals (y + 2z)* + (y - 2eJ + Bz)K He added that
he did not see a proof of FermatV assertion for the remaining case of
primes 24n — 1.
J. A. Euler15 used (a2 - IJ -f 4a1 =* (as + 1)* for a = p, q, to prove the
identity
(P* + mq* + IJ - (q2 - 1L? + 1У + *№ " IJ + D**J-
A. M. Legendre1' remarked that Fermat'sp assertion is true not only of
primes but of all odd numbers, and stated that either every number or ita
double is a CD. His proof" (pp. 545-8) was based on empirical theorems on
the quadratic divisors of P + cu\ He was led (pp. 530-542) to the empirical
theorem that, if с is a prime 8m — 3 or 8m + 1, the number of decomposi-
decompositions of с into a sum of three squares (ignoring the order and signs of the
roots) is the number of reduced quadratic divisors of the form 4я + 1
(or of the form 4n — 1); while for a prime с = 8m -h 3, it is the number of
reduced quadratic divisors.
P. Cossali1* noted that the sum of the squares of n, n + 1> n(n + 1)
equals the square of n2 + n + 1. This result has been attributed100 to
Diophantus, who in III, 5 noted that 22 + 3* + 6J *= □♦
Legendre1' proved Ctbe statement of Beguelin7* of Ch. Ц that every
positive integer, not of the form 8n + 7 or 4n, is a sum of three squares
having no common factor; the proof is by means of theorems on reciprocal
(p. 367) quadratic divisors of f + cu\ In 2Ba + 1) = ** + if + ^> two
of the squares must be odd and the third even. Hence we may set
x *= p + q, у » p — q, г = 2г and gpb2a-t-l=p*-t-qt-t- 2r*. Again,
any integer is of the form 22nBa + 1) or 2Sfl-2Ba + 1), and the latter is a
Bj; hence either any integer or ite double is a 12. The product (p. 198)
of two CIJ is not in general a C, since A + 1 + 1)A6 -f 4 + 1) is not a EL
etrop., 4,11,1780 A775\ 38; Comm. Aiith., U, 138.
14 Nouv. Mum, Acad. Roy. Berlin, «пвбе 1775, 356-7; Oeuvreer Ш, 795. In the quotation
from Format, sum of & square and a double square should read sum of three squares.
u Act* Acad. Petrop,, 3, 1779,4OS. L. Euler'e Comm. ArHh., II, 463.
** ffiat. et Mum. Acftd. Roy. Sc Fbria, 1785, 514^5.
ir Incomplete. Cf. A. G«noocbif Atti Accad. Sc_ Torino, 15,1879-80,803; Gauss.»
"Origme,Ttaflporto in ItalU, . , Algebra, 1, 1797,97.
"Ibeorie dee nombres, 1798, 398-9 (stated p. 208); ed. 2,1808, 336-9 (p. Ш); ed. 3r I,
1830,393-5 (Gennatt traiuL by Маавг, 1,1893,386-8).

262 History op the Theory op Numbers. [Chap, vii
С. F. Gauss2* determined the number Ф(т) of proper representations
X, y> z, without common factor (and counted as different from y, x, z and
from — zty,z; etc.) of an integer m as a EL Let Л be the number of classes,
in the principal genus, of the properly primitive binary quadratic forms of
determinant — тш Let ^ be the number of distinct prime factors of m.
Then
4>{m) = 3>2*+1A if m = 1, % 5, 6 (mod 8),
Ф{т) = 2Ц+ТЛ if m = 3 (mod 8).
In particular, we have Legendre's19 theorem. But the squares of x} у, г;
- j, у, г; у, x> z\ etc. give the same decomposition of m into a HI. The
resulting number of decompositions (art. 292) of m agrees with that derived
by (incomplete) induction by Legendre16 for the case m a prime.
A. Cauchy21 noted, as a corollary to Legendre's theorem,19 that if a is
any integer and if 4* is the highest power of 4 dividing a, then a is а EJ if
and only if a/4* is not of the form 8*i + 7.
J. R, Young21" solved x* + y* -\- z2 = w1 by taking w =* x +p and
finding x rationally, or by setting y* ^ 2xz. Then if w is given, take у = pz,
whence z is found in terms of p. To find (p. 346) three numbers in. har-
monical progression whose sum of squares is a square, take l/(x ± y), 1/x
as the three пшпЬегэ; the condition Зя* -h y* =* D is satisfied if % = 2,
C. Gill216 noted that the sum of the squares of 2mn(№ + J2), 2Ы(т2 - л2)
and (&1 — f)(m2 — n2) equals the square of {№ + P)(m2 -{- n*).
C. G» J. Jacobi22 proved by use of elliptic functions that
I z <- i)-
f: (- i)*Bn +
0
a result occurring also in Gauss' posthumous papers.
Jacobi2* gave an elementary proof of A). Replace x by я24 and multiply
the resulting equation by z4; we get
B) I £ (- 1)-а<*н*>' I' = 2 (- 1)(*)/V*1 (b odd, Ь > 0)-
For m positive, set a = 6m + 1; for m negative, set a = — 6m — 1; thus
where a and & range over all positive odd integers such that a is not divisible
by& The sign in the left member is -f if a = 12k ± lr — if a = 12A; d= 5.
The expansion gives the following theorem: If a number 244 + 3, not of
the form 36*, be expressed as a sum of three squares Fm i II in all possible
» Dieq. АтШц 1801, Art. 291; Werke, 1,1Ш, 343; German tranaL by H. Maeer, pp. 329-33.
Cf. H. J, S. Smithr Britiah Адаос. Report, 1865; Coll. Math. Papers, I, 324.
"Мбш. Se. Maih.Fbys.de llnstitut de France, A), 14,1813-5,177; Oeuvree, B), VI, 323.
^ Algebra, 1816; S. Wardre Amer. ed., 1832, 326-7.
** The Gentleman's Matb- Companion, London, 5, No. 29, 1826, 364,
«Fund, novafunc. ellip., 1829,166G); Werke, 1г p. 237 G)»
M Jour» fur Matb-, 21, 1Ш, 13-32; Werke, VI, 281-3(S» French tranaL^ Jour, de Math»,
7, 1842,85-100.

Chap, vii] Stjm of Three Sqtjakes. 263
ways, counting two for each case of three distinct squares, then the number
of decompositions in which one or three of the m's are even equals that in
which one or three of the m's are odd. But for 36* the first number exceeds
the second if and only if b ^ 1 {mod 4), the excess being always £b/3J
If N is any odd integer, B) shows that
3N* = Fm + 1)* + (fonj + iy + Fm2 + IJ
in more than one way if N > 1, so that the squares need not all be squab
Thus
N* = n7 -f 2n\ -f 6»J, n = 2(m + тг -f m*) -f 1,
n% = 2m — wii — tfi2, я2 = /»! — wi*,
where nx and л* are not both zero. By changing the sign of n if necessary,
we may assume that N — n is divisible by 4. Let N be a prime. Then
(N — n)/4 and (N + n)/2 are relatively prime and each divides n\ + 3nJ,
whence each are of the latter form:
n) = ** + З72, i(AT - n) - /S2 + 361
Hence every prime can be expressed in the form л3 -h 20* + 3y* + 6S2-
Since the product of two such expressions is of the same form, every number
can be expressed in that form.
G. L. Dirichlet2* remarked thatr by use of his formulas for the number
k of classes of binary quadratic forms, one can give a new expression for
the number Ф(ш) of proper representations of m as a \3\ (Gauss20). Accord-
According to G. Eisenstein,25 the result is
ф(т) =24Z(-),ifw = l (mod 4);
Ф(т) ^ 8 £ I — ], if wi ^ 3 (mod 4),
where (s/m) is Jacobi's symbol and is 0 if я, т have a common factor.
T. Weddle2* noted that, if (a, pt z), (b, q, 2f) and (c, rt z") are the extremi-
extremities of a system of conjugate semi-axes of an ellipsoid,
(a2 + b* + c2)^ + q1 + **) = {Щ - ЬрУ + (^ - cpJ + <br - c^J.
J. R> Young27 noted that the last formula follows by taking d *= s = 0,
ая + Ьд + cr^O in Euler'a formula A) of Ch. VIIL But if Ve take
d = 9 = 0, a/p = &/g, we get
(a2 + b2 + c2)^2 + g2 + r2) = (ap + &g + crJ + (<*r - cpK + (br - c5J.
G. L. Diriehlet** gave an elegant proof of Legendre's19 theorem. Let
a be a positive integer not of one of the forms 4n, 8» + 7. It suffices to
14 Jour, fur Math,, 21, 1S40, 155; Werke, 1, 1889r 496.
e JourЛШг Math-, 35,1847,388, Cf. T. Pepin, AtU Accad. Pont. Nuovi Lincei, 37, 1883-
4r44.
» Cambridge «nd Dublin Math. Jour., 2, 1847,13-19.
17 Тгкпа. 1гшп Acad,, 21, П, 1843, 330.
"Jour, fur Math., 40,1850,228-232; Werke, 2r 1897, 91. French transl. by J. Robdt Jour»
de Matb.r B), 4,1859, 233.

264 History of the Theory op Numbers. qchap. Vii
show that there exists a positive ternary quadratic form P of determinant
H- 1 whose first coefficient is a. Indeed, such a form is equivalent to
& 4. if _|_ &t go that the latter can he transformed into F by a substitution
of determinant unity; thus a is the sum of the squares of three of the
coefficients (having no common factor) of the substitution. Now the
ternary form
ax2 + by1 + cz* + 2afyz + 2xz (A = be — a'*)
has the determinant +lifb = oA — 1, The form is positive if Д > 0.
It suffices to show that a positive value of Д can be found for which — Л is
a quadratic residue of Ъ, so that с and a' may he determined to satisfy
o* — be *= — Д. For a *= 4k + 2, we take Д odd. Then b = 1 (mod 4).
We seek a suitable prune h. Since, for Jacobi symbols,
=^) - (I) - (!) - (V) - + ■■
A must be of the form 4f + 1, whence 6 ~ 4at + a — 1» The latter is the
general term of an arithmetical progression, containing primes. For
a = 8fc + 1, we take Д = & + 3, and seek a prime p for which 2p = b.
Since 2р«=йД— 1, p^l (mod 4)>
\ *л / \Д/ \p/ \ p / \ v /
There exists a prime in the progression p = 4ot + |Ca — 1). A like
iseult follows for a = 8fc + 3,A = 8* + 1, and for a = 8k+ 5, A = 8C + 3.
H, Burhenne** noted that x2 + y* + «2 = (a* + Ъ* + c*)^ if
л « mf + n1 4- p*
and
H* Faure8* noted that no number m*(8£ + 7) is a SI.
V* A, Lebeegue*1 proved that every odd number p is of the form
x2 + у* + Ъ?у where x, y, z are integers with до common factor. The
method is that of Dirichlet.18 It follows that
J, Iiouville11 denoted the number of sets of integral solutions of
tf + ^ + ^^^by \f,(ji)> Set n = 2"m, m odd, a > 0, Let u be the
greatest integer ^ л/п. Then
where c(m) is the sum of the divisors of
" Archiv Math. Phye^ 20,1853, 466-8-
11 Nouv, Ann, Matb^ 12,1853, 336.
" Jour, de Miith.y B), 2,1857,140-152.
* J d Mb., B), 5, I860,141-2.

Chap. TO] SUM OF THREE SQUARES» 265
L. Kjronecker3' proved by use of series for elliptic functions that the
number of representations of л as a G2 is 24F(n) — 12£(л), where O(n)
is the number of classes of binary quadratic forms of determinant — n, and
F(n) is the number of classes of such forms of determinant — n in which
at least one of the two outer coefficients is odd. This result gives the
theorem of Gauss* since G{n) = F(n) if n ss 1 or 2 (mod 4); G(n) « 2F(n)
if n = 7 (mod 8), SG(n) = ±F{n) + t if n m 3 (mod 8), where t = 2 if n
is the triple of an odd square, t — 0 in the remaining case.
J. Liouville18" noted that, if m н= 3 (mod 8), the number of solutions of
m ~ г2 + if + i%, where i, tif is are odd and positive, is
m Р>{7П) + 2p\m - 4 d2) + 2р'(ж - 4-20 + - • -,
where p'(n) is the excess of the number of divisors < ^n of n which are of
the form 4p + 1 over the number of such divisors of the form 4^ + 3,
while p{n) is the corresponding excess for all the divisors of n,
V. A, Lebesgue" stated that every solution of £ ** x2 + y* + zx is
given by
where (? ^ ^ + fe2, A = a2 + b2, С « с2 + (^. In the identity
|t _ xt « pi + ^
set ^ «= 1, Л = 0, and replace ae, bey qf, df by «, /3, 7, 3; we get
C) {«i + flt + y + f)*- («t + pi_-yt^ a*)i
a special case85 of Eider's formula A) of Ch. УТИ, Since every integer n
is а Й), n2 is a sum of three squares feach + 0, in general].
A, Genocchi** proved Fennat's statement that the double of any prime
&h — 1 is a EL
J. Liouville17 stated that, if m s 1 (mod 4) and F is any function,
Bummed for all the decompositions г* + «* + 16«* =* m = tj + <*[ + &e*
in which t and ti are odd and positive, while w and <*>i are even. G. Zolo-
taref18 gave a proof by use of elliptic functions*
* Jour, fur Math., 57, I860, 253. French traxtfL, Jour, de Matb., B), 5, I860, 297. Cf.
MorielL'19 For * ■ 3 (mod 8), C. Hermit*, Jour, de Mftth4 B), 7, l№, 38 j Comptee
HeoduB P&rie, 53,1861, 214; Oeuvree, IL 109.
»• Jour, de M&th,, B), 7, 1862, 43-44. Cf, liouvifleT of Cb. XI.
M Gomptofl Hendus Ptrie, 66, 186S, 396-8.
" Abo given in ВеИассЫЪ Algebra, 1, I860,165.
» Annali dl Mat., B)r 2,186S-Q, 256.
^ Jour, de Math., B), 15, Д870,133-6.
••Bull. Acad. 8c. St. raeitibouxg, 16,1870-1, 85-7.

266 History or тве Tueory op Ntjmbbks. [Chap, vn
E. Catalan" noted that the excess of the number of even values of
я *f у + z in
(fix ± \y + «Jy =b 1)' + @i =b 1)J = 3B* + 1)»
over the number of odd values of x 4- у + г is Bт* + 1)(— 1)\ There
are at least \jBn + l)/6] decompositions of 3Bn + II into а И, The
sextuple40 of an odd square is a sum of three squares, two of which are of
the form Fм dt II and the third is 4Ffc d= 1)*, The exceas of the number
of even values of x in
4*1 + 4y* + Bb + IJ = B» + II
over the number of odd values is {Bn + 1)(— 1)" — 11/4, К a prime p
is not a 121, then ps is a HI,
Catalan stated and V, A. Lebesgue" proved that the square of a SI is
a SI, since C) for 6^0 becomes
D) (a* + p + т2)8 = («* + 0* - 73)* + B<ПJ + B07)'.
This formula was employed by Euler** of Ch. XXIL
J» Neuberg4* also gave D).
Catalan40 gave the identity
(a* _|_ b* + c* + ab + be + <*с)г
^ (a 4- e)f(« + 6J + (b + cJ(a + b)* + (c1 + ac + be - obI
and by a change of notation deduced
Catalan44 stated empirically that the triple of any odd square not divis-
divisible by 5 is a sum of squares of three primes other than 2 and 3.
G* H, Halphen4* proved that every prime 8m H- 3 is a SI by means of
his1M recursion formula (Ch. VI) for the sum s{x) of the divisors of x whose
complementary divisors are odd. Let x be not a square, 121 or \S\; then
no one of the arguments x — n* is a El, so that s{x) is divisible by 8. Let
also x be a prime, so that s(x) =x + l, Hence a prime not a 121 or S)
is of the form 8m — 1.
U* Dainelli4* derived by integration the case с =* 0 of Catalan's*1 formula
(pP + ab+V)* *= (aby+ (a(a*fb)}2+ {b(a+ b)J*
8. E^aHs" noted that Ы = £ + 2\ + %ti
k = A* + & + (?,z = <£ + & + TVi = AiP + Уг - «*) - 2a(^/3 + Cy)>
11 ИесЬегеЪев vat quelques produite mdefinifl, Mem. Acad. Roy, Belgique, 40t 1873, 61-191;
extract in Nouy. Ада. Math,r B), 13,1$74, 518-^23.
« Repeated by CaUian,Nouv. Ann. Math,, B), 14r 1875r 42S.
« Nouv. Ann. Mfttb., B), 13,1874, 64,11L
* Nouv. Correep» Math., 1,1874-6,195-6.
«Jbid.t 153; 2,187ft, 117.
«• Nouv. Correep. Math., 3,1877, 29.
41 Bull. Soc. Math, France, bt 1877^ 180.
*■ Giornale di Mat., 15, 1877r 378.
« Nquv, Corrop. Math., 4,1878, 325. Cf. Malfatti11 of Cb. V1IL

Chap» VHJ StJM OF ТнКЕВ SquaBES, 267
The caee A = t, B^C = O expresses the square of a SI as a SI- The
case A = у, В = p, С = a expresses the cube of a SI as a SI.
H. S. Monck" noted that if a, b, с are integral edges of a rectangular
parallelopiped and the diagonal d is an integer, then a2 + &* + c1 = <?,
and another has the edges а + Ъ + d, a + c + d>b + c + d and diagonal
a + Ь + с + 2(L From a = l, Ъ =* — 2, с = 2, <i = 3, we get the new
edges 2, 3, 6 and diagonal 7, Cf. papers 25-20 of Cb. XIX.
S. Bealis" gave a complicated identity
rf + lft + Jt-It+ttf + *, * = «*+0* + <г'-*-«>,
said to give all solutions of the equation. He gave a similar identity which
is said to give all solutions of Ш = 31, Supplementing the theorem that N
is a SI if N has no square factor and is of one of the forms 4p + 1, 4p + 2,
8p + 3, he stated that N is the quotient of z* + y* -h 2s by the factor com-
common to x2, &*, gff where x, yy г are given above.
F. Pisani« discussed wl H- (ii + V? = (a: - I)8 + x2 + (x + II, whence
Bu + I}1 = fix* + 3. Thus 2u + l = Zy, 2x* - Зу2 = -_h An infini-
infinitude of solutions is found from the continued fraction for Vs/2.
S. Realis" expressed ад а 121 the sum of the three squares of
2(*> - {? - -у3 + *2) + 2«B0 + 3y + 45)
and two similar expressions. He gave (p. 501) expressions for 9Pn and
18P" as SI if P =: a2 + 6^.
E. Catalan stated and Realis" proved that every power of 3 is a sum of
three squares prime to 3. Kealis (p, 75) expressed n2(^ + y* + jjt2) as a
SI when n - a2 + ab + b\
Catalan58 proved that, if a = b (mod 3), a2 H- У is a sum of three
squares Ф 0; also if a = b (mod x + y) aud 2xy ** П, Also that every
power of 3 is a sum of three squares prime to 3, He" proved that every
even power oftf + ob+^isaSl and gave special identities Sb SI = SK
0- Schier*5 solved x2 + y2 *f & = u2 by setting y = x+0,z**z + y,
и = x + 6, and taking /5 + 7 = 5. Then
Ж = * - P - У\ J = $y~(y-x)(z- x),
whence x = yz/(y -f z). Multiplying the values by у *f z, we get the
identity of Dainelli,"
J, Neuberg noted that x2 + y2 + # = Л? + Y* *f ^ for
г/а **yfb = zfc = kf + S, X = a{k* - 1) + 26(fc + 1) - 2e(* - 1),
Y and 2 being derived from X by permuting a, ft, с cyclically,
* Math. Quest, Educ. Tim*, 29,1878,74. ~ ~^ ^ ~~
« Nouv. Ann, Math., B), lSr 1879, 505-6.
•• Nout. Ann. Math.r B)t 19,1880, 524-6. Same in Zeitechr. Math. Naturw. Unterrichtr 12t
1881,268. CTLionnet^ofCh. XII.
11 /ftul., B), 20r 1881,33^6.
«Matbeek, 1,1881, 73r 87.
*Atti Aocad. Pont. Nuovi lined, 34, 1880-1, вЭЧ, 135-6.
« /Wrf.t 35, 1881-2,1(№1U. Extract m Sphinx-Oedipe, 5,19ЮГ 54-55.
« Sitiunffber. Abd. Wim. Wien (Math.), 8^, U, I881r 890-1.
• Matbeeis, 2, 1882Г Ш; D), 4r 1914,116-7.

268 History of the Theory of Numbers* £Снлг. Vii
S. Realis" gave expressions involving five parameters satisfying
for A; = 7, 19, 67, and formulas to deduce solutions from a given one.
L. Kronecker8* employed the number of classes of bilinear forms in two
pairs of cogredient variables to find the number of ways any integer is а SI,
in accord with Gauss»*
E. Catalan" stated that all solutions of x2 + y2 = u* + v* + tr* are given
without repetition by и «* x + «, v = у — 0, x = sp + 00, у ~ sg + a$,
where 2s » л1 + 0* + w* and a, 0 are relatively prime integers, while
_ op « 1. If r, * - ± a +Va* + b2, and n > 1, then
is a 21 and GEL Hence the same is true of x*m — х<*-у + • • • + yi4for x, у
relatively prime integers > L
G, C» Grerono*1 noted that if N2 is a sum of squares of two consecutive
integers, N is a sum of squares of three integers of which two are consecu-
consecutive, as 29* = 20* + 21*, 20 = 2* + 3* + 41.
Catalan" noted that every power of а ЕЯ is a SI since
To solve (p. 103) x2 *f v1 = *2 + ^ + w*f set « = s + a, v = у — 0,
Then /3tf — a* = *, where a = 5{л* + 0* + to*). Take a, 0 relatively prime
and w such that * is an integer. For Pq — ар = 1, all solutions are given
without repetition by z «* sp + 00, у « *g H- ав. [Catalan"].
Catalan stated and E* Fauquembergue*1 proved that, unless x = 1 or
4aJ + 1> (a2 + l)x* =* y1 + 1 implies that x is a SI, since all solutions (if
any) of y2 — -Ax2 = — 1 are given by the terms of convergents of even
rank in the continued fraction for V3. The latter proved (p. 346} that
х* + у***и2 + 1?+1 is satisfied by 2a + 1, а — 1, a + 1, 2a and by
2*» + 1, 0* - 1, 2a1 - 0* + 1, 2a0,
J# W* L* Glaisher64 proved that, if the number of representations of
8» + lby
Bp + l)f + DтI + Ds)*, Bp + 1)> + Dr + 2M + De + 2)»
is tfi, i?t, respectively, then Rx « ^t unless 8n + 1 is a square, while if
" Matbeew, 2V1882, 64-7.
н Abb Akftd. Berlin (MathJ, 2,1883, 52; Werfce, 2,1897, 483.
и Адеос. fnmc at. bc., 12,1883, 98-101.
» Also etatod Nouv, Ann. Math,, C), 3r 1884r S42; M&tbesb, в, 1886, 05, 113.
« Nouv. Ann. Metb.v C), 2, I883r 329.
*» AUi АссвЛ. Pont. Nuori linoei, 37,1883-4, 54-6.
«i Noov. Ann. M&tk, C),3, 1884,533. a. Catalan"* of Ck XII,
•» Quar. Jour. Math., 20, 1886, 91.

Chap. УЩ &VU ОТ THREE SqTTABBS* 269
Catalan** noted that C) with £ = 0 does not give all solutions of
tti = я* + y* + s2, for example not that with ti = 27, But all primitive
solutions (u, g, у, г with no common factor) are said to be given by C).
There are several identities giving an infinitude of (but not all) solutions
of (z2 + у1+ «*)*= EL
A. Desboves" noted that the complete solution in integers of
X2 + F2 + Z1 = U*
5s given by the identity
[2(p2 + ? - s2) J + {2[(p - *J - ф + pfo - «tfI
+ C(ff " *)J - P* + 4q(p - i)J - {3|> - «)« + <?2 + 2a(P - Я)I-
Catalan" noted that, if z2 + y2 + j^ = 1, xx' + yy' + zz' = 0,
(л'1 + у* + А Цуг" - щ'У + («» ^ хг'У + (x|f" - уж'ОМ
Catalan*8 treated г** = x2 + i/1 + г2. Since a prime 4/i + X is of the
form y2 + 2*1 one solution is given by и ** 2/i + 1, x ^ 2д, We may set
u + s^rt* + 0* u — x г= 7* + fl1 and obtain a solution leading to the
identity C).
С Hermite*1 expressed the number of decompositions of an integer into
3 and 5 squares in terms of the number of classes of binary quadratic forms.
J, W* L. Glaisher70 considered the compositions a1 + Ь2 + с*, л1 + Ъ* + Ь2,
a2 + аг + a* of n as a sum of three squares when n = 3 (mod 4), a, d, с
being distinct odd numbers, and formed from them the respective quan-
quantities $a* + Sbp + $cy, 4aa + 86fc 4&a, where a = {- I)^»/1, -•->
V = {— l)^^1^1. The sum of the quantities so derived from all the com-
compositions of n equab the expression
<r(n) - 2*r(n - 4) + 2*(n - 16) - 2<r(n - 36) + - -,
where <r(k) is the sum of the divisors of к. This result holds also when
л s 1 (mod 4) if we use the quantities 8aa, 4aa, 4aa, act for the respective
compositions a* + b2 + c*, a1 + b2 + 0, a1 + У + b2, a1 + 0 + 0, where a
is odd, Ь and с are even, distinct and ф 0. The number of representations of
n as a sum of three squares is expressed in several ways as a series involving
the number of representations of к as a sum of two squares,
E. Catalan71 noted that
3{(a + 2b - 1)' + (b + 2a - 1)' + (a - 6J)
- Ca - 1)* + C6 - 1У + Ca + Sb - 2)*,
Bull. Acftd. Roy. fidpque, C), 9,18S5, 531.
Nouv. Ann. MaUL, C), 5P l$S6r 232.
Mem, Soc. Roy. So. liege, B), 13,1886, 34-0 (MO&ng* Math. Ш).
Ib&t B), 15, 1888, 73-5, 211, 250 (MelMigea Math. Ш, 1885, 120).
»М^100г1887,вО,вЗ; Oeuvree, IV, 233r 237.
, 21,1891-2,122-130.
. вс., 1S91, II, 195-7.

270 History op the Theoet op Numbess, [Chap, vii
De Rocquigny7* obtained a solution of GH • GH = Ш by use of
(a1 + ХЬ2)^! + Xtf) - (aa! + ШхJ + X(obi - *!&)*, X = с* + А
Catalan71 took the fourth variables zero in Eider's A) of Ch. VIII
and got
= («я + УУ1 + »i)f + toi
Taking x :xi — у : y\ CYoung17], we get P = GH; but the condition 5s not
necessary in view of (9 + 4 + 1){1 + 1 + 1) =25 + 16 + 1.
К. ТЪ. Vahlen74 deduced A) from the theory of partitions. The identity
c? + 20* + З72 + 65* = «' + (^ + 7 + 6)f + (- P + 7 + *)* + G - 23)J
and Jacobi's21 final result shows that every number is а Ш.
Catalan75 proved that if p 5s not a 121, then p2 is a QJ, For, if
p^tf + tf + c*, j>*-(aI
e»+ #_!_,* +<p, then
Catalan76 noted that every odd square > 1 is a sum of 2 or 3 squares.
P. Bachmann77 considered the number A of decompositions of * into
three distinct squares л* + aJ + a\ where one (or three) of a, au at is of
the form 12k ± 1 and the others are of the form 12fc db 7; the number A*
of decompositions into three distinct squares for which the reverse 5s true;
the number В of decompositions s — л* + 2a? in which <*, <*i are distinct
and a is of the form 12k rfc 1; and the number £' of such decompositions
in which a = 12fc =fc 7. He proved that 2Л + В = 2Л' + В' + D, where
2) = 0 or |(— l)'Bi + 1) — j{/3> according as * is not or is of the form
3Bг + II, andj is the absolutely least residue modulo 3 of (— l)*Bt + 1),
Bacbmann7* gave an exposition of the theory of [31,
J* F, d'Avillez™ applied Catalan's*1 formula to express the squares of
1, 3, 6, 11, 17, 25, 34, 45, • • • as SJ.
We may express 1521 as a QJ in 7 ways,80 Many identities giving equal
sums of three squares nave been noted.w
M, A. Gniber8* tabulated solutions of 31я » HJ for л ^ 6,
R, D» von SterneckM gave an elementary proof of A),
71 Mathteis, B), 2,1892,136.
"Ibid., B)r 3t 1803, 105-6»
"Jour, fur Math., П2,1895r 23.
" Mem. Acad. Hoy. Bdgjqoe, 52, !893-^r 21.
n Matheeu, B)r 4, 18M, 27, 52-53.
77 Die ADAlytieche ZaUentheorie, 18M, 37-9.
78 Arith, der Quadrat. Fonnea, 1898r 139-162, 600; Niedere Zahlentheorie. 2.1910, 320-323.
*• Jornal de Sc. Math. Kxys. e Nat.r B), 5,1897-в, 90-2.
•"Amer. Math. Monthly, 5, 1898, 214.
*/Шч6г1899, 17-20.
eM.r8f 1901r 49-50.
^r. Akad. Wise. Wieo (Malh.)t 109r I la, 1900, 2S-43.

Chap. VII] Suif OF TEREI SquaBBS* 271
H, Schubert" treated я? + у1 + з* = u\ where x, y, z have no common
factor They are not all odd, as seen by their residues modulo 4* Hence
we may assume that £ and у are even, and z and u odd* Thus (z/2J + (p/2)*
is to be factored into i(w + z)9 i(u — z)> which is done by trial,
P. Whitworth"' tabulated the number of ways each integer з 64 is a
sum of three squares each > 0. R, W, D* Christie noted cases of equal
sums of three squares.
E. Grigorief" noted that [by C) J x2 + y* + 1 = & is satisfied if
2s^pI-g*+rJ-aa, у = рд + щ 2г = ? + ф+ i*+ ?, ре - rq = 1,
when p + gH-r-fsis even. Eecott (p, 285) listed 34 values < 500 of zm
F. Hrom&lko* noted that n7 + (n + 1)* + x2 = (x -h 1)? for
з = n(n + 1),
while a* + b2 + x2 = & for z = x + a — 6, (a — Ь)з = ob.
Haag87 stated that every number not of the form (8n — IJp2 is a HL
H. B. Mathieu58 noted the identity
(«* + jS2 + 7*)[>V + b2^ + {aa + W
= Цо^ + J(jS« + ^J + [«(л2 + т5) + MIP + («tfr - bay)\
G. Humbert^9 gave theorems on the decomposition of M + Pp mto a
sum of three squares of such complex integers, where p = A + V5)/2.
A. Hurwits90 noted that, if n = 2*7ng?ffT * * *, where qiy qt, •• • are
primes 4£ + 3, and m is a product of powers of primes 4k + 1,
П2 = X2 + y* + 2*
has
solutions. It has solutions each Ф 0 except for n* = 2*^, 52*2*^, since
n* = x2 -f ^2 has 4^(n-2) solutions.
A. S, Werebmsow91 expressed a \3\ as the cube of a SI, but made errors.
G, Bisconcini" gave a table of solutions of D).
E. LandauM considered the number C(x) of integers ^ x which are ffl.
Since a positive integer is a ffl if and only if it is not of the form
/ = 4-{8b + 7), a £ 0, Л ^ 0,
* Niedere Ам1ушвг lt 1902t 165-S-
**- Math, Quest, Erfac. Tim»r B), I, 19O27 K-5.
* L'interaiediaire dee matb., 10,1903, 245.
« Zeitechr. Msth. Natorw. Unterricht,, 34, 1И», 258; 35,1904, 306.
^ТЬй.,35,1904,57.
u L'intermediatre dee math., Ц, 1904t 273. TUdng « - Д — у - i and replacing & by
Ъ + a,we get the identity од р. 163.
" Compi«s Bendne Pane, 142, 1008, 537.
"L'mtenrtduire dee math., H, 1907, l€7.
« /Ш., 15,1908, 275-6; cf, 16,1909, 135, 256.
» Periodic© rfi Mat., 22, 1907, 28-32.
» Archiv Math. Phyi.f @), 13,1908, 305.

272 Bistort or the Theoby of Ncmbebs, [Chap, vn
the number of integers ^ x of one of the forms/is И — C(z). Since there
are [(* + l)/8] integers 86 + 7
A, Gerardin" noted that
(mx - ny¥ + (nx
(* - 1)' + ** + (x + IJ = 1 + *f if ? = 3* + 1,
as for (x, t) = @, 1), (I, 2), D, 7), A5, 26), E6, 97), . - •. To Lucas is
attributed
A2m ± 2M + 1 = (8m =h 2)! + (8m =b l)a + DmI.
W. SierpinskiM noted that if к is a QJ in ъ(к) ways,
where 0 < P + m2 H- n8 < x. The number of sets of integers I, m, n
satisfying that inequality is b^n + Oix?**), for О as in Landau1" of Ch, VI.
E. Landau" proved that every positive integer not of the form
4a(8m + 7) is а И, using the equivalence of every positive ternary quadratic
form of discriminant unity to x2 + y* + z\
К J, Sanjana'7 found solutions of the system of equations
X* = y* + z* + u«, 2? + У + ^ + « = 100.
Letx *= а + Ъ,у ^а — Ь. Then^ + w2 = 4ab,2a = 100 — z — u. Hence
(z + bJ + (u + ^ = 2b2 + 200b.
He took u + b = z -bt whence г1 = 1006. Taking b = 1, 4, 9, - • -, he
found the solutions 42, 40, 10, 8 and 38, 30, 20, 12. The solution 39, 34,
14, 13 was noted by N. B, Pendse.
H. B, Mathieu" stated that the general solution of SI » SI is
U ± rE =h pD, pA + qB^ Ш, riTl5- gP.
Welsch9* gave I ± mt», n ^F pvf Im -> np =F v as the general solution»
A* Gerardin gave the identity
Ga* + W - 12Л)' = (ва* + 6Ь2 - 14abJ + (За2 - ЗЬ2J + Ba> - 2Ь*>*.
L» Aubry1™ noted the existence of an infinitude of primes each a sum of
three distinct squares» Every prime p = I2n + 5 > 17 gives a solution.
"Аюос. fnm^, 38,190&, 143-5.
•Spraw. Toward. Nauk (Proc. Sc. Soc. Warsaw), 2,1909, 117-9.
"Handbuch . . . Vertdlung det Primsabten, 1, IG09r 546-505.
n Jour. Indian Matb. Qub, 2,1910,202.
и L'intermediaire dee math., 17,1910, 28$. On pp. 72,166 it la shown that hia earlier solu-
solution, 16,1909, 220, is not general.
Ш„ 18t 1911, 62. Cku^ 21, 1914,15&-7, stated that we may need to give fractional
values to Z, я», л, pf r,
*"/«*, 17,1910, 278; Spbiox^edipe, 1907-8,27.
tn БрЫпх-Oedipe, 5,1911,25-26. Proposed bv F, Froth, Naur. Cdrresp. Matb> 4,1878,95.

We have p — a* + b2, where a and Ь are prime to 3, so that we can set
a + h = O (mod 3),
where the three squares are distinct if p > 17.
L. Aubry1** proved that not all decompositions of the square of a SI
into a \3\ are given by D). Expressions for x2 + y2 or x2 + 2y* as a SI are
given on p. 124 and 13, 1912, 11, 188-190.
H. C. Pocklmgton1™ noted that, ifJV=4m*flor4mH-2J there are
properly primitive forms of determinant — N that have the quadratic
character — 1; while if N = 8m + 3 there are improperly primitive forms
of determinant — N which have the character — 2, Transform such a
form into (b, ff c), where b is prime to N. Solve bg* sa — 1 (mod N) for g
and let Ъд* + 1 = atf. Then
N = (а, *,«,/, Л О)(Ьс -ЛЛ, - Ь»)
5s a representation of N hy a definite ternary quadratic form of determinant
unity. Reducing it in the ordinary way, we get N = И.
В. F, Davis104 noted that, if p + q + r = 1, lfp + lfq + lfr = 0, then
a* + P + c* = (pa + $Ь + тсУ + (qa + rb + pc)* + (m + pb + qc)\
E. Landau105 proved that the number of sete of integers u, «?, w for
which u* + ©* + t0*Ss is ^rtcI/1 + O(^+€), for e > 0. Application is
made to the number of classes of positive forms of given discriminant.
К Aubrym proved that pA* = B* + C* + D* implies that p is a sum of
three squares; similarly for four squares.
E. N. Barisien1" noted various special cases of C).
*G. Muhle1*" solved x2+y5+^=^J, where g is given; also, x2+y1=^J
and x2^^2^^^-^
G. Humbert,10* by use of an identity involving theta-functions, proved
that Uf(x) is any even function of z,
where t ranges over the integers occurring in the decomposition of a given
number 8Af + 3 into £ + rfj + & each t an odd integer > 0, while in the
second member the summation extends over the decompositions
8Af + 3 *= 4h2 + ddl № > d > 0).
The case/ = 1 is due to Hermite." He gave a similar result and
3f@ - 2Z№ + d - 4h)f(d + 2h),
4N + 3 = ? + /J H- ^ + 4P + 4?J = 4A' + ^ (^, |lf fe odd).
., 18,1911, 236. Cf. M. Biga&ux, 34,1917, 35^6.
ieFftxs. СвкЙн-. PhiL Soo,, 16,1911,19.
1M MatiL Quest. £dnc. Tun», B), 21,1912, 23.
im CW*tb«Bn Nacbr^ 1912, 693, 764h9. Cf. ffi
ю Spdinix-Oodipe, 7t 1Й12, SI.
*w Ш£, 8,1913,142r 175.
«"■ ESn Berto* яд- Ыт топ den pytb^gtxreui^n ZabJeo» Progr,, WoUstein, 1913L
M Compte» Beodv Park, 158,1914, 2ИЬ6; втЛц 38a Cf. 157,1913,1361-2.
19

274 History of the Theory of Numbjibs. CChap, vii
W. C.EelV" to solvex1 + y2 + a* = a*,tookz = 2MN,y = M* - N2,
a = то1 + nf, and gave to M* + 2V*, г the values ?n* — 71*, 2ящ in either
order. He tabulated 125 sets of soiotkms arranged according to the sbe of a.
A. Gerardin and E. Miot™ gave many klmtitiesx2 + y* — г? + & + tp*.
L* Aubry gave a very long, bat elementary proof, by use of theorems
on divisors of numbers x3 + my2, that every number not of the form
4Г(8я + 7) is a ED-
L. J. Mordell proved Kronecker's" theorem by uae «£ theta funetioiwk
A. S, Werebrusow118 noted that the problem to find two equal sun» of
three squares is evidently equivalent to mm* + nn' + pp' = Q, the general
solution of which is stated to be
m — off — b<x, n => ay — ca, p = a& — doL,
m' = c5 - dy, n' — d& — ЪЪ, j/ = fry — c&*
He gave long formulas said to solve xt-\-yt = tP-\-v*-\-wi completely-
E. Bahier114 found solutions of^-h^ + c2 — d*in which a + b = rf>
J = с _|_ l^ = рг _|_ y*t or a and b are given- He discussed the nature of
numbers d such that & is a sum of three squares Ф7 0,
E. Turriere1" derived D) geometrically and showed bow to deduce new
solutions of x[ + - - * + x* = Й* from a given solution.
W. deTannenberg11* found real polynomials of degree 2» in a variable В
satisfying x3 + у2 + г2 = Рг> where Pis л given polynomial oi degree 2n ш tf,
not zero for any real 0. Hence set /*= Й - O-—(^ - S, t* = i@ + bp)<
For arbitrary parameters «o, • ■ ч «щ- define two sets of functions by
«q = e*4, tfc = e"^» Let the ut г beeome и', /when tu * --, ^are changed
in sign. Define x, y, 2 by means of
P — z = 2u,tC, P + 2 = 2ряи^ x + iy = 2»^г^, x — iy —■ 2v*vi,
which are consistent since vnv'm + vnxC — P. Take ^, = г(^ + br)>
On two equal sums of three squares, see papers 19 and 86 of Ch. VIII.,
By Cesuro^of Ch. IX there are in me»n^rn1/5 representatkms of n as a SL
On a CB equal to 2**, •* or tr1, see papers 171 of Ch. M, 69 of Ch. XV, 312
of Ch. XXII^ On numbers not а Ш, papers 4> 5 of Ch- VIII. On systems
of equations including CB = □ r papers 97 of Ch. VII, 94 of Ch. IX, 32-39aT
51, 146, 165, 168 of Ch. XIX, 390-S of Оь XXI, 308^9 oi Oi- XXIL On
systems including CB =«}oruJf papers 95, 97 of Ch. XX, 353, 392, 402-3
of Ch. XXI.
"> Amer. Math. Monthlyt 21, 1914, 269-273.
ua L'interm^diaire dee math^ 21, 1914, ltt-2.
ш Spbinx-Oedlpe, nuroero врёсЫ, JTml, 2914,1-24.
ш Мею. Math., 46, 1915, 78-
ш L'interm&toiiro des math^ 23,1916,12-13; 17-18.
1U Recherche . . . Triangfee Rectanglea en Nombrea Eotfet», 1916,
ia L'enfieignetoent matb.r 18y 1918, 90-6.
ui Comptee Rendue Pat», 1G6,1&17,

CHAPTER VIII,
SUM OF FOUR SQUARES.
Diophantus, IV, 31 [32], desired foar numbers x^ such that the sum of
their squares increased Cdimmisbed] by the sum of the numbers is a given
number n. He took, n - 12 [n = 4 } Since x* ± x + i is a square,
2&£ ± 2x> + 1 is the sum of four squares, here 13 0>3- Hence we have to
divide 13 [5] into four squares and subtract \ from [add \ to] each of their
«ides to obtain the sides of the required squares. Since
134 + 9 + ++ L5 + + +
the Bides of the required equates are
11 7_ 19 13 [11 13 21 17]
10' 10' 10' 10; LlO' 10' 10' ЮУ
G. Xylander1 noted that if we take 1430 in place of 4 in the second prob-
problem, we set the solution 6*, II1, 21», 30*.
С. G. Bachet1* remarked that Diophantus apparently assumed here
and occasionally in Book V that any number is either a square or the sum
of 2, 3 or 4 squares [Sachet's theorem], and added that he himself bad
verified thie proposition for all numbers up to 325 and would welcome a
proof; he gave decompositions into 4 or fewer squares of each number up
to 120. He mentioned the generalization of Diophantus IV, 31 to the
problem to find к numbers such that the sum of their squares increased by
the sum of the numbers is a given number n. Thus л + Jfc/4 is to he the sum
of к squares. Bachet stated that if к ^ 4 there is no condition.
Fermat, in his comment quoted in Ch. Iм, stated that he possessed a
proof that every number is a sum of four squares. In stating the theorem
elsewhere, Fermat* remarked that Diophantus seems to have known the
theorem*
The reason for ascribing a knowledge of this theorem to Diophantus
lies in the fact that he made no mention of a condition on a number in order
that it be а шип of four squares, in the three cases IV, 31, 32 and V, 17, in
which he mentioned the subject, but that he gave necessary conditions for
representation as a sum of two or three squares (Chs, VI, VII)-
Diophantus V, 17, sought to divide a given number into four parts
such that the sum of any three of the parts is a square. Thus three times
the sum of the four parts is the sum of four squares. Let the given number
1 Diopbuti АклшЫЫ Renim АтНЪ., . . . , G. Xybndio, Вмйе*е, 1575, 104.
* Diopfevti Акж, Arith., 1621, 241-2.
'Oeuroe, П, 65; III, 287; letter to Методе, Sept. or ОеЦ ДО6; to be propoeed for
DoluUon to Suuto-Croiz. Ыепыше communicated it to Dttcartts, M*rch 22, 1638.
The batter Mental the theorem to St. СтЫх (Oenvw* de Dncartes, II, 256). Ferm&t,
Оеито,П,4ПМ; Ш, 315, letter to ТУфу, June, 1658, ptupomd that Braneketvid
WaDi* «eek % proof of the theorem.
275

276 History of the Theory of Numbers. CChap. vm
be 10. Then 30 is to be divided into four squares each < 10. Since
30 ^ 16 + 9 + 4 + 1, we take 9 and 4 as two of the squares and divide
17 into two squares each < \Q [the squares of 1016/349 and 1019/349}
If we subtract each of the resulting four squares from 10, we obtain the
required parts 1, 6, etc. In V, 16, the number 10 is divided into three
such parts. For a generalization to n parts, see Kausler47 of Ch, XV.
Regiomontanus5 (X Mullet) proposed in a letter the problems to find
four squares whose sum is a square and twenty squares whose sum is a
square > 300000.
Jakob von Speyer*" gave
1 + 2* + 4* + 10* =* IP, 22 + 4' + 71 + 10* = 131.
A. Girard/ in commenting on Diophantus V, 15, stated that there are
numbers, as 7, 15, 23, 28, 31, 39, not a sum of three squares, but that any
integer is a sum of four squares.
R. Descartes6 announced the theorem (" whose demonstration he judged
so difficult that he dared not undertake to find it"): Any number which
is the sum of three squares and exceeds 41 can be expressed also as the sum
of four squares, excepting only the products of 6 or 14 by 4, 4*, 4*,
There are no other numbers which are not composed of four squares, except
2-4ll> which is not a square, nor composed of three or four squares, but only
of two.
Fermat' stated that he had much trouble in finding the new principles
needed to apply his method of infinite descent to show that every number
is a square or the sum of 2, 3 or 4 squares; but stated that he had finally
proved that if a given number is not of this nature there would exist a
smaller which is not.
L, Euler7 admitted tliat he could not prove Bachet's theorem that
every integer is a S), nor give a general rule to express n* + 7 as а Ш.
Oct. 17, 1730 (p. 45), he noted that, if Fermat's theorem that every integer
x is a sum of tliree triangular numbers (a' + a)/2 is true, then Sx + 3 is
the sum of the three squares Ba + II. Hence Sx + 4 and &x -\-7 are Sh
[Cf. Beguelin* of Ch. L] Since m*{Sx + 4) - k2Bx + 1), it remains only
to prove that Ax + 2 is a 3). Oct. 15, 1743 (p. 263), Euler noted that, if
np* is а ID, n is a sum of four integral squares. Thus if it be true that
8m + 3isaCD,8m + 4isaS] and also 2m + 1, so that every integer is a SI,
May 6, 1747 (p, 419), he stated tliat Bachet's theorem depends on the
unproved fact that every number 4m + 2 is the sum of two numbers Ax + 1
and Ay -h I, neither having a factor Ap — 1 [and hence each a HI J For,
« C. T. de Murrr Memombffia BiU., 1,1786,160, 201.
,18.
« I/arith, de Simon Stevio .. . annotation» par A. Gir&rd, Lenfe, 1625, p. 626; Оотпя
math, de 9. Stevin par A. Guard, 1634, p. 157.
'Oeuvra, 2,1898, 256, 337-8, letters to Мешепде, July 27 and Aug. 23, 1638. The limit
33 given in the first letter was changed to 41 in the eecond.
4 Oeuvree, П 433, letter to Oarc&Ti, communicated Aug. 14,1659, to Huyfeos,
г Сагкяр. Math, et РЪув. (ed., P. H. Fure)p St. Petersburg, lv 1843, 24, 30, 35; letters to
Goldbtch, June 4, June 25, Aug. 30,1730.

. УШ] Sun op Four Squabeb, 277
then 2Dm + 2) is a SI and hence 2m + 1 is а Й). May 4, 1748 (p. 452),
he gave the fundamental formula (Cf. Euler1" of Ch. XIX)
Г (a1 + b2 + **
A) -js^ap + bg + cr + ds, у — о^^Ьр±(»Тб?г,
I z = orT6s- cp ±dqt v = a$±br^cq — dp,
and stated (p. 454, and Aug. 17, 1750, p. 531) that Baehet's theorem would
follow if the fourth power of 1 -|- x + x4 -j- x9 + xl* + - — contained xn
with a coefficient + 0. April 12, 1749 (pp. 495-7), he stated that he had a
proof that, if p is any prime, there exist four integers a, - -t d, each not
divisible by p, such that a* + ■ ■ • + d* is divisible by p. Set a = <xp ± x,
•fd = tpzk v, where 0 ^ x Ш ip, • > ,Q ^ v ^ ?p. Hence а^Ч- \-&
is divisible by p. If p is odd, x < %p, - - •, во that Xs + - - - + if1 < рг.
To prove that every prime ifl a SI, suppose there is a minimum prime p
not а Щ, But x* + -" + v1 = pqt q < p. Euler believed, but could not
prove, that if pq = Ш, p + S), then ;Ф1. Admitting this, we would
have a contradiction with the assumption about the minimi up p. Thus
every prime is a ID and hence by A) every integer is a UK
On the point here left in doubt that pq = 3) and q = Й) imply p = Щ,
Euler proved, July 26, 1749, pp. 505-10, that, if* m ш 7, тпА = B and
m^m imply A *= Ш. Set
twhere/, -^f ft are numerically ^mftj Then/2 + -•. + A? is divisible
by m. For m ^ 7, the quotient was verified to be а Щ,
? + ? + & + & = m£X* + У + ^ + F»J
and [In accord with, but not a consequence of, A)]
/ = aX + 6F + cZ + dV, g « bX - aY - dZ + cVf
h~cX + dY -aZ~bV, k = dX - cY + bZ - aV,
= (x + X)* + (y - Y)* + (г - Z)8 + (* - 7I
where xf - - *, f are given by A) with the upper signs. Moreover, he gave
a proof of Chr. Goldbach'e assertion of June 16 (p. 503) that the sum s of
four odd squares can be expressed as a sum of four even squares. Since
1)* + iBq + 1)* = (P + q + 1Г + (P - *)*,
The last sum involves two even and two odd squares since а * 8m + 4.
♦ For the general case Euler* admitted In 1751 that he bad no proof.

278 History of the Theory of Numbers. [Chap, vm
Hence
| = (Эр + 1)' + B2 + 1)* + 4r* + 4*»,
\ = (p + g + Dl + (p - gI + (r + «)' + (r - *)*.
As a corollary, 2A = SI implies A ~ SL
On March 24, 1750 (p. 513), Goldbach had stated that there is a definite
connection between the sets of four squares whoee sums are 2m — 1 and
2m + 1, as derived from 8m + 3 = 3h June 9, 1750 (p. 518), Euler
interpreted this as follows: From 8m — S^o^+b^ + c*. where at Ь, с
are odd,
where two of the squares are even. Set 2p = (a + l)/2, 2q - (b + c)/2.
Then
4m - 2 = BPy + B2)! + r» + 8s,
2m - I = (p + ff)« + (p
where two or four signs are +, From
where two or four signs are +. Hence, from 8m — 5 ^ ED,
2m - 1 = p1 + 9? + r* + s*,
2ш + 1 ^ (p + i)« + (q + l)^ + (r - 1)* + (s ^ l)ie
Thus r + e — p — q — 1 and we can express any odd number as a sum of
four squares the algebraic sum of whose roots is unity. £Cf. Cauchy, 1813].
Euler stated (p. 521, p. 527, and again on Dec. 4, 1751, p. 559) that while
he had proved that any rational number is the sum of four rational squares,
be had not proved the theorem for integral squares.
Goldbach (p. 526) noted that a, fit yt a + p + у + 25, and a + fi + ft,
« + T + 5, /?+7 + *, ^ anda + S, 0 + Я, 7 + *, a + 0+7 + * have
the same sum of squares.
Euler, July 3, 1751, p. 542, discussed the problem to make
* = ** + /? +72 + * + e
a Ш. Call the roots ct — Ax, fi — mxt у — nx, 6 + ar. Then
$ » A - \Bx + ^-, A = fc* + m0 + ny, Б-^ + ^ + яЧЬ

Chap. УШ] Sum OP FotTB SeUAKES. %?9
Resolve e-B into two factors a, bt both even or both odd- Then for
x « e/a, 6 = A + (a — 6)/2. Take k, tn, n arbitrarily and determine
a — b, or conversely. The case e = 8 was partially treated by Goldbaeu,
pp. 540, 546-8, 555 and by Euler, p. 557, A 3) with the sum of the roots
zero is а О since (pp. 548-9)
a1 + fr5 + e> + (a + b + c¥ - (a + by + (a + e)* + E + cI.
Goldbacb (p. 548) noted that
8л + 4 = а* +6*+ <?+#, a + & + c + d=:2.
Euler, Sept, 4, 1751, p. 551, deduced this from
8л + 3 - CB - (a + b - iy + (a + с - 1)< + F + с - l)f.
Euler8 published some results on Bachet's theorem. He proved
Theorem L There exist integers a, b for which 1 + a2 + b* is divisible
by a given prime p. For, if ^ 1 is a quadratic residue of p, there Is ли
integer a for which 1 + a3 is divisible by p* Next, let — 1 be a J*01*-
re&idue and suppose the theorem is false. Then 1 + 1—2 — 0 shows
that — 2 is a non-residue and hence + 2 a residue; then 1 + 2 — 3 = 0
shows that — 3 is a non-residue and hence + 3 a residue; and in this way
1, 2, • • ■, p — 1 would all be residues.
If A « a2 + • - -bd>, P = p* + - ^thenA/P - APjP2 = (s/PI + " "
by A), so that AjP is the sum of four rational squares. Euler admitted
he was unable to prove that, if A is divisible by P, AjP is the sum of four
integral squares. If this were proved, Bachet's theorem would follow.
But it is readily proved that every integer is a sum of four rational squares-
For, if p be the least prime not such a sum, there exists (Theorem I) an
integer A = a* + b* + & divisible by p, where ay bt с are < p/2. Then
A\v < iPt and yet A/p was seen to be the sum of four ratio? ч1 squared'
J. L. Lagrange9 gave the first proof of the theorem of Bachet
acknowledged hie indebtedness to ideas in the preceding paper by E
The steps are as follows:
(i) If p1 + $* = tp and r1 + s? = up, where p, qt rt s have no comtnon
divisor, then t and и are sums of two squares.
For, call M the g.c.d. of p = Mpx and q = Mqt; N that of r = Nri and
s ^ N$i. Then M and N are relatively prime» Call p the g.c.d. of ^f2
and p *= pplt Since
B) M*(p\ + rf) - h^
Pi divides the sum p\ + q* of two relatively prime squares. By Euler sM
theorem of Ch. VI, the quotient is a sum c? + d* of two squares. Set
ft ^ ^Vi> where ^i has no square factor. Then M is divisible by Wu
M = Ktfii. Now tf*(rf + s\) = «mpi- Since p divides Jlf2, it is prime to
№ and hence divides rf + *J, As before, /*i — e2 + p. Then, by B),
t =: ft*
1 Novi Сошш. Acad, Fteto>p4 5, l754r-5 A751), 3; Comm. Arith.r I, 230-233.
•Nouv. Mem. Acad. Roy. 8c. de Berlin, ишёе 1770, Berlin, 1772, 123-133; Oeuvree, 3,
1869,189-201. Cf. O. Werthdm^ XKoplumt^ tV^ 324-330.

280 Hibtobt of the Theory or Numbers. [Chap, vni
(ii) If 7* + & is divisible by mf + nf, the quotient f is а sum of two
squaree.
Let ! be the g.c.d* of у = lpt 6 = Iq, m = Ir, n = Is. Then p* + g* is
divisible by r* + e* = p. Hence, by (i), t = (p* + $*)/p is a sum of two
squares.
(Ш) IfP^p' + ^H-r^ + s^is divisible by a prime A > ^P, then A
is a sum of four squares.
Set P = Aa* Then a < A. A common divisor d of pt qt r, s is < A,
so that d* divides a and may be deleted from a, p1, - - >, Л Let tberefore
d= 1.
Let p be the g,c.<L of a = ftp and p1 + ff1 « ^p- Then (r* + e»)/p is an
integer «• By (i), t - ma + n\ и = Л1 + P. Thus
fo* = ^ + ^, z^mA + nZ, у ^ ттп — «А.
From P = Aa follows
ЛЬ - * + к, АЫ = P + я* + у*.
Since & is prime to f, there exist integers a, - - -, Ь such that
ж = at + <y&, у = fit + Й, [ a i < 0, | P | < J*f
C) Afe « № + 2a7» + 2fibU> + G1 + *)&*, * = 1 + a1 + Л
Hence № is divisible by b. Thus к = аД where «! < &/2 + 1/5, Thai
At = *fi + 2«7t + *№ + G1 + J%
Л1А1 - C«i^ + «7 + flV + ^(aib - a1) + ^{fllb - ff)
Replacing ai& by 1 + а1 + Л we get
By C), У + # is divisible by t = то1 + n*. By the last equation and (ii),
У + * - t(p] + tf), (ai* + ат + 0aJ + Wy - a«2 - KrJ + O,
«xA = pj + gj + rj + ej.
If a *= 6p is > 1, a! < b/2 + 1/b < a. Similarly, if ax > \f a%A is the
sum of four squares, where at < aI? etc. But each a< ^ 1, Thus a certain
a* = 1, and a%A =* A is the sum of four squares.
(iv) Any prime which divides the sum of four or fewer squares which
have no common factor is itself the sum of four or fewer squares.
If tbe prime A divides p1 + g2 + r* + s*, it divides the sum obtained by
replacing p by ± (p — mA), where я» is such that 0 ^ | p — mA \ < }A,
etc* The sum of the four new squares is < A1 and is divisible by A.
Then (iii) may be applied, even if some of the four squares are zero.
(v) If В and С are integers not divisible by the odd prime A, there
exist integers p and q such that p1 — Бд2 — С is divisible by A.
Suppose that there is no integer q which makes Ь ^ B<f + С divisible
by A (since otherwise we may take p ** 0). For
- 1 - 0И-1* - 1).

, vm] Sum of Four Sqttabis* 281
Multiply the last equation by Q *= fcH^llA + l. If p and $ can be chosen
so that pPQ is not divisible by A, then p1 — b will be divisible by A, as
shown by using Format's theorem. For q constant and p = 1, — , A — 2,
let P become Pu * >, PA^** Then by the theory of differences,
Px - {A - 3)Л + i(A - 3)<A - 4)Pt + iV, - (A - 3I
Hence at least one P, is not divisible by -4., Set m = £(A — 1). Then
Q = tfR + О + 1, R = BV + mS^V^C + - • - + mBC^-K
If О + 1 is not divisible by At it suffices to take g = 0. In the contrary
case, we note that if R becomes #f for q ** t>
fii - (А -3)Й, + К^ -3)(A -4)Jb- ••- +^_, = (A-S)!*"*,
so that at least one Ri is not divisible by A, Hence by (iv) every prime is
a SL
(vi) Every positive integer is the sum of four or fewer squares.
This follows from Eider's relation A). Lagrange added the generaliza-
generalization
tf-B?~&* + BC*)(p) ~ Bq\ -Cr]
D) ~ \wi + ^№ ± С(тгг + Бю,))*
- C{prl - Bq*t ± rpi =F Б^M + БС{дг! - рв! db spx ^F r^ij^
L, Euler^s11 proof is much simpler than Lagrange's, It is shown that if
N divides P = pI + $2 + r* + e*, but not all the numbers p, > *, &t then
N is a sum of four squares. Set P = Nn. Determine a, ht c, d, each
numerically < Jn, so that
p s* a + «a, $ «5 6 + n£ т = с + Л7, а = d + n^.
Set <r = a* + b* + c*+d*. Then с ^ n*. We readily dispose of the case*
<r = nl, pf n is odd, a, -'^d may be chosen numerically < n/2, whence
<r < nK If n is even, we have <r < n* unless a, • —9 d numerically equal
n/2, whence p db q and r ± « are divisible by n and are even. But
Яп «^ P « Sp^t whence
may be used in place of the initial multiple P of JV.] Hence let <r < n2*
Then
ЛГп =* <r + 2nA + nHt A=aa + bp + cy + dS, t s a1
Thus с is divisible by n. Set <r = ля^ so that n' < л- By A),
<rf = A* + B1 + CJ + Z>',
Multiply ,Y = n' + 2A + ni by n\ Then
ЛГп' = {nf + A)g + Д8 + С1 + 1У.
pe., 1773,193; AcUAcad. Petrop,, lr П, 1775 [17723,48; Gomm. Arith.,
t. EuleT'eOpepaportuma, lt 1862,198-201. He first repeated Ugmnge'a proof
Mid Ы»9 proof of Tbeoi^m L
* SUXed to occur onJy when ая5 — c™rf™Jn™l, whence j>? "' t r Are odd and AT ш |Р
eqaab the right member of E).

282 Надгоет of the Твяаы or Кшшжв, ЦСвдг. тхп
In the same way, Nn" (n" < nT) is the euan of foer яцпжгев; etc., iimQy
N-\ is a eum of four eqtwres-
He proved that, if N ш a prime noi dMdbig the given mtegran X, /*, r,
we can find inttLgu.!» x, у, « not divisible by N asch that* — Xx2 + 1^ + >**
is divisible by Лг, Smce X is разов to ^ we «an determine integers m
and n such that \m = — д, Xn = — r (mod АО- Then * =■ 0 is equivalent
to a = mb + ne (mod JV) lor quadratic гевйоев a, 4, с If tbe latter is
impossible, then mb + nm* non-reekhie for each of the (AT — l)/2 residues
& and hence фуев аИ the mm-reakiues. Tben if 4 is any residue, bd is a
residue e, so that ле + da must be a notwesidae, Ibis exceeds tbe
reeidue me + n by «{^ — 1) =* «u For rf Ф 1, «is prime to N. ,
if a is any mm-ma&m, 0+ ^ is a zKm-ne&klue. But щ a + w, ♦ * -,
л -f- (JV — l)u are eongrooit to 0, 1, ♦ —> # — 1 in some order and hence
are not all non-reeidaes*
Euler11 gave a d^ht modification Ы his preoecfing proof. We may
assume that p, qt г, ж in Nn = pI + 52 + ra-|-e* are numerically < \ N,
where N is a prime. Then » < N and we can find integer* a, a, *- -, ^ 5,
such that
p =* Na + tux, q = Nb + nfi, r — Nc 4- ny, ^ = ЛИ + n&,
where a, b, c, d are numericafly < ^n. Then Nn — JW + 2NnA -f- n%
so that 7 — nn\ n' < n. Multiplying by n'/*> we get
Nnf = (№1' + 4L^ + ^ + ^.
Euler11 noted that <it + ftt + e>-4(xI + 3y*)= Bl for
a = 2m(p* + дг)
Ь, с = ж(Cд ± р)я + (« =F p)rl + nCfe =F p)# - Cg d= p)r).
Euler1* remarked that the sum of two primes of the form 4n + 1 is a ffl
since each is a EJ, and verified that every number 4& + 2 ^ 110 is a sum
of two primes 4» + 1.
A. M. Legendre14 remarked that a proof of Fermat's assertion that
every prime 8» — 1 is of the form p1 -|- 5* -|- 2r* would complete tbe proof
that every number is a SL For, any prime 8» — 3 is of form p* + tf2,
any prime 8n + 3 is of form p* + 2g*, any prime 8» + 1 is simultaneously
of the last two forms.
Legendre1* reproduced Eider's" proof, using in place of Theorem I its
generalization by Lagrange.
C, F, Gaoes1* subtracted from the given number 4» + 2 any square
less than it, from An + 1 an even square, from 4» + 3 an odd square. The
remainder is ss 1, 2, 5 or 6 (mod 8) and hence is a sum of 3 squares* Thus
11 Opera poofam» 1, 1862,107-8 (about 1773).
" Novi Comm. AtvL PMropu, 18, 1773,171; Сошт. Aiith., 1,515.
i" Acts Acad. P^ivp p 4, II, 1780 A775), 38, Сотш. Antb^ U, 134^9.
" Мёш. Ac«L Hoy. 8e. FfcHe, 1785t 514 Cf. Poflock*; *bo Eukr^ Lebeaeoe» of CL YIL
u Emai нцг Ь ibtorie dee nombree, FftriB, 1798, 1Ш; ed. 2,1808,182; ed. 3,1830t I, 211-6,
Nob. 151-1 (Млеет, I, pp. 212-6).
q_ A1W4 1801, Art. 293; Wetfce, 1,1863, 348.

. тшд Suv of Гскт Зодглкв£к 283
the given number is ft sum of 4 squara. Finally, a multiple of 4 is of the
form. 4**iV, where N i» «me of the prececfag three types.
Gauss'7 noted that the theorem A) that a product of two sums of four
squares is a (U ж represented in the aimpJeeft way by
(N1 + ffin}(JV\ + ЛГ/i) = ЛГ(*Х + **) + #<k' - тУ),
where H denotes the шш and I, mf A, p, Л', у! are complex numbers,
X, X' and ^ У bong- conjugate imagmariea. He noted (p. 447) that [cf,
Glaisher," Hermite"]
F) (l + 2y + 2jf* + 2*' + -- L
He noted (p. 445) that j>f. Legendre,» Jacobi," and Genocchr"]
G) (l + % +
noted that every decomposition of a multiple of a prime p
+ & + c* 4- <^ correspond» to a solution of x1 + у2 + г1 ^ 0 (modp)
proportional to a3 + ^ ac• + bdf ad — be or to the sets derived by inter-
chaxigiiig b' and с or fr and <i. For p = 3 (mod 4), the solutions of
1 + & + У2 ^ 0 (mod p) coincide with those of 1 + (x + ty)**1 = 0,.
Froi& one value of x + щ we get all by using
(* + 4)-(» Н-г)/(и - t) (u - 0, 1, • • •, p - 1).
For p s 1 (mod 4), p = a2 + fc8; then 6(tt + г)/{<*(^ — ^0! give all values
of x + щ if we exclude the values а/Ь and Ь/а of u.
G. F. Malfatti1* did not prove as he promised to do that every integer
is a ID. After verifying this for about 50 small numbers, he considered the
equation Kn2 *= p* + $*> where it is a given integer* If we admit his
assertion that К must be a El, the equation has evident solutions with
л = 1. Taking К = a* ~f- b% he found an infinitude of solutions, with /
and ff arbitrary r by setting
The equation obtained by eliminating, p is satisfied if we take
Next, iC»r = p^ + <? + r*f in which we may limit if to be odd or the double
of an odd number, and л to be odd, is said without adequate proof to be
FMAu paper, W«ri», ^ 1989b &.
M AMFiSb Пй 8 №Ы ()v 12, pt. lv 18S6, 206^317.

284 History of the Thsobt of Numbers, [Our. УШ
impossible unless Xisa 31. Fox Я — a1 4- b* + c1, the equation beoomee
Bk(an + r) - Ж? + 6n) + Gg(p + en),
m-r q-bn p~cn
Я"ПГ' '-"Т-. «—^--
It is stated that Я = ^ = G, and that the linear equation in я, г, derived
by eliminating p, g requires n *= /* + Ф + h*, whence
P - U* ~ Ф + Юс
[For these, n\<? + V + t?) = p* + fl* + f*, identically in /, 0, A, «, b, cj
There is a similar treatment of the corresponding problem for 4 or 5 squares.
If Malfatti had proved his statement that К must be a sum of the tike
number of squares, he could have deduced Bachet's theorem from Euler's*
result that every integer is a stun of four rational squares.
P. Barlow50 gave a ** simplification of Legendre's16 proof." To show that
any prime Л divides a sum of Ш, he provedat length that s*4-*0* — 1 «=» mA
is solvable [evidently by x = 1, v> = OQ and stated that a tike proof shows
that |^Н-г* + 1 = л4 is solvable. The proof probably meant for the
latter is ae follows. If p = y* (mod A), either — (p + 1) is a quadratic
residue (^ г?) and tb^ result follows, or it is a non-residue and hence p + 1
a residue, since — 1 is a non-residue (otherwise our equation holds for
у = 0). But p, p 4-1, p + 2, - • - are not all residues. The proof is thus
only a slight modification of that by Euler-10
А* СаисЬуЪ proof in 1813 of Fermat's theorem on 3 triangular numbers,
4 squares, 5 pentagons, etc., was considered in Ch. I. It is in place to
mention here the theorems on sums of squares upon which his proof rests,
especially since special cases were cited above from the correspondence
of Euler and Goldbach. If
(9) fc = ^ + t*l4-tF2H-tp2, **=e-|-*4-t>4-a>,
then
A0) 4k - И « (t + и - v - хрУ + (* - « + * - »)* + (* - и - v + w)\
But if 4* is the highest power of 4 dividing a, then a is а 31 if and only if
a/4* is not of the fora 8xi 4- 7. If ib is even, the three sums in A0) are
even, so that* — «?/4is а И. By (9),* = e(mod2). Cauchy proved that,
if к is even, sufficient conditions for (9) are that * be even and between
Vflfe-"l and ^ and к - И/4 + 4*(8n + 7), With the exception of
я > V3fc — 1, these were seen above to be necessary conditions. For к
oddT sufficient condkitHifl for (9) are that г be odd and between ^3fc - 2 — 1
and т/5; there exists such an « for any k. As to the former case, he proved
that for any Jk there exists an integer between *Ш and ->Sfc and congruent
to * modulo 2 except when к = 1, 5, 9, 11, 17,19, 29, 41, 2, 6, S, 14,22,
24, 34. __„ __—
«• New Series of Math. Repertory (ed., Leybo^m), 2, 1800, П, 70; Theory of Numbexs,
Lod^ 1811, 212.

Chap. VIXI3 SUM OF FoUK SqTJABES. 285
Cauchy" noted that if p is a prime and a, 0 are integers for which
a + lS + i ^ Vy an<i й ^ ranges over a + 1 distinct values modulo py
and В over 0 + 1 values, then Л + В takes at least a + 0 -f- 1 distinct
values modulo p. For Л and В not divisible by p, Ax* and By* + С each
take (p + l)/2 distinct values modulo p, when p is a prime > 2. Hence
Ля1 + By* + С takes all p distinct values modulo p and therefore the value
zero. Cf. Cauchy* of Ch. L
Cauchy" noted [the case d ^ & = 0 of
= (ap + hq + cry + (aq- bpI + (or - cp)* + (br - cq)\
and a like fonnula with n squares instead of 3 [see Cauchy11 of Ch. IX}
A. M. Legendre21 gave (8) and concluded that every number of the form
8n -|- 4 is a sum of four odd squares in <rBn + I) ways, where <r(k) is the
sum of the divisors of k. It is said to follow readily that every integer is а ЭК
C. G. J. Jacobi24 proved Bachet's theorem by comparing the formulas
including G), where p ranges over the positive odd numbers, and <r(p)
denotes the sum of the divisors of p. At the same time we obtain the
theorem: The number of representations79 of 2*p as a sum of 4 squares is
8с(р) or 24<r(p), according as a = 0 or a > 0. Cf. Jacobim of Ch. III.
Jacobi* compared the formulas*
BкК/*У = W2<r(p)q», "tQkKJT « 2gJ/4 4- 2gV* 4- 2g«/* 4 ,
where p ranges over the odd positive numbers, and concluded that there
are <r(p) sets of four positive odd numbers the sum of whose squares is
Ap [see papers 23, 30, 42, 52, 69, 72, 82, 91}
V. Rouniakowsky*7 proved that, if Л, В, С are integers not divisible by
the prime pt we can give to x, у such integral values that Ai? 4- By* ~- С
is divisible by p. He first found the conditions that хоту can be a multiple
of p; then noted that, if neither can be a multiple of p, the congruence can
be written pM 4- pN — 1 s 0 (mod p), where p is a primitive root of p,
л Jour, do ГбсЫе polyt,, vol. 9 (cah. 16)» 18X3,104-116; Oeuvrea, B), I, 39-63»
n Cours d'an&Iyae de Pecole polyt., 1, 1821,457-
*>Traite dee fonctione elliptiqaes, 3, 1828, 133. Stated In Legendre's ТЫопе dee nombres,
ed. 3,1, 1830, 216, No, 154 (Мавег, I, 217); not in ede. 1, 2. a. Bouni&kowsky, Vol.
I, p. 283. Cf.J&oobi*
MWericerI,423^; Jour, fur Math-, 80,1875,241-2; Bull, dee ec, math, Mtr., 9, 1875,67-9;
letter, Sept. 9,1828, Jacobi to Legendre. Jtxobi, FimdjunenU Nor» Funct. Ellipt.,
КопфЬегк, 1829, p. 188, p. 106 C4), p. 184 (в); Werke, I> 239. Cf. J. Tannery And
J. Hoik, 06m. theorie fond, efl^ 4,1902, 260-3; J. W. L. GUieher, Quar. Jour. Math.,
88,1907, 8; papers 51-2, 81,88, 110-1.
* Jbar. ftr MaUi^ 3,1828,191; Werke, I, 247» Cf. ХдошпПе1 and Drftoui*» of Ch. XI.
* FaDdamentaNov» FmrfL EBipi, 1829,106 E5), 184 G); Werka, 1,162, 235.
" Mem, AouL So» St. PeterAourg (Math.), F), 1,1831, 565-581-

286 Hiotoby of the Theory op Numbers. [Chap, via
and M and N are odd. The latter congruence can be solved. Or the
theorem can be derived by multiplication from Lagrange's case A = L
If .N is any odd integer or the double of an odd integer, while A, Bf С
are integers prime to N, A& + By* — С = 0 (mod 2V) is solvable.
Given two arithmetical progressions whose first terms a, 0 are arbitrary
and whose common differences А, В are not divisible by the prime p,
we can choose n and n' so that the total sum of n terms of the first, n'
terms of the second, and any given integer E, is divisible by pi
\[2ct + (n- l)A)n + \{2& + {n'~ l)B)»' + tf = 0 (modp).
For, this can be reduced to the above congruence*
F» Minding28 noted that integers и and v can be chosen so that
u1 — Bt? — С is divisible by the prime p, if neither В nor С is divisible by p.
In fact, for v = 0, 1, * -, (p - l)/2, the function B& + С takes (p + l)/2
distinct values modulo p, and at least one must be congruent to one of the
(p H- l)/2 values of u*3 since otherwise there would be p +1 residues
modulo p* Hence we can choose и and v less than pj2 so that u2 + t^ + 1
is divisible by p. The proof that p is a SI is that by Euler.10
G. libri1* proved tbat there are n ± 1 sets of solutions < n of
& + ev2 + Ь s 0 (mod n),
if aT & are not divisible by the prime n. He first expressed the number
of sets of solutions as a double sum involving roots of unity.
C* G. J. Jacobi" gave an arithmetical proof of his* theorem on the
number /i of sets of positive odd solutions ti, - - -, z of
A1) i^ + ^ + y« + ^ = 4p,
where p is a given positive odd number. Two distinct permutations of the
same numbers are counted as different solutions. For such a set,
i* + & = 2p', y* + s* - 2p", p' + p" = 2p,
where p' and p" are odd. Conversely, these equations imply A1)* Hence
where N[jtp' — vfi + ^] denotes the number of positive solutions w, x of
2p' = и* + Л The latter number is iV[jp' ^ лл] — ^Cp' = o^'l where
a ranges over the factors of the form 4m -j- 1 of p' and a' over the factors
4m H- 3. Let 0 and 0' range over the factors 4m H- 1 and 4tn -]- 3, respec-
respectively, of p". Then
ЩЪр" =y* + 22] = N[p" = bff} - N\j>" = WJ
Set ВД = Щ2р = uj Then
= асГ\ >N\j>" =b&2 = Nlaa + b£\, etc.,
der huheren Arith., Bcrtio, 1832,191-3.
«Jowr.fOx Math., 9,1832,182. See Libri^^of СЬ.ХХШ.
"Jour. At M*th., l2t 1834, F7-172; Werke, 6t 1891, 245-251.

Cka*.vih] Sum of Fotm Squabes, 287
Unless a = fi, a' = p, we may set fi = a + 44, 0' = a' + 4A, Л > 0, if
the term be repeated. Thus
ft =* tf[a(a 4- 6)] H- Ща'(а + &)] - 2A1>* + fa']
H- 2JV[«(a H- b) + 4M] + 2ff[>'(a + b) 4- 46Л].
Let с range over both the a and a' numbers. Then
p. = N[c(a 4- 6)] + 2N[c(a +Ъ) + 4ЬЛ] - 2N[aa 4- fa'J
In the second term set с =* d + 4ЛВ, d < 4A, В ^ 0» Now a +
represent any even number 2C, and b + B(a + 6) any odd number
Thus
ji = JV[c(a + 6)] + 2N£2Cd + 4A«] - 2#|>a + WJ
Since a-j-^^o (mod 4), a+ft. Thus the second member of
2N[aa + bfi'2 = N[aa + 6^'] + N[ap + 6a]
is twice the like sum with Ъ > a. Set Ъ = a + 2(?, a + 0' = 4/L Then
where d < 4A. Hence ^ = N[c(a + b)J. Неге с ranges over all the
divisors of p. If p = rft the equation 2p = c(a + b) becomes %f = a + b,
which has / sets of odd solutions. But Zp/c is the sum of the divieore of p.
Thus м = <r(p)>
T. Scbonemann11 used the notation cos n, sin n for a pair of solutions
of x1 + y* = 1 (mod p)> If cos w, sin m is tbe notation for a second pair
of solutions, then the expansions of cos (n -f m), sin (n + m) give a third
pair of solutions. Then, for a an integer,
(cos n + t sin n)e = cos лп-j-i sin an (mod p).
If pis a prime, cospn = cosn, sin pn = (— l)(y)/lsinn (modp). Hence
cos (p =F l)n = 1 if p = 4A; =Ь 1. An integer a is put into u class Л "
if 1 — a3 is a quadratic residue of p, otherwise into class B. It is proved
that if cos n belongs to class Л and if a is the least integer for which
cosая = 1 (mod p), then a is a divisor of p¥ 1 when p = 4k тк 1; then
cos n is said to belong to the number a. There exist ф(р ± 2) " primitive "
cosines which belong to p ± 1. For p = 4я +1, cos n is primitive, so
that all sets of real solutions of x1 + y1 = 1 (mod p) are given by cos tn,
sin to for * ^ 1, 2, •••, p — 1; the cases of coincidence are found. The
result is that for any prime 8ro ± 1, 8ro + 3 or 8?» + 5, there are m essen-
essentially different sets of solutions, provided 0* + V = 1 is excluded. The
same ideas are applied to the determination of the quadratic character
of 2, 3, 5;
G. Eisenstein*1 stated without proof that the number of all repre-
representations of an odd integer m as a ID is 8<т(т) [Vacobi*4^ and that, if
» Jour, fur M*ih.p 19,1839, 93-110.
*Jbur. fur Mftth., 35, 1847,133; Math. Abh&ndhingen, 1847,193. Id Jour, de Math., 17,
1852, 477, the ftret result is Bttid to follow from & property of ternary quadratic form*.

288 History of the Theoky op Numbers. [Chap, viii
m = a*Jp ... f where a, 6, -are distinct primes, the number of proper
representations is
P, L. Tchebychef*1 proved that i? — Ay* — В = 0 (mod p) is solvable
if Л is not divisible by the prime p. Proof is needed only when p > 2 and
Ay* H- В is never divisible by p, whence
(Л^ + В)^1)Л + 1 = 0 (modp).
This congruence of degree p — 1 is not satisfied by all the virtues 0,1, • • -,
p — 1 of y, so that for one of them Ay* + В is a quadratic residue of p.
F, Pollock" noted that if any odd square 1W ± 8n -b 1 is increased by
3 the sum is 3Dn* =t 4n + 1) 4- Dn* =F 4n 4- 1), and hence is the sum of
four odd squares. By adding also 8, the new sum is divisible into four odd
squares, with a like result for each addition of 8* He stated that every
number 8A + 4 is reached in this way* Since every number $k + 4 is thus
a (D, Bachet's theorem is said to follow.
C. Hermite* showed that, if Л is odd or the double of an odd number,
A2) a* + 0* + 1 = 0 (mod A)
has integral solutions. First, let A = t (mod 4), * = ± 1- The arith-
arithmetical progression with the general term 4Az H- 2«A — 1 contains by
Dirichlet's theorem an infinitude of primes, each = 1 (mod 4) and hence
the sum of two squares a* + /P. Next, let A = 2 (mod 4); we employ
similarly the progression 2Az + A — 1.
For integral solutions a, 0 of A2), the definite form
/ = (Ax + az + fiu)9 + (Ay — $z + аиJ + «* + «>•
has as the numerical value of the invariant Д the value A* (being the product
of the square of the determinant A2 of the four linear functions by the value
1 of Л for the sum of 4 squares) and hence its minimum for integral values
of the variables xf • -, u is < ($)*пА1* < 2A. Since / represents only
multiples of A, the Tninfrnmn is A itself. Thus A can be represented by/
and hence is a sum of four squares.
Hermite3* repeated the preceding proof and gave the following* The
form
jf = A{i* + tf) + 2ct(zx + yu)+ 2p(xu - ly) + j (a1 -h /3* + 1)(И + *»)
has integral coefficients, and A = 1. Hence it is equivalent to
Xх + У1 + Z1 + C/»,
the single redueed definite quaternary form with Д = 1. Hence in the
four linear functions Xy - • •, V of xf • * -, «, the sum of the squares of the
coefficients of a; or of у equals A.
u Tbeorie der Congnieaxeii, in Rueaian, 1849; in Gennan, 1889, 207-9,
* Ptoc. Roy. Soc. London, 6,1851,132-3.
»CompteeRenduflPHjie,37r 1863,1334; Oeuvre, I, 288-9.
«Jour, fur Math,, 47, 1854, 343-5, 364^8; Oeuvree, I, 234-7, 258-263.

Chap. VIII3 SUM OF FOUR SQUARES. 289
For M an odd integer, the Hermitian form
with complex integral coefficients, has for the invariant A the value — 1,
and hence is equivalent to vv + ъщ the single reduced form with A = — 1.
Let the latter be transformed into the former by
v = aV + bU, u = cV + dU, ad - he = 1,
ay • •*, d being complex integers. Then M = ad + cc, where a and с are
relatively prime. Thus any odd integer is the sum of four squares such
that the sum of two of the squares is prime to the sum of the two remaining
squares."
By considering the proper and improper representations of M by
w + v>uy he obtained Jaoobi'e formula 8П(р» + 1) for the number of repre-
representations as a sum of 4 squares of M = Rpif when M is not divisible by
the equ&re of a prime.
F. Pollock18 proved Cauchy^s theorem A813) that any odd number
2p + 1 is a sum of four squares the algebraic sum of whose roots is any
assigned odd number from 1 to the maximum. For, p is a sum of three or
fewer triangular numbers. If p = (q* + g)/2, then whether q = 2л or
2n — lf we have 2p + 1 = 4n3 ± 2n + 1, which is the sum of the squares
of n, -ft, ¥n, i(n± 1). If p = fa1 + q)/2 + (r1 + r)/2, then p is of
the form a7 + a + IP, and 2p + 1 is the sum of the squares of a + I, — a,
Ъ, — Ъ. If p is the sum of three triangular numbers,
m),
2p H- 1 = 2(a4 + a + b1) + 4nl ± 2n + 1,
the latter being the sum of the squares of &тп, — Ь ^F я, ^ а ± n,
a±n + l. In every case the algebraic sum of the four roots is unity.
A. Genocchi3* "recalled" (without reference) formulae G) and (8)
and noted that the second implies that the number of representations of
4n as a ID is *(n) when n is odd, and that the first implies
where Di is the sum of the odd divisors of щ A (or £4) the sum of the even
divisors d of n with n/d odd (or even), while Nlf • ■ •, NA is the number of
solutions of x\ + • • • + ^ = n with 3, 2^ 1,0 unknowns zero. For another
similar formula see Cesaro30 of Ch. IX.
A. Desboves40 stated empirically that the double of any odd integer is
a sum of two prunes An H- 1- Such a prime is a El» Hence every integra-
integrals а ЕЕ.
" E. Ficaidj the editor of HermRe'e Oeuvrai, 1, p. 259, noted thai when a And с ate relatively
prime, ид and cc are not десеетагйу во; but that the theorem in the text U probably true.
* Phil. Тпшя. Roy. Soc. London, 144,1854, 311-4.
» Nouv. Ann. Matb.t 13,1854,100.
«N«Y. Ann. Math,, 14,1856, 293-6.

290 History of the Theohy of Numbers,
C. A. W. Berkhan41 decomposed the integers < 360 into four rational
or integral squares, and into two or three squares if possible»
G. L. Dirichlet42 gave a simplification of Jacobi's50 proof» According
as a factor a' of p' = a'q' has the form km + 1 or 4m + 3, set 5' = + 1 or
— 1. Then the number of positive solutions of 2p' = v? + x* is ZB\
Hence each couple p', p" furnishes 2tf'-2S" = 2?> solutions of A1), where
ij — 4- 1 or — 1 according as a* — a" is or is not divisible by 4. Thus
fi = 2i)t obtained by varying also p'f p", so that there is a term tj for each
set of odd solutions a\ a" of
A3) a'q' + a"q" = 2p.
Let у' be a term obtained when a' = a", q" one when a' > a". Then
д = 2*' H- 2zy. From one set of odd solutions of A3), we obtain the
new odd solutions
A' =*"(*+D+*'(* +2),
Qf = - a'x 4- a"(x + 1) = a" - {a! - a")x,
G" = a'{x + 1) - a"(x H- 2) = (a' - a")(s + I) - a".
Let a' > a". In order that Q' and Q" be positive, (a' - a")x must be the
least multiple of a' — л" less than a". Then x is uniquely determined
and Af > A" > 0. If we repeat the process, starting with A\ Q\ A'\ Q",
we obtain merely the initial set a't q\ a", q*\ since the preceding equations
hold after the interchange of a' with A', q' with Q', etc. Since
a' - a" = Q' + Q"f
two such sets of solutions give values of rj" differing in eign. Indeed, one
and but one of the even numbers a' — a" and qf + q" is divisible by 4,
since af = ± a", q' = =F q" (mod 4) contradicts A3). Hence Xtj" = 0.
Thus м ~ Zti', with each V = H- 1, so that p = К[а'№ H- 5'0] = ^(p),
as above. (Ж Pepin»7*
J, J. Sylvester" employed the lemma that, if 3M = p> + ^ + r1 + «2,
Af is a sum of four squares. We may assume that p is divisible by 3 and,
by a proper choice of the signs of 3, r, s, take q ^ r = s (mod 3). Then
M is the sum of the squares of the integers
\{q + r H- «)p |(p + r - в), \ip-q + s), t(p + q- r).
For N = 1 (mod 4), the function 3Ij*W - 2 of 2 is not rationally de-
decomposable and has no constant divisor; it is assumed to represent a prime
T for some integer x. Since T = 1 (mod 4), T is the sum of two squares.
Hence T + 2 = З^ЛГ is the sum of four squares. The same is true of N
by the lemma.
For N s 3 (mod 4), 3*W - 2 is employed similarly. For N even, it
suffices to treat N = 2 (mod 4), by use of 3*tf — 1, since the theorem is
true for 4N if true for Л^
« Lebrbuch der Unbeetimmien Analytic, ШШе, 2,1856, 2S6.
- Jour, de Math., <2), 1,1856, 210-214; Werke, 2, 1897, 201-8.
« Quar, Jour, Math., 1,1857,195-7; Coll, Math. Fapei», 2,190S, 101-2.

Chap. V1IIJ SUM OF FOUK SQUAKES. 291
J, Iiouville" considered an integer m all of whose prime factors are
= 1 (mod 4). Express 4m in all possible ways in the form (u2-b p*)(t*J + vf),
where «, > > >, Vi are odd and positive, and call two such decompositions
identical if and only if и = u\ — >, i»i = vj. Denote the first factors2 + f*
by 2a, It is stated that Za equals the number of decompositions of 16m
as a product of two sums of four positive odd squares. The latter number
exceeds 2a if m has a prime factor s 3 (mod 4).
Liouville* considered tbe N representations of a given even integer я
as a sum 8* + t\ + u\ + t^ of four squares, where sif - -, i\ may be positive»
negative or zero, and two representations are distinct unless Si — s7f • • *,
vi = vt> For the first squares s], we have
gif = о (д odd), E*J = J jv, gtj - ^jv.
The second follows from nN = 2$J + - • • + 2tr* and 2s* *= SfJ, etc. The
third was verified for small values of n [proved by Stern81]» By means of
it and nW = 2& + -. - + v\J, we get 2£J *% = nW/24.
J, G, Zehfuss46 noted the identity
F. Pollock4* stated that any odd number is the sum of four squares
the roots of two of which differ by any assigned number d from zero to the
maximum. For d = 0, we use a2 -f- V + 2c2 (Legendre, Theorie des nom-
bres, I, 186; П, 398). Next, let d « 1. Since 4n + 1 is a sum of three
squares, only one being odd,
4n + 1 = BaJ + B6)* + Bc + I)8,
2n + 1 =* (a + by + (a — by + с8 + (с -f- IM*
Tbe case in which a* is general is discussed by means of a special arithmetical
series with the general term 2n2 + 1>
C. Souillart48 proved Euler's formula (I) by multiplying
abed
— 6 a — d с
— с d a — 6
— d — с b a
by the similar determinant with p, q,r, 8 ад first row.
F. Pollock49 stated that every odd number is a sum of the squares of
a -j- p _j_ i? a — j>y a + $, a — q, the sum of two of which exceed the sum
of the remaining two by unity; also is a sum of four squares the sum of
whose roots is unity,
« Jour, de Math., B), 2,1857, 361-2.
«Ibid., B), 3, 1868r 357-360,
«Arehiv Math, Phye., 30, 1858, 466.
«* Phil. Tfcane. Roy. Soc. London, 14fl, 1859, 49-59.
"Nouv. Ann. Math., 19,1860,321.
" Pbil. Ttsda. Roy. Soc. London, 151,1861, 409421.

292 Histohy of the Theobt of Numbers* [Ckap. viii
J.IiouvilleWa proved that the number of representations Ьуз^^
of an odd number m is {4 Ч-2(— 1)<т-1>я}г(т)9 of 2m is 12<r(m), of 4m is
8o-(m), of 2cm(<*^$) is 24o-(m). The number of representations by
Xs H- 4t/2 + 4z* -f 4^ of m = U + 3 is zero, of *n = 4J + 1 is 2<r(ro), of 2m is
zero, of Am is 8r{*n), of 2"т (a ^. 3) is 24*(то). He found also the number
of proper representations by these forms. He4** expressed the number of
representations of 2лт by a? + aj/2 + Ъг* + 16*2 for (a, 6) = D, 4), A6, 16),
D, 16), A, 16), {1, 4), A, 1), in terms of <r(m) and S(- 1)(-d>% summed
for the odd integers i for which m*= i2 -f- 4s*, From Jacobi's M result, h**
derived also the number of representations by x* 4- У* + 9z* + 9^.
J. liouville50 considered an odd integer m and the decompositions
4m = * + tj + г* + 4 2тл = r2 -h rj + 4*3 + 4ef,
where г, гь ity it, r, rx are positive odd integers, and stated that
J> Plana" proved Jacobi's" formula
A + 2g + 2g* + 2g' + • • •)* e 1 + 8S^(p)(gp + Sj* + 3g«» + • • 0-
H. J. S. Smith63 discussed Jaoobi'sMT я theorems that the number of
representations of an odd number m as а Ш is 8cr(w); the number of
representations of 4m as a sum of four odd squares is Z6<r(?n).
F. Pollock" stated that the algebraic sum of the roots in some repre-
representation of a given odd number as a 33 will equal any assigned odd number
not exceeding the maximum; that the difference of some two of the roots
will equal any number not exceeding the maximum. But all that is defi-
definitely proved in this paper, dealing with numerical statements, is that
any number n is a sum of four triangular numbers, since Bachet'e theorem
gives
An -h 2 - Ba + iy + Bb + IJ -h Bc)* + Bd)\
n = (a2 + a + с*) + (Ь2 + Ь + tf).
V. Bouniakoweky" employed the known result that the quadratic
residues of a prime p = 4n + 1 may be paired so that the sum of a pair is p,
and likewise the non-residues, to obtain relations like
10* + II1 = 22 + 32 + 8* + 122, 6* + V = V + 2* + 4s -h 83 (p = 17),
13s ~ I8 + 5s + V + 12% 13* + 14> = 1* + 3» + 173
[the first from ^H-^^lS^ + l^s^l + l (mod 13) J
«• Jour, de Mftth.» B), 6,1861» 440-8» Cf, Lmuirffle* of Ch. XI»
** Ibid., {2), 7,1^73-6,77-80,105-8,117-20,167-60,165^
ш1Шч B)r 10,1865,14-24.
« Jour, de Math., B)v 8,1868, 431-2.
u Mem» AocacL Turin, B), 20,1863,130.
* British Aeeoc. Report, 1865,337; CoU. Math. Ibpere, I, 307,
« Pft>c Roy. Soc. London, 15,1867,115-137; 16,1868, 251-4; abutoct of Phi]. Tmne., 158,
1865,627-642. Hie "pxiuf" of BechetV theorem ie given in Ch. Lm
« BuB. Acftd. Sc. St. PtttariMxug, 13,1869, 25-31.

Сна*, vni] Sum of Fotjh Squares. 293
F. TJnferdinger" denoted a* -f- b: + c* + & by Sa2 and expressed
ХаР-Ъа] - • • 2a*_j algebraically as a SI in 48Л~* ways, different in general.
E. Lionnet stated and V. A. Lebesgue* proved that every odd number
is a sum of four squares of which two are consecutive. For, in -f- 1 is а ЕЕ,
necessarily 4$* + 4т2 + Bs + II, whence
2n + 1 = (q + rJ + (q - rI + s* + (« + l)a.
J. W. L. Glamher" noted that, by an identity in Jacobi's Fund. Nova,
48a + 24a, + 12a,, + 8a, + 2a< + 240 + 1^, + 40, + 67 4- Э71 + *
equals <r(JV) if tf is odd, and 3<r(JV) if Л^ is even, where a, ah ««, 03, cu is
the number of ways tf is а вит of four squares all distinct, two equal, two
pairs equal, three equal, four equal, respectively, while 0} рг or ft is the
number of ways iV is a sum of three squares, distinct, two or three equal,
and у, у*, $ are the analogous numbers for two squares and one square.
& Bealis" employed 8n + 3 = Ba - IJ 4- Bb - II + Bc - II to
show that 2n -h 1 is the sum of the squares of
whose sum is unity, where, ife^a4-&H-ck even, the upper signs are
chosen and к = 0, while if s is odd, the lower signs are taken and k = \>
More generally, every odd number N is a sum of 4 squares, the algebraic
sum of whose roots equals any odd Dumber < 2 ^N, Any number
N — An + 2 is a sum a1 + 61 + c1 + k\ where A2 is any chosen square < N;
for, according as & is even or odd, N — k* is of the form 4p + 2 or 4p + 1
and hence a SL Also ^Zehfuss4*!
# = 0*4-0*4- 7s + *, 2а = а + Ь + с-|-*, 20= -а + Ь-c + Jfc,
27 = — о — o + c + fc, 2Л=а-Ь — е + к9 аЧ-0Ч-тЧ-Л = 2&,
Hence every number N «= 4n + 2 is a sum of 4 squares the algebraic sum
of whose roots is any assigned one of the numbers 0, 2, 4, • • «■, 2/*, where y?
is the largest square < N. Every number JV = 4n -h 1 (or 4n + 3) is a
sum of 4 squares one of which can be chosen arbitrarily among the even
(or odd) squares < N,
GlaisheV*' expanded Gauss' proof of F) and gnve an arithmetical proof
by showing that, if N is odd, the number of representations of 4JV as a
sum of 4 odd squares equals double the number of representations of N
as a sum of 4 or fewer squares.
E. Catalan8* attributed to J. Neuberg the identity
{a1 + V -h c* + be + ca + ab)* - (a -h 6 + сУ(
g-, Akad. Wise, Wvm (Math,)» 59, IIr 1869» 455-464.
» Nouv. Ann. Math., B), 11, 1872, 516-9; nme by iWalie,"
u British Aaaoc. Report, 48t 1873,11 (Trane. Sect).
« Nmiv. Ann. ШИц С2), 12, 1873, 212-23.
и PhiL M«. London, D), 47,1874, 443; E), 1,1876, 44r-7.
* Nouv, Gorresp. M*th., 1,1874^5,154-5*

294 History of the Theory of Numbers. [Chaf.viii
Hence, by a change of notation,
Since every odd number is of the form P + 2g* + AJ, every odd square is *
sum of four squares.
S. Realis*1 used A) to show that, for any integer p>
and that we can find four integers whose algebraic sum is p and the sum of
whose squares is p*.
Catalan82 gave the identity
y + [aZf(hy - c0) + (by -
expressing a product of three Й) asa Ш.
lis* noted that, for every odd integer pf
the algebraic sum of three of P, • • •, S being a square. For,
P = * + У2 + 2г*
= (x + z)(x -f) + (« + ^}(z + p) + (x + z){z - tf) + ^
For, '
p = Ш ^ {as* - K2) + fa* - да) + (г2 - я») + (ад + xz + ye).
G. Torelli" proved by means of Jacobi's25 theorem the result (I) that
if 2n — 1 is not divisible by 3 and if p, q are respectively the numbers of
sets of distinct odd integral solutions, not all divisible by 3, of
2я2 + t + г* = 36<2л - 1), x* + y* + a* + ** = 36Bл - 1),
then p + 2q is the sum <rBn — 1) of all the divisors of 2л — 1. (II) When
the second members are replaced by 4>3mBm — 1), then
p + 2q = 3VBro - 1).
(Ill) If & is a prime 12X — 1 and if 2n — 1 is not divisible by k, while p, q
are respectively the numbers of sets of distinct odd integral solutions not
all divisible by к of
2x2 + y* + ** = AkT{2n - 1), хг + р* + ** + * = 4ГBп - 1),
then p + q = A^VBn - 1), (IV) If ЛГ = а'У1 ■ >, where a, b, • - - are
distinct odd primes, AM is a sum of four odd squares without a common
e Nouv. Ann» Math., B)» 14,1875,90-91.
* Nouv, Согтеяр. Math., 4,1878, 333, foot-note.
"Noirr, Ann. Math., B), 17, 1878,45»
* Giomale di Mat,, 16,1878,152-167,

Chap. VIII] SUM OF FOUR SQUARES. 295
factor in M(l -\- l/a)(l + 1/6) - - - ways. 00 If t*i, Pi> Vt are the numbers
of sets of distinct integral solutions not zero of
*■ = 2Bn - 1), **+ *,* + ** + ? - 2n- 1,
then f! = 3pi + p,. (VII) If x3 + y* + ^ + ** = 4Bn - 1) has $x sets of
distinct odd integral solutions and xi + y2 + z*=2n — 1 has p4 sets of
distinct solutions Ф 0, then Si = 2pi + p<. (IX) If Z4 denotes x2 + y*
+ 2* + t\ the number of sets of solutions of 24 = 2*Bn — 1) is expressed
in terms of the numbers of sets of solutions of 24 =•= 2n — 1 and 58 = 2n — 1
and the number of sets of solutions when two or three variables are equal.
E. Fergola" had stated the preceding theorem (V), and (I) with the
restriction that 2n — 1 is not a square.
E. Catalan" noted that 2p ~ a + & + с implies
p? + (p - ay + (p - by + (p - с)г - с1 + Ъ* + с2
and gave various identities in «, ft, c, which express the square of the sum
of three squares as a 3L
J. J. Sylvester" proved that any prime p is a divisor of s* + y2 -+■ 1.
Assume the contrary» Then p ф 4г + 1 since p does not divide z* + 1.
Let p be any primitive pth root of unity and set R - Zp**, summed for
the quadratic residues ж3 < p. Let R' be the period conjugate to R.
Expand R? as a sum of powers of p. Since p Ф 4i + 1, дс2 H- v2 + P and
no pth power of p can occur in the expansion of R2. Since, by hypothesis,
neither 2s2 nor x^ + #* is = *- 1 (mod p), no such power as p*~l can appear
in #*, while it belongs to R\ Thus no term of R' appears in i?2. As each
power of p in Й* belonging to the same period must appear a like number
of times, we have R2 = R(p — l)/2, whereas Д Ф 0 or (p - l)/2.
From this theorem follows Bachet's theorem. A similar proof shows
that At? + By2 + Сг* ^ 0 (mod p) is solvable.
H. J. S. Smith68 indicated a proof of Bachet's theorem by continued
fractions.
C. Hermite" proved F) by elliptic functions and concluded that the
number of decompositions into four squares of any odd integer n equals
8 times the number of decompositions of 4n as a sum of four squares whose
roots are odd and positive. Cf. Jacobi.*
J. W. L. Glaisher*0 considered the v(N) compositions (allowing permuta-
permutations) of 4N as a sum of 4 odd squares, took the square root of the first
square (for example) in each such composition, giving it the sign rt accord-
according as it is of the form 4wi zb 1, and formed the algebraic sum A of these
square roots. Next, consider the compositions of 2N as a sum of 2 odd
squares, take the product of the square roots of the two squares in each such
• Gioraale di Mat., 10,1872, 54.
« Nouv, Согпяр. Math., 5,1879, 92-93.
« Ашег, Jour, Math., 3,1880, 390-2; Coll. Matb, Р&регя, 3,1909, 446^8.
о CriL Math, in memorism D. Chelini, Milan, 1881,117; Coil. Math. Papers, II, 309.
rtCoure,Fic.Sc. Paris, 1882; 1883,175; ed. 4,1891,242.
"Qu*r. Jour. МвОь, 19,1883, 212-5; 36,1905, 342-3.

296 History of the Theory of Numbers. [Chap, vui
composition, determine the sign as before, and form the algebraic вши
В of the products. Then A = B, as shown by use of infinite series and
products»
Б. Catalan71 noted that
T. Pepin71 gave a purely arithmetical proof that the number of repre-
representations of m as a ffl is 8B + (— 1)~}X(*&), where Xtyi) is the sum of
the odd divisors of m. The proof is Hke that by Jacobi* and Dirichlet.42
Pepin7* gave an exposition of this proof by Dirichlet and noted (p. 173)
that the theorem is a special case of one by liouville; he proved (pp.
176-184) the theorems of Jacobi.M
M. Weill* aoted that Jacob! deduced from the formula k7 + k* = 1
in elliptic functions the result that, if N is odd, the number o! representa-
representations of 4N as a sum of 4 odd squares is double the number of representations
of^asaS], and gave a direct proof by means of the identity of Zehfuss.4*
By a similar identity, Weill proved that if N is any integer not divisible by
3, and if N and ZN admit only decompositions into four distinct squares
Ф 0, the number of decompositions of 3N as a ffl is double the number of
those of N.
G. Frattini7* proved that the number of pairs of squares for which
x* — Dy* s= x (mod p) is \\p — (D/p)}, where (Dfp) is the quadratic
character of Z> with respect to the prime p. There is given an elegant
proof, due to Biancbi, of the existence of solutions if p > 3. If X is a
residue, take у = 0. If X is a non-residue, it ie shown that, when a ranges
over the (p — l)/2 residues, a — X is not always a residue and not always
a non-residue. For, if e *= (p — l)/2 and every root of x* ^ 1 satisfies
(x — X)' ss ± l, it satisfies (x — X)« — Xя = 0 or — 2, whereas the degree
is less than the number e of the roots.
J. W. L. Glaisher™ used the term partition (resolution) of N as a sum
of squares when we disregard the order in which the squares are placed
and the signs of the roots; composition when the order of the squares is
taken into account, but not the signs of the roots; representation when both
the order and the signs are attended to. For N odd, %{N) denotes the sum
of the square roots of the distinct squares appearing in the various partitions
of 2N into two squares, the sign + or — being prefixed to each root accord-
according as its numerical value is of the form 4n + 1 or 4n + 3. An equivalent
definition (p. 98) ie that x(N) is the sum of all the primary complex numbers
a + Ы of norm N = a1 + 6*. Two odd squares are said to be of the same
class if and only if both are of the form (8n ± I}2 or both of the form
" Nouv. Ann. Math., C), 3, 1Ш, 347.
« ЛШ Aocad. Font. Noovi limreL 37,1883-4,12-20.
" /№, 38,1884-6, 140-6.
» Compfae Bendw Fbv, Щ 1884, 859^861; BuIL Soc Math. France, 13, 1884^5, 28-34.
» BexMftoonti ВЫе Aocad. lined, D), 1,1885,136-9.
* Qu*r. Jour. MaUl, 20,1885,80-167.

Chat. VUl} SUM OF FOUR SqUABES. 297
(8n d= 3)J. The following theorems were proved by use of infinite eeriee.
If N = 4n + 1 and if tf i (or Ht) denotes the number of compositions of
4N aa a sum of 4 odd squares of the same class (or not of same class), then
Hi-iH^xiN)* Asi known, Я1 + Я1 = <г(ЛГ). If N = 4n + 1 and И
of the partitions of 4ЛГ into 4 odd squares of which two are equal, P is the
number having the remaining two squares of the form (8л d= 1)' and Q
the number for which they are of the form (8m rfc 3)*f then P = Q if N is
not a square, while
Write £ for Bp + IJ + Bq + 1)*; the number of representations of
8n + 2 ae £ + Dr)J + DaI or S + Dr + 2)* -f Da + 2)* is respectively
12{erDn + 1) + xDn + 1)}, 12[*Dn + 1) - x<*» + 1I;
while there are 12<rDn + 3) representations 8n + 6 = 5 4- DrJ + Da + 2)^
Let B(N) denote the excess of the number of divisors 4n + 1 of N
over the number of divisors 4n + 3; then E(N) is the number of primary
numbers of norm N. If ft = 1 (mod 4),
x(n) = E(l)E{2n -1)- EE)EBn - 5) + E(9)EBn - 9) - ■ -
- 3) +£(9)£Dm - 7)
+
Call Ег(п) the excess of the sum of the squares of the divisors 4m + 1
of n over the eum of the squares of the divisors 4m + 3; X(n) the sum of
the squares of the primary numbers of norm n* There are given many
formulas serving to evaluate x, *, Я, Eit X, whose values are tabulated
for arguments n ^ 100, with citation to longer tables.
R. Lipechits77 found the number of sets of solutions of tf + Й + Й = 0
(mod p7), where p is a prime, and applied the result to find all integral
quaternions with a given norm and hence the solutions of m» Ш. He
discussed the real and rational automorpbs of x] + x\ + x\.
S. Reahsr* coneluded from pq ~ a1 + ■ ■ ■ + & three sets of fractional
expressions for p and q in terms of <*, • ■ •, Ь and new parameters, but ad-
admitted that he was unable to utilize them to prove Bachet'e theorem.
A. Puchta7* repeated Gauss117 derivation of Eider's formula A). To
interpret A), use the four-dimensional regular body bounded by 5 tetra-
hedra and having as vertices b equidistant points Pi, There exists a point
0 such that 0Ph - - -, OP* are perpendicular Елее, ^bile the " planes "
through 0 and any three of Plt • - -1 pA are perpendicular. We may take
О to be the point with the coordinates *i = (<*i + (h + Л1 + <**)/2, etc.,
n Uutenudiungen flber die Summen von Quad»tent Bonn, 1886. French trend, by Jt
Molk, Jour, de Math,, D), 2, i$S6t 393-43».
»■ Jour, de math. O6m,, B), 10,1886,80-91.
" Sfbr. AJmd. Wise. Wren (Math.O ^6> Ht 1887,110.

298 History of the Theory of Numbers. [Chap, viu
and get the identity 2а\-Ъх] «= Г**, where
etc. By permuting the a'e or changing the signs, we get 96 formulas A).
E. Catalan80 made an invalid criticism of Legendre'e" proof that every
prime is а ЕЕ, who is said to have assumed that every integer N bas a prime
divisor > iiN. Catalan's remark (p. 164) that if N and A are sums of
four integral squares, their quotient N/A is a sum of fractional squares,
was known to Euler.8 Catalan proved that every integer is a sum of four
fractional squares in an infinitude of ways and stated (p. 212) that every
number Ы + 4 is a sum of four odd squares of which two are equal.
M. A. Stern81 gave an elementary proof of Jacobi's*4 theorem. Let m
be odd. The number of representations of 2m as а Э! is three times that
of m, since m = p* + g* + P* + ** implies
2m =
Conversely, if 2m — a* + 0s -f- 7* -f #, two of the squares are even and
two odd, во that
Repeating the process, we get £cf. Zehfuss46]
4w - {2Py + B^)s + BrJ + Ba)*,
A4) 4m = (p + g + r ± sJ + (p + q - r =F eJ
Conversely, 4m = 2a2 implies 2ш = {J(a + 0) }* + ■ —. Hence 4m and
2m have the same number of representations as a 3L It is shown that if
Tm and 2я+1т have the вате number v of representations, then 2lm(t ^ a)
has f representations. If m — 4fc + 1, three of the numbers p, q, r, a
are even and the fourth is odd, so that the squares in A4) are all odd. If
m = 4& + 3, three of p, $, r, s are odd and one is even, and the preceding
conclusion holds. By Jacobi's30 theorem, there are 16tr(m) representations
of 4m by four odd squares» Hence if pqrs # 0, there are &r{m) representa-
representations of 4m by four even squares and hence 24o-(m) representations in all»
This result is proved to hold also if pqrs = 0» Cf. Vahlen.88
T. Pepinw proved Jacobi's* theorem that, if m is odd, the number of
decompositions of 4m as a sum of 4 odd squares with positive roots is
ff(m), by taking I ^ */2 in a formula involving sums of sines of multiples
of L The number of representations of 2m by хг + уг -f- 4^ + 4& is 4r(m).
The number of representations of 2m by хг + v2 + & + ^ or ж1 + уг + г1
-h 4?, with z + y = l (mod 2), is 16>(m) or 8a(m) respectively. He gave
w Mem. Soc. Roy. Sc. de Ubgt, B)t 15, 1888, 160 (Melange* Math., Ш). " *"
n Jour, fur Math., 105, 1889, 251-262»
"Jour, de Math. ,D), 6, 1890,10-20.

Chap. VIII^ Sum OF FotTR SQUARES. 299
various theorems on the representations of 2km by forms
I + (A)* + B4p)«.
E. Catalan" noted that, if к - 2tf + 3, к is a 31 and кг a IS.
A. Matrot84 duplicated in essence the proof by Euler10 except as regards
the theorem that every prime p divides a sum of 2 or 3 squares. Let
p ~ 2ft + 1. Consider the couples j, 2Л - j (j =- 1, ^ , ft - 1). If both
terms a, «i of some couple are quadratic residues of pt a — Л2, <*i = Af,
Л* + A? + 1 =i 0 (mod p). But if no couple is composed of two quadratic
residues, the number of residues contained in the couples is ^ ft — 1.
Hence one of the numbers ft, 2Л, not lying in a couple, is a quadratic residue
(there being A such). If ft ~ A\ A* + A1 + 1 = 0 (mod p). If 2ft = A\
A* + 1 =0 (modp).
E. Humbert85 proved that if p is odd and Ф 3, 9, at least one of the
numbers i(p -f 1), %(p + 3), — -, p — 1 is a square. Hence if the abso-
absolutely least quadratic residues of a prime p > 3 be arranged ш increasing
order of numerical value, the series contains negative terms. Hence if
p = 4n -f- 3r there exsits a positive residue a followed by the residue — a — 1.
Then а = ^*а-1^^;^Ч^Ч1^0 (mod p).
R.R Davis" noted that, if s = a + 6 -f с + diseven,a2 + b* + c^ + d*
is expressible as a sum of four new squares by means of the identity of
Zehfuss4* (divided by 4). If я is odd, add ?n* to each member and transform
into a 3L R. W. D. Christie made use of various formulas expressing a Gfl
as a G2 after proper selection of three of four squares.
A> Matrot** noted that, if p = 2h -f- 1 is a prime, we can rind two
consecutive integers ct and a -f- 1 satisfying xK ^ 1 and xh s- — 1 (mod p)y
respectively» For, otherwise 1, 2, • f p — 1 would all satisfy the first.
Hence
*m + (a + l)m + ls«^(« + l) + IsO (mod p).
For p = 3 (mod 4), A + 1 is even, and p divides a SL His proof that
every prime p = 1 (mod 4) divides a G3 was quoted under that topic.
K. Th. Vahlen88 gave essentially the same argument as had Stern.*1
His proof of Bachet's theorem is given in Ch. VII.74
E. Catalan" gave Legendre's16 proof of Bachet'e theorem. Euler8 gave
the empirical theorem that an integer is not a sum of four fractional squares
unless it is a sum of four integral squares. This is said to be false since
every integer is a sum of four fractional squares in an infinitude of ways.
« Aeeoc> fran?. av. ec., 20,1891, II, 198.
" Аваос. fiang. av. не, (Ыйюкпеб), 19, 1890, II, 79-81 [20, 1891, II, 185-191 for biHtorical
remarks on tbe proofs by Lagrange and Euler J; Jour, de math, elem», C), 5, l89l,
169-74] pamphlet, Paris, Nony, 1891. Reproduced by E. Humbert, Arithmetique,
Paris, 1893,284, and by G. Wertheim, Zeit. Math. Naturw. Unterricht, 22, 1891,422-3.
«BulLdteSc. Math., B), 15,1, 1891,51-2»
* Math. Quest. Educ. Time», 67,1892,120-2,
ет Jour, de math, diem., D), 2, 1893, 73-6.
"Jour, fur Math., П2,1893, 29.
" Mem, Acad. Hoy. Sc. Belgique, 52,1893-4, 22-28.

300 History of the Theory or Numbers. CChaf. viu
F. J. Studnicka" noted that Euler's A) includes the formula of Cauchy,**
and deduced the like formula expressing a product of three sums of 3
squares as a 3L
I* Gegenbauer" proved new expressions of Jacobi's theorems. The
number of representations of an odd number nasa Щ equals 8 times the
number of divisors of the various g.c.d.'e of n with the numbers ^ n; also
equals 8 times the sum of the products obtained by multiplying the number
of divisors of every factor of n by the number of integers not exceeding the
complementary factor and relatively prune to it. The number of proper
representations of an odd number n as а ЕЕ equals 8 times the number of
decompositions, into two relatively prime factors, of the various g.c.d.'s of n
with the integers ^ n; also equals 8 times the sum of the products obtained
by multiplying the number of decompositions of every divisor of n mto two
relatively prime factors by the number of integers relatively prime to and
not exceeding the complementary divisor.
B. Sollertinski*2 noted [Catalan"] that a SI is а Ш:
E< N. Barisien" noted that ^isafSife^a^ + y2, since
[We may conelude that & is a G3, not merely a S3.]
G. Wertheim* proved that every prime p divides a SI as had Matrot,"
and also by finding how often in the series 1,2, • ••, p ^ la residue follows
a residue, or a quadratic non-residue follows a residue.
Ik E. Dickson* exhibited all solutions of z* + y2 & 1 (mod p) and of
a^ + v2 - 0 (mod 5').
K. Petr* proved two formulas by Gauss (Werke, III, 476) on theta
functions by the method outlined by Gauss. From them are derived
relations giving the number v(N), $(N), $'(N) of representations of N by
z* + y* + 92» -|- 9tt«, & + v2 + * + 9«2> & + V + 92» + 9u\
respectively. Let х(Ю be the known number for four squares. Then
*W = klxQf) + 1G2(- iyw]x], N ф 0 (mod 3),
summed for all positive odd solutions of 3z* + y2 — 4N, For N divisible
by an odd power of 3, v(N) = 0; if by an even power of 3, <p(N) =
Also,
iVrф0
N m 0
M Prag Sitztmgeber. (Math, Natui-w.), 1S&4, XV.
11 Sitsungeber. Akad. Wise. Wien (MAth-), 103, Щ1 1894; 121.
» И Progreeo MateopAtioQj 4,1894, 237.
m 1л maiematiche pure ed applicikte, 1,1901,182-5.
M ^fangagrtinde der ZaUenlehie, Braunschweig, 1902, 396.
*Ашег. Matb. Monthly, 11,19(H, 176; 18j 1911, 43-4, 118.
n Prag Sttronffiber. (M&th. NftturwJ, 1904, No. 37, в pp.

Stjm of Four Squabbs 301
Now the third form represents N only if N is a quadratic residue 0, 1, 4, 7
of 9- But in these cases, the first form represents N only when x or у is
divisible by 3. Thus V(N) is zero except in the following cases:
ftfO = ЫЮ if N - 1,4, 7 (mod 9); *'(#) - х(ЛГ/9) if AT « 0.
Thus V and hence also ф is fully determined.
R. D. von Steraeck9* gave an elementary proof that every prime p
divides the sum of two or three squares, no one divisible by p. Let Rj
denote a quadratic residue and Nj a non-residue of p. If — 1 is a residue
of py a sum 1 + ** is divisible by p. If — I is a non-residue of p, there
exist two residues whose sum is a non-residue. For, if not, the sum of j
residues is a residue; in particular, jR = Rj (mod p), which is false when j
is a ndbresidue. From
R + R^N, -N^Rz (modp)
follows R + Ri + R* ^ 0 (mod p).
B. Bolaano" proved the existence of integers tf и such that
^-Би'-СеО (mod p),
В and С not being divisible by the prime p [Lagrange9]. For* = 0,1, >■ ■,
i(P - !)> its square £ takes £(p + I) incongruent values modulo p. For
w = 0, 1, ■ ■, *(p - 1), the sum Вч* + С takes }(p + 1) incongruent
values. Hence at least one of the first values is congruent to one of the
latter, since otherwise there would be p + 1 incongruent numbers modulo p.
J. W. L. Glaisher" noted that all the partitions a» + &* + т* + *' of
4m into 4 odd squares can be derived from the partitions a2 + b2 + c* + #
of the odd number m by the transformations [cf. Stem81]:
o = (iiHc + d, 0 = a=Fb-c + d,
7 = oTb + c-d, В = a±b — c — d.
A partition of trc produces twice as many representations of Am as of m,
and every partition of 4m can be derived from one of m by such a transforma-
transformation. Hence the number of representations of m as a 3] ie 8 times the
number of compositions of 4m as a sum of 4 odd squares. Here and later,
he100 made a further study of the function \(m) QGlaisher7*] and the related
functions P(m)t Q(m)f fi(m), defined as the sums of the products of the
roots (taken in the form 4л + 1) of the first 2, 3, 4 squares in each composi-
composition of 4m as a sum of 4 odd squares, X(m) itself being the sum of the roots
of the first square in the various compositions.
Glaisher101 applied elliptic function formulas to find the number of
representations of a number as а вит of four squares of which r are even,
for r ** 0, 1, 2, Z, 4.
Math. Phye., Щ 1904,
n Ibid., 237-8 (posthumous paper).
" Qu*r. Jour- Math., 36, 1905, 305-353. Extract! by P. ВасЬтыш, Nkdere Zahlentheorie,
II, 287-292, 319,
7^37,1906,3*4*.
1ШЧ 38,1907,8-9.

302 History of the Theory of Numbers. CChap. vni
A. Martin102 noted that, if t = 2p* + 2$* — n*t the sum of the squares
of t -f 4npt t — 4np, t + 4715, t — 4ng equals the square of 4p* + 4(f + 2n*.
Also [Aidaw of Ch. IX],
P. BachmannIW gave an exposition of papers by Glaisher,70 Dirichlet,4*
and Stern.81
L. Aubrylw proved that every integer N is a HI. It evidently suffices
to treat the case N odd or double an odd number. It is first shown that
N divides a certain X9 + Y* + 1, where we may take X ^ N/2, Y ^ N/2.
Consider therefore the numbers N\t N*, • • - defined by
X) + Y\ + 1 = NJTnu Xi - ВД Yt ^ NJ2.
The N*& form a decreasing series of positive integers. Hence a certain Nn
is unity. Then N^ - X^i + FJUi + 1- But if
X2 + Г* + 1 - DE, Ж = рг + q* + Iя + $\ - pX + rY + 9 = aE,
sX + qY + p =cEf qX -*Y + r = dE, rX + pY - q = ЪЕ,
then D = a2 + b2 + cJ + cP. Applying this theorem for p = 1, r = 0,
s - X*.!, g = F^j, X = X^2, Г = Г„_2, whence D - AV_* ^ = ЛГ^Ь
we see that iVn_^ is a 3]. By the same theorem we see by induction that
every Ni is a S3. Hence N = Ni is a HI. ^There is no explicit proof that
c, • •», d may be taken to be integers and hence that the decomposition is
not merely into four rational squares.]
E. Dubouis106 proved that Descartes* * statements are true. The
numbers not a sum of 4 squares > 0 are 1, 3, 5, 9, 11, 17, 29, 41 and
4nX (X = 2, 6, 14), n ^ 0.
S. A» Corey106 gave a vector interpretation of A) by use of four pentagons
with a common vertex and four consecutive sides in one pentagon parallel
to corresponding sides of the others.
С van E. Tengbergen107 proved that x* + у* + г* = О (mod p) has
(p — l)(p — &)/48 sets of solutions < p/2, where к ^ — 1, 5, 11, 17
according as the prime p = 8v — 1, 8t> — 3, Sv + 3, Sv -f- 1.
E. Landau108 proved that the number of sets of integral solutions of
u2 + t? + v? + y* ^ x is £Л: + 0(sl+f), for « > 0 and О as by Landau,1"
Ch. VI.
G. MetrodJ<* solved л* + {a? + J/J + (« + 2j/J + (* + 3y)« = zl for л;
the radical is rational if 2* — 5y2 = u* and hence if z =* a1 + 5b2, у = 2ofc,
w = a1 - 56*.
L. Aubry110 showed how to find all solutions of a* + b* + c? + cP — N,
first when a2 + б2, ac + Ы and N have no common factor, and next when
'« Amer. Math> Monthly, 16,1909,19-20.
1M Niedera Zablentheorie, 2, 1910, 286, 323, 34^-358.
i« Aaeoc .fnmj.! 40, 1911, Ir6l-6.
inb L*interm€diaire des math,, 18, 1911, 55-6, 224-5.
'"Amer, Math. Monthly, 18, 1911, 183,
w Wiskundige Opgavcn, Amelerdam, 11,1913, 244-7,
1M Gottiogen Nachrichten, 1912, 766-6. *
'« SfAinx-Oedipe, 8,1913, 129-130.
и* 1Ш., numgro special, March, 1914,1-14; errata, 39.

Yin] SUM OF FOUR SquaKES. 303
their g.c.d. is m, but a, ■ • -, d have no common factor. Combining numer-
numerous cases, he obtained Jacobi's" theorem on the total number of solutions,
and the theorem that, if N = 2"pf - p\ and a ^ 2, the number of
solutions in which a, ■ -,d have no common factor is
where A = lif<* = 0, A = 3ifa = l, fc = 2if«^2. He showed how
to find the 4я sets of solutions of хг -f- y* -f- 1 s 0 (mod p), where p is a
prime 4л ^ 1, also the solutions for any composite modulus.
L. J. Mordell111 proved by use of theta functions that the number of
solutions of х* + ^Ч2Ч^ = « is 8{26 — S(- l)'c}, where Ъ and с
range over those divisors of m whose complementary divisors are odd and
even respectively [equivalent to Jacobi's** result].
Mordell111 proved the conjecture by Glaisher (p. 48) on the derivation
of all representations of 4771^1 as a ffl from those of imi and 4m*.
A. S. Werebrusow113 gave the general solution of Э1 = 3].
L. E. Dickson114 gave a history of the proofs of Euler's7 formula A),
its interpretations and generalization to 8 squares.
For Pellet's proof that A& + By* + С = 0 (mod p) is solvable see
paper 104 of Ch. XXVI.
For minor results, see papers 12 (end), 31, 49,106 of Ch. VII; 13,26,30,
39, 52, 76, 84, 94, 95 of Ch. IX; 159 of Ch. XIX; 434 of Ch. XXI.
"i Mew. Math., 45, 1915, 78.
ш Ibid., 47\ 1918, 142-*.
i« Uiirterm&toure dee math., 25, 1918, 50-51; ertr. from Math» Soc. Moecow,
»" АшадЬ of Math., B), 20, 1919, 156-171, 297.

CHAPTER IX.
SUM OF n SQUARES.
Representation as a Sum of Five ok Mobs Squabes.
C. G. J. Jacobi1 remarked that a comparison of the sixth and eighth
powers of two series for BЯ/тIЛ would yield arithmetical theorems (for
that from the fourth powers see Jacobs * of Ch. VIII).
G. Eisenstein1 stated that he had obtained purely arithmetical proofs
of these theorems of Jacobi on the representation* of number* as the sum of
six or eight squares and stated the generalizations:
The number of representations of 4r + 1 as a sum of six squares is 12*
and that of 4r + 3 is — 20s, where 8 = Z(df — dj), dt ranging over the
divisors of the form 4k + 1 of the given number, d% over the divisors 4Jc + 3,
The number of representations of ад odd number as a sum of eight
squares equals 16 times the sum of the cubes of its divisors.
He stated that there is no analogue for 4r + 1 of the theorem that the
number of representations of 4r -f- 3 as a sura of ten squares is 122 (rfj ~ df),
Eisenstein' stated that, if m ie an odd number > 1 having no square
factor, the number \p(m) of representations of m as a sum of rive squares id
- 80e, — 80<r, — 112a, 80<r, according as m = 1, 3, 5, 7 (mod 8), where
.-ж-
the symbol being Jacobi's. For proofs see Smith18'tl and MinkowskL1*
Eisenstein4 stated that the number of solutions of xj -f- • • - -f- x) = m is
-16 -372 (^
282(- l)(it/l(-)j»Bm-M), ju odd and < да, ifm^l (mod 4);
\rn/
provided m has no square factor.
V. A. Lebesgue* discussed the decomposition of a prime p or its double
into m squares, where m ia a divisor > 2 of p — 1. Using indices relative
to a primitive root of p, divide the indices of e(a + 1) for a «- 1, 2, > • -,
p — 2 by ?» and let oo, Hi, * • -F o^j be the number of the indices with the
i Fundament* Nova Fuse. ЕШр., 1829, p. 188; Werke, 1, 1881, 239. a. H, JT 8» Smith,
CoU. Ma.tb. Papers, 1,1894, 306-И. a. J^oobi* of Ch. IIL
* Jour, fur Math,, 85,1847,135; Math. Abb., Berfm, 1847,195.
* One representation yields a new one if the root? of the equates are permuted or changed
in sign, while a composition is unaltered.
' Jour, for Mftth., 85, 1847, 368.
< Jour, fur Math., Щ 1850,180-2.
' Compfe* Rendufl Pane, 39,1854,
305

306 History of the Theory of Numbers, [Chat, ix
residues 0,1, — -, m — 1 respectively. Write о„н-< — a<- For m odd,
E a] - p = Zuiui+i = 2а.-ас+1 = - - = 2aiO1+m^i, 2p ^ 21 (&i — «ч-*)*>
when A; = 1, - ■♦,*?* — 1. For w even, 2а<а»+у — 2a^a»+* if j — A: is even, and
2p = E («> - а^к)\ km > к > 0.
Lebesgue* proved his preceding results,
LebesgueY noted that tables of indices lead to integers a, such that
V - /ОЖр'1), /(p) = «о + ад + ' • • + ^iP^1»
where p is a prime ты -j- 1, m > 2, Set
Then p* = F(p)F(p~1)^ Hence if in the decomposition of 2p into a sum
of тп squares we change д» into Aif we get a decomposition of 2pk.
J. Liouville8 stated that the number of representations of the double of
an odd number masa sum of 12 squares is 2642d5, where d ranges over
the divisors of m. The number of proper representations is 264Zfi(m), where
{) \ + } - -. \cny + <?<^l}\t m = аш& > - cy,
a, - • •, с being distinct primes. If D* ranges over the square divisors of mt
Liouville9 stated that the number of representations of 2*m (a > 0) fls
a sum of 12 squares is
summed for the divisors d of m. Proof by Humbert.4*
Liouville10 denoted hy N(nt py q) the number of decompositions of n
into p squares of which the roots of the first q are taken odd and positive,
while the last p — q are even and the roots are taken positive or negative
or zero; by N{n, p) the number of representations of n as a sum of p
squares. It is stated that
A) NBm, 12) = 26Ч[ЛГBш, 12, 2} + 224tfBw, 12, 6)
+ 25bNBmt 12, 10) j (m odd).
Let m be odd, d any divisor of m, В = m/dt and set
* Jour, de Math., 1», 1854, 2»8; B), 2, 1857, 152.
* 7WdfJ 19, 1854, 334-6; Comptee Rendua Paris, 39y 1854, 106S-71.
» Jour, de Math,, B), 5, 1860, 143-6.
* Ibid.t B)r 9, 1864, 295-8.
^/Wd., B), 6,1861, 233-8- Fnot by B«U «

Chap. IX] Sum OF FlVE OB MORE SQUAfiES. 307
The following formula is stated:
T 2),
The cases v = 1, v = 2 correspond to theorems proved by Jacobi,1 For
и = 3, A) gives NBmf 12) = 204f6(m). It is stated that
N(mt 12) = 8fi(m) - 16т'Г,(т) + 162^ = 24ft(m) - 2uNDmt 12, 12),
where 2*1 is the sum of the squares of the first terms in the various repre-
representations of m as a sum of 4 squares s1 + ej + si + &
It is stated that
4* + 2, 4i + 2), ft - h *. = ° fr > 0),
B, being independent of mt but dependent on f;
Po(m) = NBm, 2, 2), *,(*) - ATBw, 10, 2) + 6ЧЛГBш, 10, 6).
From the latter, NBm, 10) = 12-17*(m), when wi = 3 (mod 4). For
such an m, Eisenstein* had given N(m, 10) = 12p4(m),
Iiouville11 noted the existence of numbers oo = 1, ah - •, а„_! =* 16""\
[^=-1^ fti, - -, 6,.!, independent of m and «, but depending on yt such that,
for every odd integer m and every integer a ^ 0,
E 4r + 4, 4. + 4),
= EW2-+4 4^ + 2,4s + 4).
These results and those in his10 preceding paper hold also if N be replaced
by Mt where Af (w, p, q) is the number of solutions of
(i's odd and positive, a>'s even) for which ilf -, ш^^ have no common
factor, and if Г„ р, be replaced by
ZMA) = i2M«) - 1, m = ПР',
where P ranges over the distinct prime factors of m*
Iiouvillelle noted that, if m is odd, the number of representations of
2"+Sn by Q - tr2 4- 4(y* -f z1 -f ^ + u1 -h y8) evidently equals the number
4D*+1 _ (_ 1у**г-и(*\р1(т) of representations of 2*яг as a sum of six squares
(Jacobi1). The number of representations of n = 1 (mod 4) by Q is
pt(n) + г^г1 — лдо(п), summed for the odd integers г for which n ^ г*Ч-4Л
Corresponding results are found for forms like Q in which however only 4,
3, 2, or 1 of the coefficients are 4, and for & + My2 + г* + Я + «*) 4- ^
11 Jour, d* Math-, B), 6,1861, 369-37^
""• /Ы^ B), 10,1865, 65-70, 71-2, 77-80,151-4,161-8, 203-8.

308 History or the Theory of Numbers. ССяар. IX
Liouville12 stated that the number of representations of n = 2*ти (тл odd)
as a sum of 10 squares is
where X is the positive value of Z(d\ — (Й), where d\ ranges over the divisors
41 + 1 of n and d* over the divisors 4J + 3 (Л being the same for m as for n),
while ^ is the number of integral solutions, positive, negative or zero, of
n = e* + e'Yand v is the sum of the products aV1 for all the solutions.
When m s 3 (mod 4), м == v *■ 0 and the formula becomes that of
Eiseoetein' if a. — 0, and that of Liouville10 for a = 1, In the notation of
that paper, X = р*(т)т Thus
4, *r
The last sum is multiplied by — 4 when a is replaced by a 4- 1* Hence
N(T+lmt 10) + 42VB"m, 10) - {16"*1 + 4(- ly^JptO»)-
The values of N4 « AT{2^Jw, 10, 4) and ATe = N{T^mt 10, 8) follow from
4( - iym-mpt(m) = 5ЛГ{2*т, 10) -
Н, J, S. Smith18 stated that the principles indicated in his paper enable
one to deduce by a uniform, method the theorems of Jacobi, Eisenstem
and the numerous recent ones by Liouville on the representation of numbers
by a sum of four squares and other simple quadratic forme; also the
theorems of Jacobi1 on six and eight squares. In view of Eisenstein's
remark that there is a single class of quadratic forms of discriminant unity
in n ^ 8 variables, but always more than one class if n > 8, the series of
theorems relating to representation by sums of n squares ceases when
n > 8. There remain the cases n = 5, 7, Smith gave a description of the
general theory on which are based the formulas for the numbers N6 and N7
of primitive representations of 4*w*5 as a sum of 5 and 7 squares, respec-
respectively, where <*> is odd and £ has no square factor:
where, as in AT*, the product extends over every prime dividing u but not 3,
while Ft is denned as follows: For 6 = 1 (mod 4),
and rj = 12 if В ^ 1 (mod 8); v = 28 or 20, if В в 5 (mod 8), according
as« = 0or«>0; while,* if 6 = 1, чП is to be replaced by 2. But, if
8+1 (mod 4),
41 /*
•(•-«),
" Comptee Rendufi Ffcria, 60,1865, 1257; Jour, de Math., B), 11,1866, 1-8.
"Pro. Roy. Soc. Loodoa, 16,1867, 207; CoH. Math. Paper», 1,1894, 531.
' The ifB here need wtm replaced by qH in hie*1 later paper giving proofs.

Chap. IX] STJM OF FlVE OB MORE SQUARBS, 309
where ^ = 1 or i, according as a = 0 or л > 0, Next,
For* = 3 (modi),
where 17 = 30if*-0,A ^3(mod8); ij = 74/3 if a - 0, Л = 7 (mod 8);
* = 140/3 if a > 0. For * ф 3 (mod 4),
—-)s(s-2&Xs~4S),
S f
where 17 =* 1/3 or 5/12 according as a = 0, a > 0.
J< Iiouville14 stated that, if m is of the form Sk -(- 7,
= 0, />,(*) = 2(-
where г ranges over the positive odd integers < ^mt and d ranges over the
divisore of the odd number n.
E. Catalan16 obtained by means of elliptic functions the result that the
number of solutions of ij 4- ■ • ■ -f г^ = 8n in odd integers ilt - - -, i* equals
the sum of the cubes of the divisors of n.
J, W, Ia Glaisber16 stated that^ if Rm is the number of representations
of N as a sum of m squares (attention being paid to the signs of the roots of
the squares), and if P is the sum of the reciprocals of the odd divisors of Nt
then
Д1 - iR* + iR* ± £ftr - (- 1)'~*ZP.
C. Sardi1? stated that the numbers of the form 4Om + 63 are decompos-
decomposable into seven squares which end with the digit 9. Of. Santomauro,"
G» Torelli18 noted that the preceding result follows from Fermat's
theorem that every number ia a sum of m m-gonal numbers, in the equivalent
formulation by Barlow90 of Ch, I, which implies also that 200wi + 14283
is a sum of 27 squares ending in 29, of which 23 equal 529 or 729.
E. Santomauro" proved that every integer 4Om + 9k is a sum of к
squares which end with the digit 9 [if к > 1, as it fails for m ^ 2, к = 1].
Cf. SardL1'
E« Lemoine* called ЛГ = aj + • • • + e* a decomposition of N into
maximum squares and n the index of N if a\ is the largest square = Nt
" Jour, de Math., B), 14,1869, 302-4.
u Recirerchee mir quelquee produit» indefinie, Mem. Ac. Roy. Belgique, 40,1873, 61-191.
Кёвцтё in Nouv» Am>» Math., B), 13,1874, 518-23- Cf» dlle"
» М«в, Math., Б, 1876, 91.
n Giorn*le di M»t., 7t 1859,115.
"Hid., 1С, 1878,167,
u IJn teofemA d'an^fim, 1879,8 pp.
MComptes Rendus Paris, 95, 1883, 71d-22.

310 History of the Theory op Numbers, [Сиаг. ix
a\ the largest square Ш N ~ a\t etc. Let y« be the least number of index n.
For n even, ym ends with 67; for n odd, with 23. Also,
M, d'Ocagne21 stated the empirical generalization that, if то ^ 3, the
last \_{m — l)/2] digits of ym are the same and in the same order as those
of Vm+i> Lemoine added the remark that the only possible final squares
are Л2, /F+l, Д' + l + l, /P+l + l + I, iF + 22, fl* + 2* + 22,
Д* + 2я + 1, Д2 + 2я + 1 + 1, Д2 + 2я + 1 + 1 + 1, where R > %
T. J, Stieltjes2* noted that, in view of Jacobi* of Ch. VIII, the number
of decompositions of N ^ 5 (mod 8) as a sum of 5 positive odd squares is
where <r(n) is the sum of the divisors of nf and 4f(m) = — 2(
summed for the divisors d of m.
С Hennite28 proved by use of elliptic functions that the number of
decompositions of N = 5 (mod 8) as a sum of 5 positive odd squares is
ЫЮ + X(N - 2я) + x(^ - 4?i + xW - б5) + - • -,
X(n) = 2Cd + dO/4,
summed for all factorizations n = dd\ df > 3d-
Stieltjes24 noted that the total number F(n) of solutions of
n = x) + •••+«!
is 24Л(п) + 16B(n) for n even, and $A(n) + 4BB(n) for n odd, where
Л(п) - X(n) + 2X(n - 4) -h 2Х(л - 16) + 2X(n - 36) + • • •,
B{n) = X(n - 1) + JC(n - 9) + Х(я - 25) 4- • • •,
X(n) being the sum of the odd divisors of n> He expressed A{n) in terms
of B(n)t and BDn) in terms of #(«), and therefore FD») in terms of F(n)*
He verified for each odd prime p < 100 that F(p*) = 10{p3 — p + 1), and
for p = 3, 5, 7 that
T, Pepin25 expressed the number N(m, 5) of representations of m as a
sum of 5 squares in terms of N(mt 4) in the evident way of considering
m — x2 as a 3). By use of elliptic functions he evaluated Ni — ЛГг, where
ЛГх (or JVi) is the number of representations of m as a sum of 5 squares of
which the first is even (or odd); also P — Q, where -P (or Q) is the number
of representations of m as a sum of 5 squares of which the first two have an
even (or odd) sum; he also proved that
N'(m) - N"{m) - 2(- l)m(Xb - 2a),
u L'intennedtftire des math., 1, 1894, 232,
» Comptee Rendua Paris, 97,1883,981.
и I bid., 982.
*/Wd., 1545.
* Atli Accad. Pont. Nuovi Lined, 37,18834, 9-i8.

Ска*. IX] Sum of Five on More Squares, 311
where a ranges over the divisors SI =fc 1 of mi and о over the divisors Ы ± 3,
while N' (or 2V") is the number of solutions of w=x2-ftf*-fs2-f2*t
with s* even (or odd). For m = 8/ ± 1, ЛГ' = 2tf", He noted the recur-
recursion formulae
mN{m, 5) = 2 £ Fn* - m)tf(™ - n2, 5) = 102nW(m - n\ 4).
He proved (p- 48) for any odd prime p the statements by Stieltjes* con-
concerning F(p>), F{pA)>
E. Cesaro* stated that the number of ways of decomposing n into a
sum of p squares is in mean Cnpf*~l, where
2{p - 2)(p - 4)(p - 6)
с =
For p = 3, С = r/4. For p = 4, С = /
A. Hurwitz27 proved and generalized the conjectured results by Stieltjes24
concerning F(p2) and F(pl). If m = 2*pV • * •» where 2, pj q, —» are
distinct primes, the number of decompositions of m* into 5 squares is
F(m>) = К Jj,, «][g, /8] . • •, X = 10
For proof, set ™ = 2*n, Then by Stieltjes* formula, Ffm2) is if times the
sum, for all positive odd integral solutions a, b of a -f Ь = 2n,
XX{a$ h) = X(n2) + 2X{r? - 22) + 2X(n5 - 4s) + - —
But if 7, 5, €, - — are the odd primes dividing both a and b,
e, b) = Б
where the summation with respect to at, bt extends over all positive odd
integers at, bt whose sum is 2n/L By the known formula
X(l)XBn - 1) + X(Z)XBn - 3) + ХE)ХBл - 5) + - • -
+ XBn - l)Z(l) -
viz», the sum of the cubes of the divisors of the odd number n, we get
a, b) - ZUp4 - Pb(p
and hence equals {j>, a][g, 0] ■ • ia the desired formula for F(m*)> Part
of Stieltjes' formulas follow from those of Liouville5 of Ch, XI.
» Comptee Rendus Pane, 98,1884, 504-7.

312 HlBTOEY OF THE THEORY OF NUMBERS CCaAP. IX
T. J. Stieltjes*7" wrote Fi(n) for the number of decompositions of n into
7-squares and stated that F^injlF^m) equals
1
3!
H, Minkowski25 proved that the numbers of the form 8n + 5 are sums of
5 odd squares» The number of proper representations of d as a sum of 5
squares, not ail odd, is
summed for the integers m prime to 2B. A number d = 5 (mod 8) has
proper representations ae a sum of 5 odd squares.
P. S. Naaimoff" proved that the number of decompositions of n *= 2*m
(m odd) as a sum of 8 squares is -^(8*+1 — 15){i{t»), where him) is the sum
of the cubes of the divisors of mm He determined the number of decomposi-
decompositions of any integer into 12 squares.
E. Cesarow noted that the number of decompositions of n into v squares
is JVt — N* — Ni -f N4 — N6 -h N9 4- • ■ •, where Nv is the number of
positive integral solutions of the system of equations
«A •••*,«= P, xtyx + h *.V* ^ »-
The numbers of decompositions of n into two and four squares increased
by double the number into three squares is Mx — Mz + Ms — M7 -h • - <t
where Mp is the number of positive integral solutions of ту = p, x% + yi\ = n,
H- J, S, Smith11 proved the formula for the number of representations
as a sum of five squares which bad been stated by him in 1867, and deduced
therefrom the formulas of Eisenstein,3 The subject proposed by the Paris
Academy of Sciences for the Grand Prix des Sciences Math, for 1882 was
the theory of the representation of integers as a sum of 5 squares (with
citation of resulta of Eieenstein), Apparently no member of the commission
which proposed the subject of the prize knew of the earlier paper by Smith;
nor was the latter mentioned in the report33 of the commission which
recommended that prizes of the full amount be awarded to Smith and to
*• Comptes Rendue Pane, 98,1884, 663-1.
11 M6m. prfeen«e A l'Acad. Sc. Inet. France, B), 29, 1884, No» ± Gesamnu Abh., I 1911,
118^9,133-1.
" Application of Elliptic Functions to Number Theory, Moscow, 1885. French i&ume in
Annalcs ec. de i'ecole norm. aup6r.r C), 5, 1888, 36-7.
M Giomale di Mat., 23, 1885, 175,
11 Mem. SavwiflEtr. Pane Ac. 8c,, B), 29,1887, No. 1; Coll. Math, Papere, 2r 1894, 623-680;
cf»p. G77.
* Smith's Coll. Math Papers, 1,1B94, lxviMxrii.

Сидр, 1X3 Sum of Five or More Squares, 313
Minkowski," then a student of 18 years of age at the University of K5nigs-
berg.
Ch< Berdelle" proved that any multiple of 8 is a sum of 8 odd squares.
From
tt^tf + ^-f-c^ + d8, 8a2 = 4a* — 4a + 4a* + 4a,
8 -h 8n is the sum of the squares of
2a+1, 2a - 1, 26 + 1, 26-1, 2c + 1, 2c - 1, Ы + 1, 2d - L
If & of the integers a, &, c, d are zero, 2A of the 8 squares are unity.
J, W, L, Glaieher14 noted that, if <r{n) is the sum of the divisors of n,
the number of representations of n as a sum of five squares Is
10{<r(n) + 2<r{n- 4) + 2*(n- 16) -h - j ifn^l (mod 8),
but twice that expression if n = 3 (mod 4).
L. Gegenbauer16 proved that the number of representations of an odd
number n as a sum of eight squares equals IGAf y where M Is the number of
divisors of the various g.c.d.'s of n with all triples chosen from 1, * * •, n.
Also M is the sum of the products of the number of divisors of every factor
of n by the number of those triples whoee elements do not exceed the com-
complementary divisor and form a system relatively prime to it. There are
three further theorems on sums of 8 squares, five on sums of 12 squares
and two on sums of 6 and 10 squares each» The number of all [[or proper]]
representations of an odd number n as a sum of three squares and double
a square is 2D — B/я) j/i, where the symbol is Jacobi's and ц is the num-
number of all for proper] representations a* — 2y*, у ^ 0, 2x > 3y, of the
various g.c.d/s of n and the numbers ^ n. There is a similar theorem
on a sum of five squares and double a square*
G< B. Mathewe" noted that the number of sets of solutions of
is the coefficient сл of gn in the expansion of
^-A + 2^ + ^ + 2,.+ .-.)', ,ш
By logarithmic differentiation,
\t = - t Ф(.п)<Г-\ #(n) - 2
о aq
summed for all odd divisors fx of n. For" n *= 2*mt ^(n) = 2e+l<r(m). By
the logarithmic differentiation of 0* — 1 + Ciq + erf + - - - and comparison
»BuILSoc.MatIi.deFniace, 17,1888-9,102,205- Cf. Catalan »
* Messenger Math., 2\> 1891-2,129-130.
« SitituigBber. Akfld. Wk Wien (Math.), 103, Ш, 1894,
« Proo, London Math, Soc.t 27, 1895-6, 55-60.

314 History of the Theory of Numbers. [Chap.ix
of coefficients, we get linear equations for the c's, from which
{n - 1) кф(п -2) * - кфA) n
( - 2) кф(п - 3) ■ ■« n - 2 О
1
P. Bachmann17 gave an exposition of the work of Smith"**l and Min-
kovraki2* on sums of 5 squares, and Eisenstein1 on sums of 5, 6, 7, 8 squares.
E. Lemoine stated and L. Ripert" proved that every integer equals
the вит of p and certain distinct squares, where p ^ 0, 1, 2 or 4.
H. Delannoy18 proved that every even square > 4 and every 4th
power > 1 is a sum of five squares > 0, and that a(a + 2) is a sum of 4 or
5 squares > 0.
E. E. Moritz40 considered the representation of numbers as quotients of
sums and differences of squares.
O, Meissner41 considered the representation of numbers of an algebraic
field as a sum of л squares. In particular, the numbers of the field denned
by t ^ are sums of 5 squares, 4 of which are rationel.
J. W. L* Glaisher48 employed the sums P(m) and Q(m) of the products
of the loots (taken in the form An -+- 1) of the first two and three squares,
respectively, in each composition of 4m as a sum of 4 odd squares, and
proved the following theorems when m is odd. The sum of the odd roots
in all the representations of m as a sum of б squares, 3 of which are odd and
3 even, is rfc 120P{*ra), the sign being + or — according as m = 7 or 3
(mod 8). If a3 + - - + f2 is any partition of 2m into 6 odd squares,
where a, - ^ Г are taken in the farm 4n + 1, and if s is the sum of the 15
products of a, ■ -, J" taken two at a time, then 2* = — 120 Q(m)t summed
for all the representations of 2m by 6 odd squares. For the partitions of &N
into 8 odd squares, where N is even, the corresponding sum 2e is zero.
The number of compositions of Sm as a sum of 8 odd squares is the sum of
the cubes of the divisors of m.
K. Petr" proved, by use of theta functions, two hitherto unproved
theorems stated by Liouville on the representation of even numbers as a
sum of 12 or 10 squares.
E. Jacobsthal" proved that every prime p = 4n + 1 is a sum
of Ь squares, where $ is the g.c.d. of n and p — 1, and g is a primitive root
of p, while p ranges over a complete set of residues modulo 5.
« Arith. der Quad. Fonnen, 18Й8, 608-22, 652-68- "~ ~~
« Nouv. Ann. Math., <S), 17,1898, 19M; 19.1900, 335-6.
» L'intermediaiie des math., 7r 1900, 392* 9, 1902, 237, 245.
« Univ. Nebraska Studies, 3, 1903, 355. Cf. Moritx»4» of CIl VI,
л Archiv Math. Phya., C), 5,1903, 175-6; 7,1904, 266-8-
« Quar. Jour. Math., 36,1905, 349-354.
«Canopiii,Fr*g,34,1905,224-9. Ftetr/»
«AnweDduDgen . . . quadxntiechen Rwte, Di«. Berlin, 1906,20. Cf. Jacobetbal,^ Ck VL

Chap. IX] 8тШ OP FlVE OR MoK» SQUARES. 315
J. W. L. Glauber1* evaluated the number R^(n) of representations of
л as a sum of t squares for each even integer t ^ 18. The simplest results
are
(n) =${E4(n)
the first two of which are due to Eisenstein2 for n odd and to H. J. S. Smith1
for any п. Неге Ег(п) [от Eft.(n)^ is the excess of the sum of the rth powers
of the divisors of n which [or whose conjugates] are of the form 4k -f- 1
over the sum of the rth powers of the divisors of n which [or whose conju-
conjugates J are of the form 4£ + 3; also,
Un) - S(- I)-*, Ш = Z<- 1)**** Ш' = я);
while 4xi(n) is the sum of the fourth powers of all the complex numbers
having n as norm. In an addition to this paper, Glaisher4* evaluated by
elliptic modular functions the sum of the rth powers of all primary com-
complex numbers of norm n and (p. 274) evaluated R^(n).
W. Sierpinski47 noted that the number of representations of n as a sum
of r squares is
where Дг(и) is a polynomial of degree 2i with rational coefficients.
G. Humbert4* derived the formula, in which ^ = Я\@), 6i = G1@),
in Jacobi's notations for elliptic functions of the variable qt
B) 4rf«! + „Ж = 4Ё Bm + l)y*-(
bet GPt q(a) be the number of decompositions of a into p + g squares of
which the first p are odd and the last q are even» By equating the coeffi-
coefficients of д*+1/* in the two members of B) and in the formula obtained by
changing q to — qt we get
4<?e, 4Dtf + 2) + (?*, *<4ЛГ + 2) = 4{- 1)*2(- l)-Bm + 1)S
5(?10p оDЛГ + 2) - 6ft, 4DЛГ + 2) + ft. tDY + 2) = 4Z(- l)-Bm + 1)*,
the summations extending over the odd divisors 2m + 1 of 4^V + 2. If
N is odd, N = 2Af + 1, GV oDiV + 2} is evidently zero» The preceding
equations give
G*, a(SM + 6) - ft. 8(SAf + 6) - #2(- l)^{2m + IL.
The total number of decompositions of 8Af + 6 into ten squares is
evidently
lO-fl-8-7 .10-9
G+G
"Quar. Jour. Math., 38, 1907, 1-62, 17S-236, 289-351; mimmary in Ptoc. London Math.
Soe», B), 5,1907, 479-400.
«• Quar. Jour. Math., 39, 1908, 2G6-300.
" Wiadomoeci Mat., Warww, 11,1907, 225-231.
*• Comptce Reudus Parie, 144, 1907, 874-8.

316 Histoby of the Theory of NumbjbrS.
and this number equals 204Z(<£ — d[)t where dt ranges over the divisors
4Л + 3 of Ш + 6, and di over the divisors 4h + 1.
In B) replace irffo) by SfctfWrt **<* «<*) by tfft1) + irffo1). Then
change g* into g. We get
Equating the coefficients of g*+W4 and those of g*+l*t we get
3) «f 10G,. 7Dtf + 3) = 2(- l)
where 2m + 1 ranges over the odd divisors of 1Y + 3 and 4N + 1, respec-
respectively. The first formula gives for the total number 120(ftp • + ft. i) of
decompositions of 4N + 3 into ten squares the value 122(<£ - d}), due
toEisenstein.1
For 12 squares, it is shown that
Bm +
Tbus the total number в6(<7Мр, -J- 14Ge> t, + Gt. ю) of decompositions of
AN + 2 as a sum of 12 squares equals 2642<f, d ranging over the divisors of
47V -j- 2. Changing q into g*, we find that
ft.4(8M) = Gipl(810, G>ti{m + 4) + 0i^8M +4) = I6ZBm + II,
summed for the divisors 2m + 1 of 8Af + 4. Next,
v\e\ + f JlJ = ieSmW(l - g4-)
gives Gi.4(8Jlf) H- С4,в(8Д£) * 162?»*, m being such that 2Mfm is odd.
By these and a more complex relation, one may obtain the total number
of decompositions of 4ЛГ into 12 squares, and thus prove IiouvilleV theorem.
K. Petr4* proved Iiouville's11 formula for the number of representations
of 2*m as a sum of ten squares by use of the theta functions with the char-
characteristics A,1), A, 0), @, 1), @, 0) and formulas in Jacobi's Fimdamenta
Nova (p. 101). Also, LiouvilleV result on 12 squares by use of the fourth
derivatives of f(«).
E. Dubouis* wrote Sn for & sum of n squares each > 0. For к > 45,
the odd number * — 1 or fc — 4 is a £4j whence к is а Я* The only numbers
not 5Л are stated to be A = 0,1, 2, 3,4, 6,7,9,10,12,15,18,33. Every
number + Л + 1 is a S*. The numbers not 8ш* are stated to be В - 1,
2, 3, 4, 5, 7, 8,10,11,13,16,19, The only numbers not a S^.n are the
В + n and the first n integers*
* J. V. Uspensldj" discussed the representation of numbers as sums of
squares,
« AjfchiT Math. Phy*., (8), 11,1907, 83-b Ffetr •
« L'intenn^diaire dn math., 18,1911, 66-66.
« Math. Soo. Kharkov, B), Ut ШЗ, 31-64.

Chap. IX] SUM OF FlVE OR MORE SQUARES. 317
B. Bonlyguine52 employed the notations
gummed for all the Nv(n) integral solutions (positive, negative, or zero)
of x\ + • • • + x$ = n. Write ak(m) for the sum of the fcth powers of the
divisors of m and
pk(m) «2(- l)<-"-u'y*
for the difFerence between the sum of the fcth powers of the divisors 4h + 1
of m and the sum of the Arth powers of the divisors 4h + 3. By use of
elliptic functions, it is shown that, if n = 2?mt where m is odd,
+ <•?> E (я) +
ТЪеге is given a similar expression for tfar+jftt). Also,
t
kr—14
with a similar expression for N&+4(n). Here the a's and d's are rational
numbers not depending on n. It is stated that there result the known
formulas for the number of decompositions into 2,4, 6, 8,10, or 12 squares
and apparently new formulas for 14 or 16 squares.
Boulyguine** stated a recursion formula for his63 2(rc);
ArNJn) = Fr(n) + AfX £ (л) + Ал 2 (л) + Ar» 2 (n) + • * •,
for r = 2, 3, • ■ •, where Ar, Arl, ■ ■ • are independent of nf while FT(n) is a
specified function differing in the three cases r odd, r=*4& + 2, r — 4fc + 4.
S. RamanujanM studied the function ^(n) for which
Special cases of ^ are the functions х(л), Р(л), х<Сл)> в(п), G(n) of Glaisher"
of Ch. VIII. He touched (pp. 179,183-4) on the number of representations
of n as a sum of s squares, s = 10, 16, etc.
L. J. Mordell56 proved that various empirical results of Ramanujan"
follow from expansions of elliptic modular functions.
41 Comptea Rendua Рдги, 15S, 1914, 328-830.
М/Ш., 161,1915, 28-30.
м Тгаад. Cambr Fhfl. Soc^ 22, 1916,173-9.
* Proc. Cambr. Phil. Soc, 19, 1917, Ц7-124.

318 History of thk Thboby op Numbers. [Сна*, ix
R. Goormaghtigh" proved that every power of an. even [odd] integer
with ал exponent ^ 3 ie а шип of 5 [j6] squares > 0. If n is odd and > 1
and if a > 0, n4**1 is a sum of 5 squares > 0.
Mordell" employed the theory of modular functions to find the number
of representations as a sum of 2r squares.
G. H* Hardy4* deduced from the theory of elliptic functions the number
of representations as a sum of 5 or 7 squares. This investigation, con-
continued by S. Ramanujan/** led to a complete solution, of the problem of
the representation of a number as a sum of n squares for n < 8, and to
asymptotic formulae for any n. The method used is an application of the
general theory cited in СЬ. Ш.ш
E. Т. Bell1** proved LiouvnVs10»ll formulas by use of series for elliptic
functions and stated that they are only the first cases of an infinitude of
similar results which may be found by using higher powers than the first
and second, or products, of the series.
On 10 odd squares, see Pollock117 of Ch. I. On 8 squares, see Sier-
pinski*" of Ch. VI. For 5 squares, see Hermite69 and Humbert1** of Ch.
VII. In Ch. XI are noted Iiouville's results on sums of n squares for
n = 8 and 12 and in papers 6 and 7 minor results for n ^ 5 and 7,
Relations between squares.
The Japanese Aida Aminei59 proved between 1807 and 1817 that
xx = - a\ + a; + a\ + ■.. + <£ xT - 2alaT (r = 2, - < -, n),
v te a\ + • ■ • + a\
satisfy sj + \-xi = y\ This result was known to Euler- "*• "* of Ch.
XXIL Ajima Chokuyen,68* in a manuscript dated 1791, had solved
x\ + •. - -f x\ = v1 in integers.
It was proved by J. R. Young,*0 who proved also the identity
(i, j = 1, --,n;i< j).
The latter was proved otherwise by A. Cauchy."
Aida's result has been published also by D. S. Hart and A. Martin,"
К Catalan,6* A» Martin,* and G. Bisconcini* (by geometrical considerations
math,, 23,1916, 1524?.
17 Quar. Jour, Math., 48,1917, 93-104.
"Froc- Nat. Acad, Sc., 4, 1918, 189-193, Proc. London Math» Soc., Records of Meeting,
March, 14, 1918.
«- Trans. Cambr. HuL Sqcl, 22,1918, 259-276.
* ВЫ1. Amer. Math, Soc., 26,1919, 19-25.
"Y. Mikami, Abh. Gtecb. Math. Win., 30, 1912, 247, Bftsed on С Hitomi'e article in
Jour» Phye. School of Toltyo, 16,1906, 359-6*
»• Jour. Phye. Sdiool of Tokyo, 22. 1913, 51.
« Tt*n*. Eoy- Iriah, Acad., 21, TI, 1848, 333,
* Coin* d'aaalyee de l'eoole polyt., lt 1821, 455-7.
* Math. Quest. Educ. Timefl, 20,1874, 83; 63, 1895, 49,112.
« BuH. Acwl. Roy. 8c. Bdgique, <3), 27, 1894,10-15.
11 Рюе. Edinbirgh Math. Soc.T 14, 1896,1X3-5; Math. M*g-, 2,1898, 209.
«Periodioodi Mat., 22,1907, 28*

СялгЛХ] Relations Between Squares. 319
in n-space). By multiplying «< by ^] for г = 2, - - -, n, we get
(а* + 2«<а!I = (- a* + 2*,<ф* + Sa<Baa;J,
a formula noted by G. Candido.*
M. Moureaux47 noted that successive applications of Aida's formula
gives
(*; + --' + <#* = *; + ■■> + &:.
J. CunlifiV* noted that we can find any number of rational squares
whose sum is a rational square since n + fe* = О, к = Dr* — n)/Dr).
Thus> ifn = l+4 + 9 + 16, taker = 3, whence*; = 1/2, and we have five
squares whose sum is a square.
L. Calzolari8* found special solutions of
*?+ ••• +А = уг
by setting Xi = k -\- a^y = к + 2a<* Tbe new equation is linear in each а±
E. Lucas™ noted that the sum of x consecutive squares may be a square
for x = 2, 11, 23, 24, but for no further value 1 < x ^ 24; the sum of n
consecutive odd squares is Ф □ if 1 < n< 16» Cf. papers 76,81,86,87,100,
and 103 below; also papers 80,130-8 of Ch. I; and Brocard^of Ch. ХХШ.
H. S. Monck71 noted that & ~ (a? + b1J ~ BabJ + {2bc + c2I if
a = 6 + с Hence if
a' = cj H- h «I, ^ = №cj + • ■ • + 46^ + Bbc + c1I
is a sum of n + 1 squares. Also,72
S«5 = {2s + (л + l)o}f, ш = 2ch <a ^ 2s + 2a - (л - l)c*
F. P. Ruffini73 discussed the positive integral eolutions it ш Vi = ■ — = it
of
»T + " - + % = M/ <i + ■ ■ ■ + i ^ p.
Let Xi be the number of i's with the value 1, and x2 the number with the
value 2. Set a — r — xY — x2. Then
xi + 4лг2 + Zi2 ^ u, «! + 2x2 + 2г « ff Csi£vi'"e »i).
Solve for a?j and x2, and require that the values be ^ 0» By Xi,
«I - % s F = и - 2» •* 2*1* + 22г,
where the summations extend over s — 1 values of i. Hence
г, ^ 1 + Vf+T,
The condition 1 + V ^ 0 is treated sunilarly, first solving for i^\. For
14 SuppL al Periodico di Mat,, Щ 1916, 97-100. Case ar = r by Aid*1** of Ch, XIII.
" Comptes Rendue Paris, Ц8, 1894, 700-L
w The Gentleman's Math. Companion, London, 3, No» 14,1811,281-2.
" Giornale di Mat.r 7,1369, 313. Cf. Ch. ХШ.1"
7C Recherches eur Tanalyse indetermin^, Moulins, 1873, 91. Extract from Bull. Soc.
drEmulation Dept. de lTAlher, Sc» Bell, Lettnea, 12, 1873, 530.
71 Math. Quest. Educ. Times, 20r 1374, 83-4.
71 Ztaf., 30,1879, 37-8.
11 Mem. Accad. Sc. latituto Bologna, 9,1878,199-215. Simpler than Ыя paper, ibid., 8,1877.

320 Histoby or the Theory of Numbers. [Chap. IX
и = n* — 1, v = 3(n — 1), the initial pair of equations are the conditions
on a Cremona transformation. For и = пг — 2, v = Зл + 2р — 4, they
are the conditions on the transformation of R. De Paolis, Mem. Accad.
Lincei, 1877-8.
J. W, L, Glaisher™ expressed the sum 2(a< — a,J of n(n — l)/2 squares
i— 1
where v - (n — l)/2 or n/2 ~ 1, according as n is odd or even, and
Сш ^ cos Bтт{п)) 9т = sin Bтт{п).
G. Dostor" desired 2» + 1 consecutive integers such that the sum of
the squares of the first n + 1 of them equals that of the last n, and proved
that Ше first of the numbers is nBn + 1) or — n.
A. Martin* proved for n *= 3, 4, 5 that a sum of n consecutive squares is
not a square. Call x1 the middle square when n = 3 or 5; the problem
reduces to the fact that 3x* + 2 » П or b(x* + 2) = □ is impossible.
G. Dostor*7 noted that, if ax + • ■ + an ^ np/2,
the last by setting a* - 0, so thatYe a sum of n or n — 1 squares is ex-
expressed as a sum of n squares. Also
D. S. HartY> found squares whose sum is a square by subtracting
(« + m)z — x* from I2 + 2l + ■ ■ ■ + n1 and, by trial, expressing the dif-
difference as a sum of squares, which are then deleted from the n squares.
J. A. Gray™ noted that we may start with a sum S of squares, choose a
divisor a of S and set 8 + x1 = (x + a)s, whence 2s = S/a — a.
Hart81 considered the sum S of the squares of 2» — 1 consecutive
numbers the middle one of which к z and, for special values ^ 181 of я,
made S a square. Cf. Lucas™.
К Catalan81 proved there is a number equal to a sum of p squares and
having its square equal to a sum of 2p squares, by use of the identity
ft h., 3, ШОД р. 43.
Matb. Phy*., в4,187flf 350-2. a. Zeitachr. Math. Naturw. Unterricht, 12, 1881,
269; B. Collignon, Алое. fr№9. av. ec,, 25, П, 18&6,17; Ceeuro** of Ch. I.
я Matiu Visitor, l,l«80r 156. Cf. Lucae •
»» Aidiiv Math. liy*, 67,1882, 2G5^8.
« For n - 3, E, Otalan, Nouv. Oorreep. MaA., 4,1878, 3.
7i МШ. Mag&ane, 118S24 B9
в
, 119-122; «naU oorreoted by Martin, 2,1892, 94.
Matbeaa,3,1883,199.

Chap. IX] RELATIONS BETWEEN SQUARES. 321
Catalan" proved the last result and (p. 106) gave a long identity
furnishing particular solutions of u* = x\ 4- ■ ■ ■ 4- zj. If an odd number
N is a E21 and if n is the number of equal or distinct prime factors of Nt then
ЛГ* is a sum of к squares Ф 0, к = 2t 3, • ■ •, n + L
R. W. D. Christie84 noted equal sums of four or more squares.
A.Martinwnotedthat2JH-31H-61 = 71, 1* + 2* + 4* + 6* + 8s = II2,
Is + 2* H- ■ ■ + 50* - 206* =l + 2* + 22J-5t + 8J«f20t,
He88 stated that one can find several sets of 50 squares whose sum is 2312,
that Is + 2* H- ■ ■ + 24* = 70s, and similar results. Cf. Lucas ™
F. Tano's method to find an infinitude of solutions of
when к is of the form C* — l)/2, is given in Ch. XILW
A, Martin87 found many sets of squares whose sum is a square by
means of the methods of Aida" and Gray,*0 and by seeking to express
S* — b* as a sum of distinct squares ^ nt> where b* lies between n* and
8a = Is + • ■ • + n*. He noted that the sum of n consecutive squares is
not a square for 2 < n < II, and gave solutions for n = 11, 23, 24, 26,
etc. [cf, Lucas70]. He gave solutions of
Л - Xs - П, 8п + 1 = П, 8п-8т-х*=П,
and tabulated the values of Sn for n < 400.
E. Catalan»8 noted that, if N =t 1 are primes and N + 2, 2ЛГ* + 2
is a sum of 2, 3, 4, and 5 squares.
E. Fauquembergue" and others noted the identities
(af + ... + a|)J ^ (aj + oj + aj - a\ -
+
P. H. Philbrick90 noted that we may find n squares whose sum is a
square by Aida's" method or by starting with a sum S of n — 1 squares
such that 8 is a product of two factors a and b, both eveu or both odd, and
applying
R. J, Adcock" noted that, if в ■■я + у + г + р,
sV + pV + ** + № + (xy + жг + xv + yz + yv + да)* = Bл3 +
11 Atti Aocad. Fbnt. Nuovi lined, 37, 1883-4, 58.
tt Math, Quest. Educ. Timee, 49, 1888, 169-173; French tranal., Sphinx-Oedip©, 7, 1912,
177^87
teBuH. РЫ1. Soc. Wfteb., Щ 1888,107 (SmRtani&ii MboeL OoQ, 33,1Я88).
Ы1Ш., 11, 1892,580-1-
•T Math. Mag., 2,1891^3, 60-76, 89-96,137-140.
" MatbesiB, B), 3,1893, 235.
" МяЛЪеяв, B) 4, 18W, 277; в, 1896, 101,
"Amer. Math. Monthly, 1,1894, 25fr8
л /Ы., 2,1895, 285.

322 History of the Theory of Numbers. [Chap. IX
Several vrriters*2 found nine integers in arithmetical progression whose
sum of squares is a square.
A, MartiaM noted that the sum of the squares of the nine numbers
x — Ay у x — 3j/, • ■ •, x -f 4y in arithmetical progression is a square if
9x* + 6(V = □ ♦ Так* у = 3z, x1 + 60** - (x + гр/q)*; hence ф is
found rationally.
Various writers94 made 2£ll s< and Xx] squares for n ^ 2, 3, 4, 5, 9»
A. Boutin" noted values n — 4, 9, - ■ >, 50 such that the sum of the
squares of n integers in arithmetical progression is a square.
A. Martin» solved &M hii = e}+ h <£ by setting cn = a + b~
and finding b* rationally.
T. Meyer97 gave solutions of a2 + &* + • ■ • + n* + «* - Л
G. La Marea" proved that SaJ = П if aXj > •>, a* are integers such that
at : oi = 3 : 4, a, : a*+i = 3:5 (г — 2, — >, n — 1). For, by <*i — 3gi,
at ^ 4#i, я^ = 3gi, as = 5<fcj we have aj + a\ = E^02, 5^i : аз = 3 : 4,
E#0* + flj — Eг)*, e^c-j where z ^ 5#»/4 is stated erroneously to be q%.
Ed. Collignon*9 noted that x = 2ak(k + 1) is a solution of
Xs + {x - аУ + - - + (x - ЫУ = (x + ay + ■ - - + {x + bI.
E» N. Barisien10* noted that a sum of p consecutive squares is not a
square for p < 20, except for p = 2, 11, without treating the case p = 13*
First, let p ^ 2n + 1 and denote the middle square by x* and the least
square by (x ■— п)*. The sum of the squares is Bn + 1)\х* + п(п + 1)/3},
which is not a square for n ^ 4, n = 7, 8, 9» For ti = 5, llfx2 + 10) is
to be a square, whence x = Uh ± 1. Tben ^+10^ llwi*, k *= 2l7l ^ Q
or I (mod 3). A table of 8 solutions includes
(я?, h, m) = B3, 2, 7), D3, 4, 13), D61, 42, 139), (859, 78, 259).
For p = 2n7 let (x + n)* be the largest square. Their sum is
N = 2nx(x + 1) + nBn2 + l)/3.
For n = 1, N = 2хг + 2x + 1 = 471 + 1, where 71 is a triangular number.
Thus T =* 6, 210, 7158, - -, giving
32 + 42 = 5^ 20* + 21* ^ 29s, 119* + 1202 - 1692.
The cases 1 < n ^ 9 are impossible. Cf. Lucas.70
E. N. Barisien101 gave the identity
(a* + Ъ* + с2K = [а(Ь* + с3 - о2) J -J- [Ь(Ь2 + а1 -
11 Ajner. Math» Monthly, 2, 1895, 129^30, 163.
"Math, Quest. Educ, Time», 63r l895r Ш-2.
M Lrinterm6diaire dee math,, 1, 1897, 42-i.
»/W(fM 5, 1898,75.
» Math. Magazine, 2, 1898, 212-3,
>v Zeitochr. Math. Naturw. Unterricht, 36r 1906, 337-340,
» П Boll. M&L Giornale Sc» Didat. (ed.t Conti), 5, 1906,152-5.
" Sphinx-Oedipe, l906-7r 129. Саве а ^ |r Dostor,"
»<» Sphinx-Oedipe, 1907-8, 121-6. <X Martin. *
la Bull, de math. Шт., 15,1909-10,18b

Chap, IXJ RELATIONS BETWEEN SotaBES. 323
and obtained102 ten decompositions of 2663 into nine squares by multiplying,
two by two, five decompositions of 266 as a sum of three squares.
E. Barbette103 used the method of Martin87 to find squares whose sum
is a square. He gave (pp. 87, 96) many sets of consecutive squareswhose
sum is a square, [cf. Lucas7()J
E. Miot1M stated that, if 2* < m < 2*+1, the square of a sum of m squares
is a sum of 2* + 1 squares.
E. N. Barisien105 noted that the sum of the squares of x*, 4sVj 4xyb,
2yb and 2xy{2& + 5xV + 2^) equals the square of z* + $x*y2 -f Sz*y* + 2yb,
and gave seven squares whose sum is a square.
L. E. Dickson1M gave a history of the problem to express the product of
two sums of n squares as a sum of n squares»
Onl4 " +x2 = ^TseeLucas161ofCh.I. Onxf + >■> +xl = R\
see Turriere115 of Ch. VH, Escott2W of Ch. XXI and paper 94, p. 322. On
x\+ .>.+*;,= !,*, see papera 96a, 98 of Ch. XX; 268 of Ch. XXI (p=3);
and papers near the end of Ch. XXII (p=4). By Landau21 of Ch, XXV,
every definite polynomial in a; is a sum of the squares of 8 polynomials.
1Л Mathesis, 10, l910r 185.
i* Leg вошли» de p-ietnee puissances distinotes egales a une p-ieme puissance, Liege, 1910,
77-104.
'« LJbterm$diaira des math., 19, 1912, 195.
i* Sphinx-Oedtpe, 8, 1913, 142.
'«Annals of Math., B), 20, 1919,156-171, 297.

CHAPTER X.
NUMBER OF SOLUTIONS OF QUADRATIC CONGRUENCES IN
ft UNKNOWNS.
For n Ш 4, report was made in Ch* VIII on the papers by Libri," Schone-
mann,11 FrattinJ/* Lipachitz,77 Dickson,» Tengbergen,1'7 and L. АиЬгу,ш
and to many papers proving merely the existence of solutions. See also
Hermite,11 Lebesgue," and Pepin» of Ch. VIII, Vol. I of this History.
V. A. Lebesgue1 noted that F = 2a*rf = 0 (mod p), where p is a prime
2Л + 1, may be reduced by multiplication of the variables by constanta to
a form
A) *!+■•■+•?-»«+■■■+# (*<***>)>
where n — 1 it p = 4q — 1, and n is a quadratic non-residue of p if
p = 4q -\-1. Let N*k, N*, N'k denote ttie number of sets of solutions of
Vi + ■ ■ ■ + Vl = *
according as a s 0, a is a quadratic residue or non-residue of p. In view
of his2 general theorem, the number of sets of solutions of A) is
according as n = 1 or a quadratic non-residue of p. Also, if P* is the
number of solutions of F = 0 and t the number of F — <га* ^ 0, the
number for F ^ a is (т — Pp)/(j> — 1)- It is proved that, if A is odd,
Nl = p*-\ Nk = p^1 + t, K = p^ -t, t = (- iy*-w*'p/y*-»i»;
while, if A; is even,
ATS = pM + (p - 1)Z, Nk = N*k = p* -I, I - (- 1)(Р-ой-*Лр<*«-«#
Lebesgue1 gave a simpler proof of the last results and also found the
number of sets of solutions prime to p.
C. Jordan4 proved by induction from n = I and n = mton = l-\-m
that, if ax - • * о** ф 0, aiz? + ■ ■ ■ + as»*!» — Л (mod p), where p is an
odd prime, has р*я~х — p*^** sets of solutions if к Ф 0 (mod p), and
р*^1 + (p* - p^)»' sets if A: = 0, where
/(
are Legendre symbols. Also, aixf + ■ ■ ■ + Oi^ixS^i ^ к (mod p) has
p** + pV sets of solutions. As a corollary, there are (p — l)/2 variations
1 Jour, de Math., 2,1837, 266^-275.
"Vol. I, pp. 224-5of thieffietory.
9 Jour, de bfAtib, 12,1847, 407^71.
ACoinptw Bmdae Psne, 62,1866, 687-90; Tnfte d^ eobetitutioos, 1870, 156-61 (with m
mwpriat of mgu in. the theoram on p. 610).
325

326 History of the Theokt of Numbers. CCha*. x
of signs in
\p)' "■• V~V~r
V. A. Lebesgue* gave two proofs of Jordan's formulas, not using induc-
induction* The first proof uses hia1 results for reduced congruences. The second
proof is based on his3 amplification of Libri's method,
EL J. S. Smith* proved that if p is an odd prime and m any integer,
xz — y* = m (mod p) has p\p + (~ mfp)} solutions. Each of the con-
congruences xz — y* ^ 1, 3, 5, 7 (mod 8) has 48 solutions in which x and у
are not both even. If p is any prime and г > 0, i* ^ 0,
xz — y1 = mpi (mod p<+0
has p****** A — 1/p2) solutions in which z, z are not both divisible by p.
C. Jordan' proved that xtyl+ ••- + xKy« ^ 0 (mod 2) has 21* + 2-
sets of solutions, while Xj + Vi + *tfi + - - * + ЗД» = 0 has 2**^ — 2*
sets of solutions.
Jordan8 determined the number of sets of solutions of / ^ с (mod M),
whsre / is any homogeneous quadratic function of xi, * • -, x*> The number
is the product of the numbers of solutions for moduli which are the powers
of primes whose product is M. Consider
/ « P"(a,rf + ■ ■ ■ + a*&l + bi^ixt + > - •) ^ с (mod I»),
where at least one coefficient аъ • - •, о», bit, - • - is not divisible by the
prime P. First, let P > 2. By means of a linear transformation, we may
remove the terms Xixt} etc., not squares. The problem is reduced to
The number of sets of solutions, in which хи * - ■, x, are not all divisible by
P, is P^tT, where r * 0* — l)(n — l)+n — p, л = p + j + ■ ■ •, and CT
is the number of sets of solutions of A\x\ + • ■ • + Apt} = ^ (mod P),
given above.4 For solutions in which x1? • - *, xv are divisible by P, we can
remove a power of P and are led to the preceding case.
For P =^ 2, we can transform/ linearly into 2*2Л + 2*2Э + " s where
each 2P is of one of the four types Sp — x^fi + — - 4- ХрУр,
S, + il^, S, + Лг3 + Л^ g^ + м* + адг + tP,
where Л and Aj are odd integers, A ^ 7, and p may be zero. The number
of solutions is found by treating these four cases in turn.
T. Pepin1 proved Jordan's4 results by expressing the number of solu-
solutions in terms of the number for the congruence in which the number of
unknowns is less by two.
* Comptw RenduB Pane, 62,1866,868^72.
* Trane. FblL Soc. London» 157,1867, 286-7, $ 18; Coll» Mirth. Paper*, I, 492-4.
7 Ttaite dee ndMtttutHms, 1870, 198.
' Jour. dB M&Uu, B), 17,1872,368-402, Gompti» Rendue Pkn, 74,1872,1СШ.
* N A МО B) 101871,227-234.

Chap. X] QtfADBATlC CONGRUENCE Ш П UNKNOWNS. 327
H, Minkowaki10 found the number}[m\ N\ of sets of solutions of
Ш
f — Л ai&iXt = m (mod N)-
If
then
f{m>,N} =jf£
so that the problem remains to find j(m\ N) whose determination depends
upon that of 2/Г', where хъ —, з„ range each over a complete set of
residues modulo N. The problem is reduced to the case of a power of
prime modulus. The paper is too complicated to admit of a brief report.
L. Gegenbauer11 considered / = «ix] + • ■ ■ + a*x*, with r of the a's
quadratic residues of the odd prime p. Let trl(r) be the number of sets of
solutions of / s 0 (mod p), and <r*(r) the number of those in which no x = 0.
Let s' and s be the corresponding numbers for / = 1 (to which we may
reduce/i s Ъ Ф 0 by multiplication). For r > 0,
with more compUcated recursion formulas for si(r), sn(r)t which with
<x[(r) = 1> ^(r) - 0, si(t) = $[(t) = 1 + Br - 1)(- 1/p) determine the 8
and <r as by Jordan.4
K. Zsigmondy12 proved the final results of Lebesgue.1
P. Bachmann13 gave an exposition of the subject.
L. E. Dickson" gave a generalization of Jordan's4'7 work to any finite
field and a derivation of canonical forms.
R. Le Vavasseur1* discussed/ = и (mod p), where p is a prime and
/ = ax* + bxy -h aV + ex + c'y + d, A = Aaa'd + ЪссГ - ас'2 - oV - db\
Ь - 4дл' - Ъ*.
If 5 is a quadratic non-residue of p, f = Д/5 has one and but one solution;
for и ф Л/5, / = « has p + 1 solutions. If 5 is a quadratic residue of p,
/ ^ Д/5 has 2p — 1 solutions, / = и ф Д/5 has p — 1 solutions. If 5 s 0,
/ = и has p solutions.
J. Klotzie found the number of sets of solutions of the general quadratic
congruence in any algebraic field.
i* Mem. presentee U'Acad. Sc. Inet. Fiance, <?), 29, 1884, No, 2, Arts, 7, 8, 9; Acta Matb.,
7,1885, 201-258, eapec,, pp. 210-37. Ck*amtn. AbhM 1,1911, 3,167.
и Sitzunpber. Akad, Wiffi. Wkn (Math.), 99, Ila, 1890, 795-9-
a Monatehdle Math. Phy^, 8,1807, 38.
13 Arith. der QiudrsL Formen, I, 1898, 478-515.
i* linear Groups, 1901, 46-9, 158, 197-9, 20ДО; Madiacm Colloquium Lectures, Amer.
Math. Soc, 1914. d. i. К AIcAtoe, Amer. Jour. MaUl, 4lt 1919, 225-12, on Jordan.»
u Mem. And. 8c. Toolooee, (Ю), 3, 190ft, 44^8.
u Vkateiialondiita d. natvif. GwdL Zurich, 58» 1913, 239-6».

CHAPTER XI.
LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES.
J. Iiouville enunciated without proof numerous results in a series of
eighteen articles, " Sur quelques tommies generates qui peuvent e*tre utiles
dans la theorie des nombres."
Let m be an odd integer, a an integer ^ 1. Set
2Tm - m' + m", m = db, m' = d'B% m" = d"b'\
where m' and m" are odd positive integers. Let f(x) = /(— x) be an even
single-valued function. He1 stated that
(a) E {.£[/<<*'-d") -Ж + О1И -2r*-£di№ -fi?d)\,
where d, d', dn range over all the divisors of m, m\ m'\ respectively, and
the first summation extends over all the pairs of positive odd integers m%
m" whose sum is Tm. Taking f(x) — x**, we get
where the coefficients are those of even rank in the binomial formula, and
Г,(т) denotes the sum of the pth powers of the divisors of m. For д = 1
and p. = 2, we have
The first gives the number of decompositions of 4-2*m as a sum of 8 odd
squares; the second gives the number of decompositions of 8-2*ж into
« + 2<r, where e is a sum of 8 odd squares such that «/8 is odd, while * is a
sum of 4 odd squares.
For f(x) •" cos art, (a) gives
XB sin eft ■ 2 sin <*"*) = ^^d sin8 B-VH).
Taking л = 1, t = t/2, or by settmg/Ca?) * (- l)x/J, we get
which yields Jacobi's*-л theorem of Ch» VHI that 4m has f i{m) representa-
representations bs a sum of four odd squares*
For a function f(x, y) which is unaltered by the change of the sign of x
or of y, Iiouville stated that
f Z,E/( - d", V + S") -f(d' + d", $' - «"
1 Jour, de MitlL, B), 3, 1868,143-153,199-200. Firat and eeeond articUe.
329

330 History of the Theory of Numbers. CCbaf. Xi
Set
/fo y) =" cos xf* cos yz,
Ф(т) = £ sin Л-cos Ъг> w(m) =* £ coacft-sin bz (dB = m).
Then (c) yields the result
We now include the case in which a ** 0 and set
Tm = 2*V + 2*V' (a' ^ 0, «" ^ 0, m', m" odd).
Let m = dB, etc., as before* Liouville* stated the formula £a case of (eM]
where dt d'7 d4 range over all the divisors of m, mf, mf/, respectively, and the
first summation extends over all the pairs of even or odd integers 2"W,
2*"W whose sum is 2ewi. Consider the case a = 0; then a' or a" is zero;
but, by introducing the factor 2 before the first member of (G), we may
restrict attention to the case m = m' + 2*W'. Since 25 =* 2d, we get
(F) 2 £ f
a case of (d). For example, if f(z) = x*,
For /(x) = x2 or x4 in (G) we get
Again using the notation m — mf -\- 2''W, Liouville stated the follow-
following two cases of (d):
ДО){ЧН = E {/@) + 2/B) + 2/D) + ... +2f{d~ 1)]
{D) ' +22! j£№ ~ d") - M + d")]l-
Z i £,№ - <*" + 1) - ^С<*' - d"' - 1) - ft** + d" + 1)
(E) *' ' + F(d' + d" - 1)]} - ГA)Г,(я0 - Z F(d),
where F ie an odd function:- f (- x) = — F(x). For /(x) ■> (- II'1,
(D) gives
i(M " p(»I - 2pK)pK'}, p(m) m S(- l)«-»tt.
The first expression is therefore the number of decompositions of 2m into
A) ^ + ^ + 2'(u! + ^,
with y, z, u, v odd positive integers and a > 0; it is also the number of
deoompositionB of m into the form A) with у and z positive odd numbers,
> Jour, de Math,, B), 3t 185S, 201-S, 211-250. Third and fourth articles.

Chaf.xt] Liouville's Sheies or Eighteen Articles. 331
and «, v any even integers. For f(x) ^ x1, we deduce from (D) that
which gives the number of decompositions of 4w into в + 2V, where a
and a- are sums of 4 odd squares.
For m any integer > 1, let m — m' + m". Liouville stated the follow-
following case of ( f ):
where f(/») Ц the number of factors of w and the accent on the final sum-
summation sign signifies that a term f(k) is to be suppressed when к is a divisor
of d. For f(x) = s* and m a prime, (H) gives
(№) 2ri(«Ofi(«'O - АИ2 - l){5m - 6).
This result may be used to prove the theorem of Bouniakowsky that
any prime m of tie form 16A; + 7 can be decomposed into 2x* + p"+y
in an odd number of ways, where p is a prime 4Л+1 not dividing y.
Liouville4 stated that, if f(x, y) is unaltered by the change of the sign
of x or yt
where the first summation extends over all decompositions m' + 2*'W of
w- If /(ж, у) reduces to a function/(x) of x only, (d) becomes (F). If it
reduces to/Qj), (d) becomes (D). To pass from (D) to (E), take
/(*) - F{x + 1) - F(x - 1).
In (d) take/ to be (- l)^^(x), where/(x) is an even function. Then
For 2Vi
tf, 0) + ^(W, 2) + #B4 4) + • ■ ■
-1)] - 2-Z#B4 0),
which reduces to (G) for /(ar, j^) = /(a;). Formula (H) is a special case of
^+E_ {E да - d", r +«") - /(if + d", 5' ^ г")]]
(f) +Il:(d-l}j/@,il)-MO)j+2E4/ft2)+---
+/(«, rf- 1I - 2£'{/B, «) + ■■• +/(d - 1, 5)),
where the accent indicates that /E, y) is to be suppressed if у is a divisor
1 Jour.de Math., <2), 3, 1858, 273-283. Fifth article. ' ~"

332 Histoby or the Theohy op Nxtmbebs. [;cha*. xi
of d, and f(x, h)#x divides d. Set Д(*, у) = f(x, у) - /(у, x). Then
*' - d», *' + П! = 2(c* - 1)A(O, d)
where ДE, у) is to be suppressed from the final sum if у divides d. The
last formula is valid for any function Д for which A(x, y) = — A(y, x).
Liouville4 employed in his sixth article two simultaneous partitions
2то =* то' + m"f m — nh + 2"rrh (m's odd and > 0)*
get то,- = di&i, etc. Let F(x) be a function for which
F@) -0, F(-x) - -F(x).
He stated that
where d, di, d\ d" range over the divisors of m, mi, m', m'\ and the first
summation extends over the mf and m" whose sum is 2m. For F(x) =* xf
so that there are Tit»») + 4& decompositions of 8wi into a + 2<r, where ^ is
the sum of the squares of four odd positive numbers and <r is the sum of the
squares of two such, while В is the number of decompositions of 4?» into
For a like function F(xO another formula was stated:
h d" + d'tr) + F{d' - d" - d'") - F(d' + d" - dfff)
- F{d' - d" + d»*)l) - S(eP - l)F
where the two members relate to the respective modes of partitions
то ^ m* + m" + f»'", m = mi +
For F(x) = x? there results the formula
Hence if G is the number of decompositions of 4m into a sum of 12 odd
squares, and Я that of 8m into e 4- 2V, where « is a sum of 8 odd squares
with e/8 odd, and a is a sum of 4 odd squares, then
From (M) and <F), with/(x) - xF(«) is derived
F(^^ 2-4K1
1 ' = 2(eP - 1)FW + 822B- -
each relating to the single mode of partition щ = m\ + 2*Vo,, тп<
Jour, de ММЬ., B), 3t 1S58, 326-396. Sixth article.

Chap. XI] LlOCVILL&'s Se&IEB OF EIGHTEEN ARTICLES. 333
liouYitle* remarked that if we multiply the members of (a) by x*f
where p = 2щт, and sum for p = 2, 4, 6, •• -, we get
-lf(s'- s") -/(I' +
which includes various formulas of the theoiy of elliptic functions* He
stated that it is easy to prove (a) and then deduce (a), and that be had
ш his lectured at the College de France given a direct, elementary proof of
(a), based on Dirichlet41 of Ch. VIII, the method applying to (b) and with
slight changed to the other formulas*
For any integer m, let
B) m = m* + i»", m" ~ У4ГФ > 0 (<T, *" odd and > 0),
while m' may be negative. Then for F(- x) = - Fix), F@) = 0,
0 цщф square.
A discussion of the case F{x) = ж shows that, if we set
«r - fi(m) - 2f,(m - 1) + 2f,(m - 4) - 2ft(» - 9) + 2fl(m - 16) ,
continued as long as the argument of f i is positive, then for m even,
~f42"/~ Jr4 2 /~1 0 ifm
> (\ o* (\- 1 = equare,
*~f42"/~ Jr4 2 /~1 0 ifm + square,
while for wt odd,
^-Л | 2r f'"-9^ fOT if m= square,
2 ^+2f4 2 ^ 10 ifm+square.
Using the same partitions of m and a function such that
*(* - У) - *[*, V)> *i- *,»)-- '№ Л *@, у) - 0,
liouville stated in his eighth article that
5" -
J 0ог^{лй1) + '(<An3)+ +^(^2^- 1),
according as *n is not or is a square. As a special case,
p(m) - 2P(m - 4) + 2p(m - 16) - - - - = 0 or (- l)
For ^(x, y) a function of г only, G) reduces to @).
Set &(x, y) = (- l)*t*F{x, y), so that F is an odd function with respect
to x and to y. Then G) gives
W = 0 or {- 1)^{^(^ 1) - F(^ 3)
according us mm not or is a equare.
k Jonr. de MAth., B)v 4,1859,1-8, 72-80. Seventh and eigbth tittda.

334 History of the Theoby of Numbers. [Сна*, xi
Iiouville4 stated that, for a function f(x) ~ /(— x),
where the summations relate to the partitions B) and m = m] 4-
respectively.
Fop m = 8? + 5, Дх) ^ л; вш (xx/2), he derived the relation
p(m - 4) - 4p(m - 16) -f 9p{m - 36) - - - -
It follows that, if we effect in all possible ways the decompositions
m = & + ^ + s;7 m ^ n2 H- 4(nf 4 + nj) (e>0, n odd and
If, in place of the second type of decomposition, we employ
m = r* + ri+ >>• +ii,
where rf n, - - -, и are positive and odd, then
For the same two types of partitions and for a function f{x, y), even
with respect to x and to y, Uouville stated in his tenth article that
(ту) = 0 or /(VS, 2^ - 1) 4- 3/(^m, 2^ - 3)
according as m is not or is a square. If Дх, у) is a function of 7 only> this
reduces to (£).
For the same partitions and for a function &{x> у, г, t), even with respect
to x, yt z and odd with respect to Z, it ш stated that
l" H- ^', ^" - 2m\ 2T"d" -f m' - 5", a")
according as m is not or is a square. К & — ty(z, y), W becomes {ij)«
Other noteworthy cases are & - tf(z) and & ^ F@«
Liouville7 stated in his eleventh article that, if / is an even function,
(в SZ(- Xf-**Sifr - 2mO = /(l)pBm - 1) + /C)рBж - 9) + - .,
the summation extending over all integers m' and all divisors 5" of m", where
C) 771 = гш'1 + m"t m" » rf" (m" odd and > 0).
' Jour, de Matb4 B), 4r 1859,111-120,195-204. Ninth and tenth articles.
» J<mr. de Math.r B), 4» 1859, 281-^304. Eleventh article.

Liotjville's Series of Eighteen Articles. 335
The second member of ({) equals 2/(i), the summation extended over all
the decompositions
D) 2m = г2 + *i + P* (h 4 odd and > 0, p even).
Torf(z) = (- ly^Px, the first member of (# is 2(- lI"'^™ - 2т*)
and equals i^y where E is the excess of the number of cases in which mf
ia even over the odd cases in
m — 2m'* + m* 4- - • • H- w* (m', wi^ any integers),
since Sf i(m) is the number of representations of яг as a sum of 4 squares for
m odd.
Let ctfj be the number of sets of solutions of
m = 2m'1 -f ™f + * " + ml
in which m' is odd, $?z the number in which m' is even. Then a discussion
of (£) for the case/(x) = x* leads to the result, relating to D),
- 2г* - 2pl if m = 1 (mod 4),
- 2t» - Zp1 if m ^ 3 (mod 4).
If Jlf is the number of solutions of D),
2mM = 22Я2 H- 2^
Let Да?, у) be a function even with respect to x and to y. Then
(x) Z2(- i)^-«*f(|" - 2m'f 2rf" H- 4mO - Z2(- I)^"^!, 4 H- W,
where the flummation on the left relates to C) and that oa the right to
2m » m* 4- я**, m2 = <ii5t (?»j> i«2, d% odd and > 0)*
If/ reduces to /(s), (x) becomes (£). Also,
(p) 4S2(-l)*'+<l"-«*/B^d" + mO - 2Ж(-1)у<«') ^ 2(-l)"-y(-Jm) or 0,
according as m is a square or not, where rn is any integer and
m ^ m'1 4- m", w" = 2^d", ro = ^ H- e^ 4- s"*,
m", d", 3" being positive and the last two odd*
A discussion of the case m - 8v + 7,f(z) » ^, shows that Ni/N% = 17/20,
where iV\ is the number of representations of m as a sum of 7 squares in
which the first square is odd, and N* the number in which the first square
is even, including zero»
For m odd and f(x) any even function,
0 if ro+ square,
ш * im'» ^ d"i" (d", a" odd and > 0).
For m = 4^ + 1, this formula holds for any function/(*)♦
Iiouville8 stated that for F(z, у, г) odd with reepeci to x, y, and г,
(и) 22FB^(i" + m', «" - 2m', 2^'+ЧГ' -f 2m# - i") ^ 0 or 2

336 History of the Theory of Ntjmbbrs. [Chap, xi
according as m is not or is a square, where j = 1, 3, 5, «••, 2Vin — 1,
ж - я* + 2*сГ1" (<*", 5" odd and >0).
This becomes («) for F - (- l)b-**F(x, y). Next,
22F(<T -f m'> *" - 2m', 2tf" -f 2m' - 5")
^ 0 or £ F(^, «, «) - £ *■(*, 2чЦ 20,
according as m is not or is a square, the summation on the left relating to
m = mft 4- d" (d" > 0, I" > 0).
For m, #', 5" odd and positive,
(x) 2ZF(d" H- 21»', 5" - 2m', 2m' H- d" - в") = 0 (w = гт'1 + d'0.
laouville* stated that for a function F(xt y, z) odd with respect to xy yf
d
(A) 2F(fc - 2mt, dt H- 2mt - mu ^ + 2т» + mj = 0,
the summation extending over all partitions of a given integer m = 3
(mod 4):
m « ffiJ H- 4wJ + 2d»5, (mi, *, 5t odd, d» > 0, fc > 0).
Take
^ B, u) being odd with respect to x> even with respect to u. Then (A)
becomes
(A,) S^(fc - 2nht h H- 2w,) - 2?(h - 2тг, тоО-
With the same notations, Liouville stated in the fourteenth article that
(B) 2^E, - 2mt> di + 2mi-mlth + «0 = 0,
and if &(x, y> z, t) is changed in sign by a change of sign of x only, or of
у onlyf or of both z and t,
(C) 2&(h - 2mtf dt + 2m* - «u d, H- 2w, + «ix, ^ H- *»i) =0»
When ^ is independent of i or г, (С) becomes (A) or (B), respectively.
In the fifteenth article is given the following generalization of (C):
where a > 1 and the sign of /3 is chosen so that H<* + Л » odd, while
the summation in the first member applies to the partition
m = mf + 4*$ H- 2*4fcfc (тъ<4, «tOdd,A > 0, J^ > 0),
ж being a given odd integer > 1.

Chap, XI] LlOtTVILLE'S SERIES OF EIGHTEEN AbTTCLEB, 337
Iiouville1* stated that, if &(x, у, г, f) changes sign with xt or yt or both
*and J,
2тг, dj -f *»t — тг, d* -f wij + mh h -f 2*ii)
^- I,a-i,e + 6, » + 2e + l) (a'H-fr* = *i, a>0),
where the summation on the left relates to the partitions (of any given
integer m)
m ^ m\ -f m\ -f <Wi (* > 0, 5, > 0, 5, odd).
IiouviHe11 stated that if -ф{х, у) is symmetric and even with respect to xf
H- 42(- l)cfc-i>**-cH-D^BdljF 2*+14),
where the summations relate to the partitions, hi which m is odd:
2m = df6' H- d"b'\ m^dt, m = <*Л H- rfj5i,
all the symbols being positive integere and, with the exception of a2, odd.
In the eighteenth article, Iiouvifle employed a function W(xt y), odd
with respect to x and even with respect to y, and stated that
z{- 1)^-«*{^(* и- d", a' - a") + *<<*' - #', v + Щ
= 2ГB<*, 0)
For ^(j5, y) ^ x, the latter gives
SnfmXw") = fi(tn) H-
the summations relating to 2m = m' 4- *л", m = ffli+ й11^, where the ?n'e
are all odd aad positive*
G» L. Dirichlet12 proved (a) of LiouviBe1 for a — 1. G, Humbert" gave
a proof by use of infinite series. G. B. Mathewe" gave a proof.
J, Liouvifle11 stated his8 formula (?) and that
Z2(~ 1)MB^ -f m' - *)№d H- m\ 2m' - a) = 22B*<* - 5)/(mr, 2arf + 5),
where the double accents on m, <%, d, ^ have been dropped»
Iiouville1* considered two arbitrary functions f(m) and F(m) having
definite values for m = lf 2, 3, - - -, and set
where each summation extends over all divisors d of m. For any real or
complex numbers я» '>
XU~X9{d)Z£*) = 1*-Z.WXJ$ E - mid).
If we take/(m) aad F{m) to be powers of m, we obtain a formula concerning
"Jour. deM*tb,g), 9,18*4*38*400. enteenth article
u Jour. 4? МдОь, <X), 10,1885,1Э5-14^ 169-176 A7th and 18th articles).
"BuILdeeSe. MsiiL, C2), 33,1, l«»y S8-60; fetter to IiouviHe, Aug. 27, 1858.
11 /Wd^ B), 34,1, 1910,29-3L
u Proc London U&tk Боец ЗД 1ШЗ-4,85-ft^
ц BolL dee BclM*A^B>, 33,1,19»f«l-tj lrtt»to Diriohlet, Oct. 21r 1868.
u Jour, de Mirtb,, C), S, 1668, №G&

338 Histoiiy of the Theory or Numbers. CCrap. XI
the sum ^(fr) of the /*th powers of the divisors of к and given in Ch. X of
Vol, I of this History, From the above formula we readily pass to
V. A, Lebesgue17 noted that for any integer mt
which reduces for the case m a prime to the final formula (H') of Liouville.2
Liouville18 gave formulas of the type of those in his series of articles-
Liouville19 noted that, for any integer m,
m - m\) = 0 or mDm -
according as m is not or is a square» This follows from (ф) of liouville8
with F(x, y, z) = xyz.
H. J. S. Smith19* gave a proof of (a) and
the summations extended respectively over all solutions of
where d't b', d\, Ь\, ту Ш\ are positive and odd, while f(x) is an odd
function.
С. М. Piuma20 proved (e), (L), (N), G), and (,).
E. Fergola21 stated and G» Toreill22 proved a theorem related to one in
Liouvffle's seventh article. Let a4 denote the product of the highest
power of 2 dividing n by the sum of the odd divisors of tl Then
a* - 2а^г H- 2ап-ч - 2a*_9 + 2a^w - 2а„_« + - - = (- I)" n or 0,
according aa n is or is not a square.
S. J. Baskakov43 proved the formulas in Liouville's twelfth article.
T. PepinM proved all the formulas in Liouville's first five articles except
(f) and its speciaUzations (H), (g).
N. V» Bougaief26 proved some of the theorems in Iiouville's series of
articles by showing that, if F(x) is an even function, an identity
Jl in сое mx - У\Вп cos nx
implies SAtJ?(m) = SBnF(n)t and a similar theorem involving sines and
an odd function Ft(n).
~~ и Jour- <Je M»th^ B), 7, 1862,256. " "^
" Ibid., 4iwg, To be eoneidered under cbu» number in Vol. Ш.
i» Jour, de Math., B), 7,1862 375-6.
^Report British Амос, for 1866, art. 136; Coll. Math. Рарегн, I, 346.
» GionuJe di Mat., 4,1866,1-14, 66-75, 193-201.
^ Gioniale di Mat., Ю, 1872, И.
»Jbtd., 16,1878,166-7.
tt Math. Soc, Moeoowr 10,1,18Й2-^, 313.
" Atti Accad, Pont. Nuoyi Lined, 38,1884-5,146-162.
» Math. Soc, Moecow, 12,1885,1-21.

Снлг. XI] LlOUVlLLE'S SEBIfiS OF EIGHTEEN ARTICLES. 339
Pepin2* proved all the theorems in Iiouville's first five and last two
articles, and (L) of the sixth.
E. Meissner27 proved all the theorems in Liouville's articles VII-XVL
Thus there remain unproved essentially only (N) and (Q) of the sixth
article. [Piuma,M pp. 197-201, proved (N).]
P. Bachmann2* gave an exposition of selected formulas from Liouville's
series»
A. Deltour2* proved (a) and recalled how it implies that, if m is odd,
the number of decompositions of 4m (or Вт) into a sum of 4 (or 8) odd
squares equals the sum (or sum of cubes) of the divisors of m.
P, S. Nasimoff30 proved formulas (a) and (c) of Liouville,1 (F) of Liou-
(P) of Liouville/ one of Liouville,18 and related results,
» Jour, de Math., D), 4, 1888, 83-127,
" Zurich Vierteljahr Naturf. G«a^ 52,1907,156-216 {Dim., Zurich).
" Niedere Zahlenthoorier 2, 1910» 365-433,
" Nouv. Ann. Math., {4), 11, lflll, 123-9.
w Application of Elliptic Functions to Number Theory, Moscow, 1885. Frenc
Annaleg sc» de 1тёсо1е norm, euper^ C)j б, 1в88, 147-64»

CHAPTER XII.
PELL EQUATION;
MADE A SQUARE.
The very important equation x* — Dp* — 1, which has long borne the
name of Pell, due to a confusion originating with Euler, shonld have been
designated as Fennat's equation (cf. papers 41, 62-64).
There appeared in India and Greece as early as 400 B.C. approxima-
approximations a/6 to V2 such that a2 — 2b* = 1, and similarly for other square
roots, the derivation of successive approximations being in effect a method
of solving the Pell equation. For example, Baudhayana, the Hindu author
of the oldest of the works, Sulva-sutras, gave the approximations 17/12
and 577/408 to V2. Note that
5 + 2^-21 17I-2-12' = l. 577-2.408'=!-
Proclus1 D1(M85 A,D.) noted that the Pythagoreans made the fol-
following construction: On the prolongation of the side AB of a square
with the diagonal BE lay off ВС - AB, CD = BE. Then
•- 2AB2 4- 2BD*.
But CD2 = BE2 = 2AB2. Hence
AD2 = 2BD* = FD\ FD = AD = 2AB H- EB,
E
\
for the sides ABf BD,
Then
Also BD = AB + EB. Write sLf $*,
du dh - - for the diagonals BE, FD,
8»+i = S» 4- dnt dn+i = 2s* + d^
Now let sj = 1 and replace di = V2 by the integral approximation d
and employ our recursion formula with dn replaced by 6*. We get
si ^ sx H- «i = 2, ^ = 2s1-^-6l = 3,
»8 = s2 + «i = 5, *i ~ 28, + £« - 7, -»»
Then 6n, sn give a solution of 6* — 2s2 = {— 1)".
and
1 In Platonis iwn publicam commentarii, ed», G» КЫ1, 2,
Hultoch), 393-400.
341
lj 24-^9; excur» II (by F.

342 Histohy of the Theory of Ntjhbers, [Chap, xn
Theon of Smyrna* (about 130 A.D.) called the a'a and 6's side and
diametral (diagonal) numbers and gave tbe above recursion formulae without
the geometrical interpretation.
Archimedes (third century B.C) gave the approximations 265/153 and
1351/780 to -d§, which can be explained in connection with x* — 3v* — — 2,
x2 - dy* = 1.
Heron of Alexandria used the approximation a + r/Ba) for Va2 + t.
For a more detailed account than what precedes of the connection be-
between the knowledge of the early Greeks and Hindus of approximation to
square roots and Pell equations, see H, Konsn* and E. E, Whitford,4
The history of the cattle problem of Archimedes will now be discussed
hi detail.
In 1773, Gotthold Ephraim Leasing* published a Greek epigram in 24
verses, from a manuscript in the Wolfenbuttel library, stating a problem
purporting to be one proposed by Archimedes,8 ш a letter to Eratosthenes,
to the mathematicians of Alexandria, as well as a scholium giving a false
answer, and a long mathematical discussion by Chi\ Leiste. The problem
is to find the numbers Wt X, K, Z of the white, black (or blue), piebald (or
spotted); and yellow (or red) bulls, and the numbers u>, xt y} z of the cows
of the corresponding colors, when
(I) W = (I + i)X + Z, B) X = a + i)Y + Z,
C) Y = ft + № + Z, D) » = (i + J)(X + x),
E) x = A + t>(K + y), F) у - (| + £)(Z + z),
G) г = (J + \)(W + to), (8) W + X = О,
(9) У + Z = Д,
the final notations being those for a square and a triangular number»
Leiste found at once the integral solutions of A), B), C):
A0) Y - 1580m, Z - 89Ы, W = 2226m, X - 1602m.
Then, by D), m - 2p, x = 12a. By E), a = 3/5, у =; 20D/3 - 158p).
1 Pl&tonici . . . ezpoeitio, 1544j 67. Theon Smyrnaeus» ed-> Б. НШег, Ldpog, 1878, 43;
French traneL, by J. Dupuie, Paris, l$93j 71-5.
•Geechichte der Gleichung i» — Dw1 = 1, Leipaig, 1901j 2-17. Reviews by Wertheim,
В1Ы. Matb., C), 3,1902, 248-251; and Tannery, Bull, dee 8c. Math», 27, П, 1903, 47»
1 Tbe РеЙ Equation, Columbia Univ. Die*., Ne* York, 1912,3-22. The following related
papereait: not mentioned in the pages juat cited: Б. S. linger, Kuuer Abrise der Geach.
Z. von Pythagorae bie Diophant, P*>gr.r Erfurt, lS43j C- Henry, Boll, des Sc. M«th.
Antr., B), 3, I, 1379, 515-20; H, Weiesenbom, Die uration*iea Quadmtwurtdn bei
Archimedes und Heron, BeHin, 1884; Zeitech*. Math. Phya., Hitrt.-lit, Abt., 28, 1Ш,
81; E. MahleTj ibid.t 29, 1S84j 4I-.3; W. Schoenbarn, 30,1885, 81-90; C. Demme, 31,
1886, 1-27; K. Hunrath, 33,1888,1-11; V» V. ВоЬушп» 41,189вк 193-2П; M.Curtse.
42,1897,113,145; F. Hultech, Gottib^en Nachr., 1893,367; Q. Wertheim, Abb. G«ech,
Math., Vm, 146-160 (in Z^tecbr. Math. Pbys., 42, 1№T); Zeibchr. Math. Naturw.
XJDterrieht, 30, 1899, 253; T, L, Heath, Euclid1» SJemente, 1, 190S, 396-40L
1 Zur Gescbichte der titemtnr, Braunschweig, 2,1773, No. 13,421-446. Leering, Sammtliche
Schriften, Leip»ig, 22,1Й02, 221; 9,1855, 285-302; I2r 1897,10CV-15; Open, XIV, 232»
' Archimedes opera, ed., J. L. Heibeigj 2,1881, 450-5; new e*L, 2,1913, 528-34.

Chap.xu] Pell Equation, си?-\-Ьх-\-с = Пщ 343
By F), p - 5g, z = 3Oy, У = 11B97? + -у), whence П7 - 800 - 19067g.
Then G) gives 30y =* A505g + /5I3/2, g - 2r, 0 - 25. Comparing the
resulting у with the earlier 7, we get a linear equation in dt r, whence
r = 4657u, 5 = 1359235w.
By substitutions, we get m = 93140«, whence
W - 207329640«, w = 144127200u
X - 14921028Ou, x= 97S6492Ou
{W) Y - 147161200w, у = 70316400*
Z * 82987740«, z = 108784260u.
For « = 4, we get the numbers in the scholium; but they satisfy neither
(8) nor (9), since neither W H- X nor 8(У -f Z) H- 1 is a square.
Returning to {10'), we note that the greatest common divisor of the
numerical factors is 20, whence и - »/20, where v w an integer» Then
W + X = 4-957>4657», p - 957^4657n2r
йшсе ТГ + X is to be a square. Then Y H- Z = C^2 + Q/2 gives
B* + IM = 8(F + Z) + 1 - an2 + 1, a = 410286423278424.
Since a is positive and not a square it is possible to choose an integer n
so that an2 + 1 = П by Euler.81 If the resulting square is even, we can
deduce one making an* + 1 an odd square (Еи1ег,м § 86, § 88)»
J. J. I. Hoffmann7 said the problem was due to a much later computer»
J. Struve* gave a 36 page discussion making no advance over Leiste.
Gottfried Hermann9 made an interpretation which led, not to (8) and
(9), but to W + X ** a square whose side is of the form a*{a — ЪO
У + Я=Д,ТГ + Х+У + Я = Д1. Thus if we take the numbers A0),
we must make
3828m = {a\a - Ъ)\\ 247Ы = C(C ^ ^ f d® + l\
He stated on the authority of К. В. Mollweide that С. К Gauss had com-
completely solved the problem under the earlier interpretation, but had not
published the solution,
J. Ft. Wunn,w in a review of Hermann's paper, replaced (p. 201) condi-
condition (8) by the condition that W + X shall be a product of two approxi-
approximately equal factors. Without returning to this condition, he passed to (9):
Y + Z = 2471m = 2471 15U - A.
? Ueber dio Arith. der Griechen, Mains, 1817, Intnorf.j p. xvi (tranaL )
8 Altce priechischca Epigramm^ maihemaiischen Inhalta, von Leasing eret einmal zum Druoke
befOrdertj jetrt neu abgtxiruckt und mataematiech und kritisch behandelt voa Dr. JP
Struve UDd Dr. K. L- Strave, Vater und Sohn. Aitona, 1821, 47 pp.
'Ad tnemoriam Kregclio^St^rnbachianam in and. jur. die 17 Julii 1828: De Axchimedia
Problemate Bovmor Umveraitute programm, Leipzig, 1828. Reprinted in Godofredi
Hermanni, Opvscvbj lipeiae, 4, 1831, iii-v7 228-238.
10 Jahrbucher fur PhUobgie u. Paedagogik (ed.r J. С Jfthn), 14, 1830,194-202.

344 Нютовт о* the Тнвовт of Numbers, [Chap, xn
The least t ia 990, the aide of A being then 27180. He considered also
higher values of t, but save no final answer to (l)-(9).
G, H. I\ Neeeelmann" argued that the final part of the epigram leading
to conditions (8) and (9) was a later addition, partly since he believed that
triangular numbers were not employed in Archimedes' time (a view already
expressed by G. S. Khlgel").
0. Terquem1* stated that the tenth condition added by Hermann is
incompatible with the earlier conditions.
A. X H, Vincent14 regarded as spurious the conditions relating to the
cows. By the first three conditions, we have A0). Then Y + Z = 2471m
is to be a A and this ia the case if m = 99-122314, the side o! the A being
244628. Then 4JW + X jb approximately 861182, which jb very nearly
the area of Sicily in square *fcufe«, in accord with Vincent's interpretation
of the condition to replace (8).
C. F, Meyer11 duplicated the paper by Leasing and discussion by Leiste,
adding merely that, in attempting to make cm' + 1 a square by the con-
convenient method of Kausler, he had carried the development of Va into a
continued fraction to the 240th quotient without finding the period»
A.Amthor16 showed that Wunn's problem (l)-G), (9) is satisfied by
taking и =* t>/20, v — 117423 in Leiste'e values of TF, * - ♦, z> вшое then
Y + Z ~ 1643921 1643922/2, W+X = 1485583-1409076.
For the main problem (lb(9), be satisfied (8) by taking v - /-4657л1,
/=; 3-11-29 = 957, as in Leiete, Then in (9), viz., Y + Z = q(q + l)/2,
set t = 2q + 1, u = 2-4657n. We obtain the Pell equation
? - Du* ~ 1, D = 2-7-/-353 = 4729494.
He found that the continued fraction for Vd has a period of 91 terms
and obtained as the least solutions
T = 109 931 986 732 829 734 979 866 232 821 433 543 901 088 049,
U = 50 549 485 234 315 033 074 477 819 735 540 408 986 340.
It remains to derive the least solutions tt и in which и is divisible by
2-4657, so that n shall be integral. By proving and applying general
lemmas concerning tk + щ^О = (T + U*0)k, he found that, for к — 2329,
tkj uk is the desired pair. He verified that W has 206545 digits.
B. Krumhiegel" made a historical and philological discussion of the
problem and concluded that, while the epigram itself ш probably subsequent
to Archimedes, the problem itself is due to him. This accords with the
"Die Algebmder Grieche*, Berfm, 1842,488. On p. 485, hia g = 57 - ahouU be 54 ^
и bf&tb. WMerboch, 1,1803,184. a. M. Cantor, G^echichte Math., <xL 2,1, 297; ed. 3,1,
31Я
« Noov. Ann. Math. 14,1855, Bull. BibL, 113-124,130-1. He &t first attributed incorrectly
Hennann'e paper to F. E. ТЪешю.
"Nooy. Ann. Math., 14, 1855, Bull. BibL, 165-473; 15, 1856, Bull. BibL, 39-42 (ratond
Greek text *nd French tzud.).
» Em diophimtbcbi* Problem, Prop., Pot*kmf 1867,14 pp.
"Zeitechrift Math, Phya, 25, 1880, Hvt.-Iit. Abt,, 153-171.
11 ZeiteeJjrift MMh, Pby».t 25,1880, Hkt.-Iit. Abt, 121-136,

PKLL EQUATION,
view of J. L. Heiberg," P. Tannery,1» R Hultech,* T. L. Heath,11 and
S. Gunther*
A, H. Bell" found a " complete solution," baaed on the an* + 1 = П of
Leiste, involving numbers of 206545 digits, as by Amthor.18
G, Loria,*" M> Merriman,*16 and R. C- Archibald118 gave accounts of the
cattle problem.
Diophantus (about 250 A,D.) was frequently led to special Fell equa-
equations in solving problems in his Arithmetic*. In II, 12, 13, 14, 29, be
made i1 + lt if + 13, if - 1, if + 1, V + 9 equal to a square s1, by
taking z =y — 4, у ~ 4, у — 2, у — 2, Zy — 4, respectively, and similarly
in II, 30. In III, 12, 13, he avoided the initial equations 52s1 + 12 » D,
266V — 10 = D, since 52 and 266 are not squares [though x = 1 is
a solution of each], and, beginning anew, was led to y* + 12 = D,
77^* — 160 « D, which he solved by equating them to (у + 3I and
G7*-2)f, respectively. InlV, 8,33,he treated 2a? + 4 = D = Bz - 2)*
and 7m* + 81 = D =* (8m + 9)*. In V, 12, 14, he discussed
26V + 1 = D = {5x + 1)*
and 30s* + 1 = D =* Ez + II. §o far, the problems solved are all of
the form ex* + b — О with either a or 6 a square. In VI, 12, he stated the
lemma: Given two numbers whose sum is a square, we can find an infinitude
of squares 8 such that, when the equare 8 is multiplied by one of the given
numbers and the product is added to the other, the result is a equare.
Thus, given the numbers 3 and 6, let з = (s + IJ; then shall
3(a? + IJ + 6 = Зя1 + 6r + 9 = D,
say C — 3a;I, whence x = 4; and an infinitude of other solutions can be
found. This lemma is applied in VI, 13, 14 to 12s* + 24 « О to obtain
the solutions x = 1, 5* In VI, 15, 15V - 36 = D is said to be impossible
since 15 is not a sum of two squares. In VI, 16, be made the important
statement that, given one solution of A& — В ~ jf, we can find a second
solution; thus, given 3*5* — 11 = 8l, set x = 5 + zt whence
3E + z)% - 11 = 3z7 + ЗОг + 81
will be the equare of 8 - 2г for z ^ 62. In VI, 12, he had made the more
u Question** Archimedeae, Dim. Hauniae, 1879, 25-27; Philok*iM, 43,1884,486.
" Mem, вое. ее. рЬув. xa&. Bordeaux, B)r 3,1880, 370; Bull d* Sc. Math, et Aatr,, B), 5,1,
1S81>26-3O; Bibl.Math.,3,1902,Ш. BeprmtedinTiuimfciybMemoira»aeientifiquee,
1,1912,103-5,118-23.
*> An&imedflB, in Fbuly-Wiswwa'e Heftl-Encydopidie, Пх, 1896, 634,1110.
11 Diophantus, ed. 2,1900,11-12,122, 279; Archimedes, 1897,319; Archimedes1 Werke, 1914,
471-7,
9 Die qpadr. IzntkinfllitSten, etc^ Zettachrift Math. Phys., Abb. G*ech. Math., 27,1882,92.
Tbia and K» Hmu»th's TJeber das Anerohen der Quidratwarid bet Qriechen und Indern,
1883, wra reviewed in La Revue Saentifiqae, 1884, I, 81-3, 499-502.
"Math* Мдакте, Wwhington, 2, 1895, 163-4; Amer. M&tlb Monthly, 2, 1805, 140-1
A,1894,240).
"*Ье science eeattenen'utica Gieda, ed. 2,1914,932-9.
mThe PopuJar Science Monthly, 67,1905, 660-6.
^ Amer. Math, Monthly, 26,1918, 4111

346 History от the Тяеовт of Numbers. [Chap, xii
special remark that fix2 + 3 =* □ has an infinitude of solutions, since it
has one solution x = 1.
Diophantus solved Ax* + Bx + 0 =* у- only in the following cases.
(a) If A is a square, a*, set у = ax + тп, whence x is found rationally;
examples in II, 20, 21, 23, 24, 33, III, 9, 16, 18, IV, 15, 21, V, 3, 4, 18, 20.
(b) If С = e2, set у =* mx + c; examples in II, 17, IV, 9, 10, 12, 14, 45.
(c) In IV, 33, 18 + 3x — x* is to be made a square, say mV, where
(m* + 1) 18 + (fI - П. Then, multiplying by 4, 72m2 + 81 - Q, say
(Ы + 9J, whence m = 18, 18 + 3z - 325x* = 0, x « 6/25, In general,
as remarked by Nesselmaim11 (pp. 333-4), the corresponding condition that
tho root x of Ax1 -f Bx + С - m*z2 be rational is £#2 - AC + Cm2 = П,
and, as in (b), can be satisfied if IB2 — AC is a square.
While H. HankeP* believed that Diophantus was influenced by Indian
sources, M, Cantor15 took the opposite view except as to integral solutions.
P. Tannery26 went to the extreme of believing that the Greeks influenced
the Indians also in the question of integral solutions, while even the cyclic
method Cnext explained] is only a variation of the Greek method of solv-
solving t2 — Du* — 1, since from the Greek method of deriving from one ap-
approximation to V# a closer approximation it is easy to pass to the Indian
method.
E. B. Crowell17 compared the work of Diophantus with that of Brahme-
gupta,*8 and the first solution by Brouncker" with that of Bh&scara.50
Brahmegupta24 (born 598 A.D.) gave a rule to find x m that Cz* + 1
shall be a square. Assume any "least root" L and add to Cl? such an
"additive" number A that the sum is a square G2; call G the "greatest
root" [L and G are values of x, у satisfying Cx* + A = #2J. Write L, G,
A twice» By cross multiplication, we obtain a least root LG -\- GL, while
CLL -h GG is a greatest root, for additive AA; dividing these new roots
by A, we get roots for additive unity. For details, see Bhascara.M
For example (§ 67), let С ^ 92» Take L = 1, A = 8, whence G = 10.
Then 2LG *= 20, 92L2 -f G2 - 192 are least and greatest roots for additive
64. Dividing them by 8T we get 5/2 and 24 as roots for additive unity.
By composition of the last pair with itself, we get other roots 120 and 1151
for additive unity.
By composition of the roots for additive unity with the roots for additive
A, we get roots for additive A (§ 68, p. 364)» For example (§ 77, p. 368),
from 3-302 + 900 = 602, 3-11 + 1 = 22, we get the least root
30-2+ 1-60 « 120
and greatest root 3-30-1 +60-2^ 210for3>120* + 900 - 2102.
" Zur CeBchkbte der Math, in AJterthum und Mitteblterr 1874, 2Q4.
*Vorl«e. Uber Geecbichte Math,, 1, 1880, 533; e<L 2, 556; ed. 3r 596,
» Mem. Soc. Sc. Phy* Nat» Bordeaux, B), 4, 1882, 325,
« M. Elphinstone'a History of India, ed. 9 1906, 142, Note 16 (ed., Crowell),
" Brahm&ephutVfliddli&itA, Ch* IS (algebra), ($ 65-66. Algebra, with arith. and теояига-
tion, from the 8a£L9crit of Braimegupta and Dhdscarar tranel.by H.T, Colebrooke, 1817,
p, 363, Cf» Simon.1**

Pell Equation, <*z4-kr+c=CL 347
We may deduce roots for additive unity from roots for additive
=b4(§§ 69-72, pp. 36S-6). If CL* + 4-<?T then L(G2 - l)/2 and
Q(Qi _ 3)/2 are corresponding least and greatest roots for additive unity.
If £L* - 4 = <?, and we set j> = (<? + 1)((? + 3)/2, then pZ<? and
(p — 1)(<? + 2) are corresponding least and greatest roots for additive
unity.
If the coefficient С be a square (§ 73, p. 366), divide the additive by any
assumed number Ъ. To the quotient add Ъ and from it subtract b and di-
divide by 2. The first result is a greatest root; the second, divided by the
square root of C, is the corresponding least root.
If the coefficient be divisible by a square f (§ 75, p. 367), use the quo-
quotient as a new coefficient and find roots» If the least root so found is divided
by t, we get the desired least root. The greatest root rema us the same.
For С = 3, A = -800 (§ 77, p. 068), remove the factor 202. For the
new additive — 2, we get roots 1 and 1. Their products by 20 are the roots
desired»
Alkarkhi* (about 1010) solved x* + 5 =* y2 by setting у ** x + 1, and
x* - 10 = y2 by setting у ~ x - I. To solve 77V ~ 160 *- to2, set
w = 77s — 2. To solve (pp. 72-4) x1 + 4x = y2, set у = 2х\ to solve
4z* + 16z + 9 = y2, set |/ = 2x — и, where n* > 9, say n = 5. As the
condition (p. 113) for rational solutions of ± (ax — b) — x1 ~ D, he found
that |a* T b must be a sum of two squares. Finally (p. 121), & — w2 = ap
for v « (a -f 0)/2, w =» (a — Й/2.
Alkarkhi2*1 used the approximation a + r/Ba -h 1) for Va* + r.
Ibn Albaima2* (born about 1255) used the same approximation when
r > a, but for r ^ a employed a + r/Ba). The latter was used by Heron
of Alexandria and by Elia Misrachi A455-1526) in hie Arithmetic (ed.,
G. Wertheim, 1893, 1896).
Bh&scara Acharyaw (born 1114) gave a method of deducing new sets of
solutions of Cx2 + 1 = y2 from one set found by trial. Take any number
+ 0 and call it the "least root" L [for additive A J By the positive or
negative additive quantity A is meant a number which added to or sub-
subtracted from CI? makes the sum or difference a perfect square, its root being
called the "greatest root" G. Thus if С = 8, L = 1, A = 1, then G - 3.
Composition (§5 76-77, p. 171). From these roots Ц G and the same
or a new set of roots I, g, we obtain by cross multiplication and addition a
new least root X = Lg + lGy while у — CLl + Og is the corresponding new
greatest root. The product of the two additives gives the new addit ve.
Thus (§82) for the former example, take I = 1, g = 3, A » i; then
* = 6, у = 17. Next, from L = 1, G = 3 and X = б, у = 17, we get the
new roots 35, 99 and so on indefinitely by means of composit:oii.
" Extnut du Fakhrf, Tndtt d'blgdbre par Ben Alhacan Albrkht (Arab MS.), Ftench trazul.
by F. Woepcke, Pat», 1853, 84f 120.
iU КДП Ш Huftb, Germao tranal. by A, Hocfahram, П, 14.
mLe Talfchye, p. 23» French transb by А. Мат, АШ Acnd. Pont. Nuovi Iinoei, 17f
1634, 311.
M Vlj**an1ta (aleebra), Cb. 3, §|75-Ю, ^Affected square.^ Colebrooke» 17(blg4.

348 History or the Theoby of Numbers. £Chaf~ хц
Or (§ 78, p. 171) we may take hg — Ю and СЫ — Gg на new roots.
A second method (§5 80-81, p. 172) lor additive unity consists in taking
the least root to be 2a/(л? — С), where a is arbitrary, and finding the greatest
root. Thus (end erf § 82, p. 174), for С « 8, take a = 3; the least root is
6 and the greatest is the square root 17 of 8-61 + 1.
Cyclic method (§§ 83-86, pp. 175-6). Taking the least root, greatest
root and additive as dividend, additive and divisor, find the multiplier
by use of the pulveriser (see papers 2, 4 of Ch. Д), If the excess of the
square of that multiplier over the given coefficient С be divided by the
original additive, we get a new additive. The quotient corresponding to
the multiplier and found from it will be the new least root, from which a
greatest root may be deduced. The operation may be repeated. We find
integral roots with 4, 2 or 1 for additive, and by composition deduce roots
for additive unity from those for additives 4 and 2-
For example E 87, pp. 176-8), to make 67x* + 1 a square, take 1 aa a
least root, — 3 as additive, whence 8 is the greatest root. Thus divi-
dividend » 1, divisor = — 3, additive ** 8. By the pulveriser, a multiplier
is 7 and the quotient is — 5, a new to** TO°^' ^Ibe *ew additive is
6 ~ G1 - 67)/(~ 3). By 67(- 5)* + 6 = 41J, 41 is the new greatest root.
Now start with dividend 5, divisor 6, additive 41, get the multiplier 5,
quotient 11 = least root, new additive — 7 = Es — 67)/6, and greatest
root 90. Next, start with drvidend 11, divisor- 7, additive 90. Reducing
the last by multiplee of the divisor, we get the abraded additive 6. The
multiplier is 2. Adding the negative of the divisor, we get the new multi-
multiplier 9 and the quotient 27, giving a least root. The new additive is
(91 - 67)/(- 7) - - 2, and greatest root Is 221. By composition of this
set of roots with itself, we get L ^ 11934, G = 97684, A = 4, Divide
the roots by the square root of 4* We get I = 5967, g *= 48842 for addi-
additive 1.
When unity is subtractive (§§ №-89, p. 179), the problem is impossible
if the coefficient С he not a sum of two squares. In the contrary case, we
may take as two least roots the reciprocals of the roots of the two component
squares. Thus (§90) if С = 13 = 2* + 31, the least root i gives the
greatest root f ♦ Doubling and applying the cyclic method, we have divi-
dividend 1, divisor — 2, additive 3. We deduce the multiplier 3 and quotient
— 3, the least root* The new additive is 4 and greatest root is 11- Repeat-
Repeating the operation, we got L ^ 5, G = 18, Л — — 1.
When € is a square a* (§95, p. 182) and the additive is A, least and
greatest roots are (for b arbitrary)
=(f-0- itt+0-
Bhascara solved various problems by the method of the affected square*
For 6V + 2y = c* (| 177, p. 247), Fy + II = 6c* + 1 for с ~ 2 or 20,
у = i or 8, To find E 178, p. 218) two numbers the square of whose
sum added to the cube of their sum equals twice the sum of their cubes,

Pell Equation ах*+Ъх+с= О. 349
take у — с and у + с as the number*, whence
с = 2, 28; j/ = 3,48.
For V - *W «= с1 (S 181, p. 249), divide by yK To find (§ 182) two
numbers whose difference is a square, and sum of squares a cube, take с
and с — n* as the numbers; the sum 2c* — 2спг + n4 of their squares is
equated to n* (a restriction), whence Bc — n8I = п*Bпг — 1), and 2nl — 1
is made a square. To make (§ 188, p* 253) уг + s1 and у + г both squares,
treat the first condition by § 95 with 21 as the additive and z as the arbitrary
number b; we get у = (г* — z)/2; the second condition now becomes
\(z* + z) = p*, or Bz + 1)* = &p* + 1, which is a square for p = б от 35,
The sum (§ 189, p. 254) of the squares of two numbers increased by their
product is to be a square; on adding unity to the product of their sum by
the root of that square, the sum shall be a square. The first condition is
found to be satisfied by the numbers fe and c; then the second condition
(h)(h) + 1 = П holds if с = 6 or ISO.
E. Strachey*1 translated into English the Persian manuscript of 1634
of Bhascara, To solve Ax1 + В = у*, take any square/2 and find a number
§ such that Ap + 0 is a square, say 0*. Then xf = %fg, y' « Af + 0*
satisfy Ax*% + p' = j^for /^ = 01; and
x" = x'g ± //, y" - 1^^ ± АхУ
satisfy
A*" + P" = ^ Э" - ffp.
If ^7/ s« Bp2, remove the factor p from xr\ yrf; we get a solution of the pro-
proposed equation (if firf = B/p2, multiply by p). Otherwise, we proceed as
before» For example, consider 8x* + 1 — y\ Take / — 1; then
8/* + l = 3J,
so that we take 0-1. Then
x> = 213 = 6, ^ = 8-P + 3J- 17, 8-ff + l = 171.
A new set of solutions is given by
s" = 6 3 + 17 >1 =35, y"= 17*3+ 8-6-1 =99.
For the cyclic method ("operation of circulation"), choose as before
[relatively prime] numbers / and g such that Ap + ($ — &. Then by an
earlier rule рог solving a linear Dioph&ntine equation] choose integers X,
Y such that (fX + g)j& ^ F. Chooae an integer m so that the difference
between (mfi + ^X}1 and Л shall be as small as possible numerically. Now
{тгф + Х)г — A is divisible by P; call the quotient 0\ Set x* ^ mf + F.
Then Jjc^ + ^ k a square, say j^. Unless f? = Bj? or B/p2, proceed as
before- For example, let A = 67, В = V Take / = 1, 0 = — 3; then
0 = 8, X - 1, Y 3, m 2,
11 Kj* GcraU, or tEe «l^Acm <rf t^ Himla^ Ьэшкмц 1ЫЗ, Infaodiictioii, pp. Э6-5Е,

350 History от the Theohy of Xtjmbers. cchaf. xii
The next step gives 0" — —7; the third, f?"•= — 2. His solution of
Xх — 61y* = 1 is quoted by Whitford* (pp. 37-8), who remarked that the
wording is clearer than in Colebrooke's translation.
El-Hassar** A432) obtained fop -Va^H- r, when a — %r — 1, the approxi-
approximations a + p = 9/4, where p = r/Ba), and
a + p- p?l\2(a + P)\ -161/72.
[Note that (9, 4) and A61, 72) are solutions of ж» - V = L]
Nicolas Chuquet33 obtained, in 1484, successive approximations to Vn
for n ^ 14» He began, by noting that VS lies between 2 and 3. Their
arithmetical mean is 2£; its square 6J exceeds 6 by \. Take the next
smaller term \ in the series £, f, Jt t, • • •. We have 2$, whose square is
less than 6- We now have an approximation exceeding the root and one
less than it. Adding the numerators and denominators of \ and \> we get
the new approximation 2$, whose square < 6» Similarly from 2| and 2%
we get 2$. In this way he obtained the approximations 2 4- r, where
r - ii #, f f Л-, A, «, », it, 0, ift, Jft.
[Tor r & 0, J, £, A, ^, ^^, 2 + r gives the successive convergents to the
continued fraction for -J8- To deduce a third convergent p2/£2 from two
successive ones pg/flo, Pi/^ij the law is pt = p9 + 2!pj, ^ = jo + Щ\> Thus
Chnquet^s process produced also intermediate fractions, obtained by re-
replacing z by smaller numbers.} Chuquet34 gave answers to the following
problems, but with no details as to solution» Find a square which increased
by 7 (or 4) gives a square; answer, 9 (or 9/4). Find three squares whose
sum is 13; answer, 11$, 1$, t Find three cubes whose sum is 20; answer,
15f, 3|, 1.
Jortianua Nemorarius8* noted that x{x + 1) is neither a square nor a
cube С if x is an integer Ф 0, — 1; for x — ^, it equals (f )*].
Estienne de la Roche" copied the above method of approximation from
Chuquet's manuscript.
Juan de Ortega in the later editions A534,1537,1542) of his Arithmetic»
gave the approximations
which correspond87 to the first solution of z* — Dy1 *= 1, and
which correspond to the second solution.
H. Suter, Bibliotbeca M&tb^ C), 2,1901, 37. Abo atmultwusouely by Alkakftdl, French
traiMl in Atli Aoe«d. Pont. Nwvi Lincei, 12» 1858-9, 402-4.
Le UipKrtj en Ь «заме <кя rombra, BWL ВОЛ. Stoau Sc Mat^ 13,1880, Й97-9. Bia-
cDa*d b^ B. GfcUier, Zcttochrift fiir 4a» HeUechuhrt^n, 2,1877, 430; К Rodet, Bull.
Soe. ЫтШ. de France, 7,1879, Ifi2; F. Tannery, BGibothtscs bbih., B), lr 1Й87,17.
Letriparty , . , , Appendb; BeBL BW. StonaSc. Mat., 14,1881,455,
ЕЗетши Ля&- deeem Hbt^ danos^r. Jeoibf Fmbri StepoVae», fWk, 1514, YI, 26.
L
,,9, Cf. F, Tannery, BftJkAbeta Mfttb,,
B), 1, Ш7,19^0

Pell Equation, ах*+Ъх-\-с^О* 351
J. Buteo** gave several approximations for V66 all of which give solu-
solutions of sl - 6%3 = 1, the last one being z/y,x - 8449,y = 1040. He also
made use of Chuquet's method»
P. A. Cataldi*9 gave approximations to VS by the two formulas used
by El-Hassar8* and used implicitly approximations by continued fractions.
Nicolas Rhabdas" used the first approximation by El-Hassar. It was
used later by Luca Paciuolo, Cardan and Tartagha (references, VoL I, Ch.I).
Fermat" stated February, 1657, that if D is any number not a square
there exists an infinitude of integral solutions of x2 — Dy2 = 1; for ex-
example, 2l - 3 I2 = 1, 7* - 3-42 -= 1. He asked for the least solution of
61^ + l = D and of 109y* + 1 = O, and a general rule for finding the
solutions of Dy* + 1 = П.
Although Fermat, in the introductory remarks to his "Second defi,"
had expressly called for solutions in integers, this introduction was omitted42
in the copy made for Lord Brouncker by the secretary of К Digby. This
explains why W. Brouncker and John Wallis4* first gave merely the rational
solution
of na* + 1 = y2y the case p = 1, e = 2rt giving Brouncker^s solution
x = 2r/(r* — n). ThelatterhadbeengivenbyBria^caTa^fsecoiidmethod),
and was obtained by Ren6 Francois de Sluse44 A622-1685) by setting
nz* + 1 = A - тх)\
Fermat" was not satisfied with these evident solutions in fractions.
W. Brouncker46 gave an infinitude of integral solutions x for n = 2, 3,
5, 6 and their products by squares; thus, for n = 2,
each numerator being equal to the corresponding denominator diminished
by the preceding denominator, while each denominator equals the numerator
of the term immediately preceding when reduced to an improper fraction.
[The formula ©ves \x = 1, 6, 35, 204,1189, •. >, with the recursion formula
tn+t = Gin — k~l»]
Waffis47 noted that if x — f is one solution, so that nP + 1 = P, then
x - 2fl is a second: nB/Z)* + 1 = BP — IJ, so that one can get an
"loan. BuieeniH Logistic*, quae et ftrith» » » ♦ , Lyons, 1B39, 76.
^ Ti^tUto del Modo Brevisaimo di trouare 1a Radice qu^ira deffi oumeri I6l3, 12,
rtP. Tannery, Notice 8arl«deaiarithm6tiqueedeNt Rbabdae, Fftrie, 18S6, 40,68.
"Oeuvne, П, 333-6, fetter to Frenicfe and " Second d6fi am matbemaiidenB " [WalHa and
Brouncker]; French tiwaL of biter, Ш, 312-3.
« G. Wertbeim, АЬЬшД. Gwhichte Math., 9,1899, 563.
«Commfiraum epietolieum de Wallis, Oxford^ 1QG&, 767; bound with Wattia' Algebra,
Oxford, 1665; WftUie" Орелц Oxford, 2,1603. French tnmal. En Oeuvree de Fermatt
m, 417-8; letter IX, W&ffis to Di^by, Oct. 7t 1657»
44 MS. 10247, f. 286 vereo, dn fcmde latin, KbliothAqtie Nat. de Paris.
41 Oeuvre», U, 342, 377; lettera to TX&y, June 6t 16571 April 7,1668.
m Оншпегсшш, 775, fetter XIV, Nov. 1,1657; Oeuvraj de Fermat, Ш, 423.
47 Letter XVI. XVIII to Digby, Dec. 1, and Dee. 26,1657; Oeuvree de Fermat, Ш, 434-5;
480^9.

352 History op the Theory of Хшшевв. [Chap. XII
infinitude of solutions In this way, but not all. He stated that all solu-
solutions are obtained from Brouncker's rule by setting г = a/e, whence
x — 2oe/(a* — ne1), and choosing integers a,e such that a* — ne2 divides
2ae.
Wallis" gave a long exposition of results which he implied are essentially
due to Brouncker. Be gave a tentative method to solve na* + 1 =^ D.
For n = 7, take the square 32 just > 7; then 7 = 3* - 2, 7-2* = 6* - 8,
7 >3l *= 9* — 18, whence we have a number 18 which is double the root 9;
hence 7-3* = (9 — IJ — 1. In general, use the square c2 just > n and
exceeding n by К Employ na* =* (caI — ba* for a = 1, 2, 3, • •»? until we
reach a value a of a for which ba* ^ 2ca, and then replace ca by (ca — 1) + 1.
For each a ^ or7 we thus have two values of яЛ Presently we can make
a further reduction of ca — 1 to ca — 2, etc., etc. It is stated that we finally
reach an equation in which the number subtracted is unity and hence a
solution. Devices are suggested (pp. 465-74) to abbreviate the long calcu-
calculations.*
Given (pp, 474-8) one solution, nr* + 1 = s*, set i = 2s; then the
values of x in the successive solutions of nx2 + 1 = О are r, rt, т{& — 1),
r(f — 2t)y *• *, while if га, г0 are any two consecutive terms, the next term is
riffi - a).
W&Uis" explained in an example Brouncker's method of finding a funda-
fundamental solution. The example chosen was 13a3 + 1 = □, Since 13 lies
between the squares 9 and 16, set 13a2 + 1 — Ca + &)*, whence
4a2 + l = 6a6 + bI, 2b > a > b.
Hence set a = b + c> whence 2bc + 4c2 + 1 = ЗЬ3, 2с> Ъ > с Set
Ь = с + d, с = d + e, d = e +/. Then e' + l-oV + 'tf2, 7f > e > fy.
Hence sete = 67 + 0,/ = 0-|-A, 0 = A+i. Then 4At + 3*1 + 1 = 3ft*.
Thus Л > г. Taking* ft = 2г, we see that the last equation becomes
lli* + 1 = 12г* and holds for г = 1, whence h = 2, .. >, a = 180. It is
noted (pp. 492-3) that, since 6, c, d, - * * are decreasing integers, we finally
reach a term which divides the preceding, as in Euclid's process to find the
g.c.d*, a process entirely analogous to the present one» If we had proposed
the example 13a8 + 9 *= D, we would get Иг3 + 9 = 12i*, whence i «= 3,
and similarly for any square in place of 1 or 9. But if & is not a square,
13а5 + к = D is not always solvable, but when solvable the solution can
be found by the above method.
As noted by H. J. & Smith,60 Brouncker's method is the same as that
given by Euler66'*2*81 and really consists in the successive determination
«■ Commercium, 789, letter XVII to Brouncker, Dec. 17, 1667; Oeuvree de Fermat, III,
157-480.
« Commereiuiii, 804, letter XIX to Brouncker, Jan. 30,1658; Oeuvra de Fermat, П1, 490-
503. Cf. WalliH, Algebra, 1S93, Ch. 98.
♦ To proceed ae would later writers, set A — i + jr whence — 4£ + 2ц + ^ — 1; then
i = j -H A, whence j1 — Qjk — 4J^ = 1, with unity ae coefficient of & square term, во
that j — 1, к ** Oia an evident solution.
»BritialiAffiOG.Report,l861,3l3; Coll, Math. Papery 1,193. (ХЕлпеп/р.ЗО; Whitford/
pp. 52-6; Wertheim."

Chap. XII} PELL EQUATION, ОХг+Ъх+С~ D. 353
of the integral quotients in the development* of T/U into a continued
fraction, where T = 649, U = 180, is the fundamental solution of
T* — 1317* = 1. But" Brouncker did not prove that hifl method will
alwaye lead to a solution of Г* — DIP = L
Frenicle" cited his table" of solutions of x* — Dy2 = 1 for all values
of D up to 150 which are not squares and suggested that Wallis extend it
to 200 or at least solve it for D ^ 151, not to speak of D = 313 which is
perhaps beyond his ability. In reply, Brouncker54 stated that within an
hour or two he had found by his method that 313o* — 1 = fr2 for
a = 7170685, b = 126862368, whence x = 2ab is the desired solution.
Wallie" gave the last solution and 151{140634693J + l =* A728148040)*.
Fermat* was at first satisfied with the solution of an2 H- 1 = П by
Brouncker and Wallis, Later, Far-mat" stated that he had proved by
the method of descent the existence of an infinitude of solutions n of
an2 + 1 ^ П when a is any number not a square. He admitted that
Frenicle and Wallis had given various special solutions, though not a
proof and general construction.
In an anonymous letter to Digby, either by Frenicle5* or inspired by
him, it is etated that Wallis0 affirmed that he could easily prove the exis-
existence of an infinitude of integral solutions of an2 + 1 = П and implied
that the proof is expressly contained in that passage; "but our analysts
recognize no trace of proof there".
N. Malebranche" A638-1715), after stating that he had not seen the
work in the Commerrium Epist. of Format and Wallis on Ax* + 1 = П,
remarked that we can find a solution if A = a2 ± kay к ^ 1, 2, or J (no
details given), or if the difference between A and some squared divides 2£.
Thus, if A = 33 or 39, t -= 6, A - ? = Jz 3, a divisor of 2t We have
39*2 + 1 ^ (fix + 1)\ x~4; 33z* + 1 = (fix - IJ, x = 4. He treated
by a tentative process the new types A = 13, 19, 21- For 13, multiply by
the squares 1, 4, 9, * • -, until we get a product whose difference from the
square divides double the root of the same square; since 13 - 25 — 1 = 18l,
set 325*1 + 1 = (l&c + IJ, whence x «• 36. Again, 19 9 - 132 = 2,
whence 171x* + 1 = (Ш + IJ, x « 13. He noted that if Ax4 + 1 = y2,
* Aleo noted Sept. 6,1658, by Chr. Huygens, Oeume completes, II, 1889, 211.
^Commercium, 821, letter XXVI to Ifeby, eent by the latter to Warns Feb. 20, 1658;
Oeuvrw de Fbnoat, Ш, 530-8.
"SdutioduorimipitiUemAtuiD . * .., 1657 fretwork).
u Commercium, 823, letter XXVII to Digby, March 23,1058; Oeuvree de Fermat, Ш, 536-7.
щ letter XXIX to Впншскег, March 29,1668; Oeuvrea de Format, III, 542.
11 Lettera from Fermat, June, 1658, and Frenicle to Digby, Oeuvies, Ш, 314, 577; II, 402
(Latki).
n Oeuvree, П, 433, letter to Careavi, Aug. 1659.
u Oeuvrea de Fermrt, Ш, 604-5 (Trench tranal., 607-8)»
w С Henry, Bull. Bibl. Storia Be. Mat. Ffa,, 12,1879, 696-8»

354 History of the Theory of Numbers» [Chap. XII
then АBхуУ -j- 1 = Q, so that we obtain an infinitude of solutions, but
not all, from one solution. A. MarreM stated that the last result was
copied from a letter written by Claude Jaquemet, who gave the second
solution X = 2xy, Y = 2Ax* -f 1. *
Wallis61 attempted to prove that t2 — Du2 = 1 always has positive
integral solutions, but made use of a lemma which is false [Lagrange74- M
and Gaussw]: Let m be the integer just > VZ>, whence m — VZ> < 1> and
set p — m — <Jd, r = 1/B VZ>); then it is possible to find two integers z
and a such that
But the difference of the fractions in this inequality approaches zero as
z and a increase, so that their ratio approaches p>
The name Pell equation for x2 — Dy* - 1 originated in the erroneous,
notion of L. Euler52 that John Pell was the author of the unique method of
solution explained ш WalUV Opera, whereas WalUs gave only Brouncker's
method. Nor, as stated by Hankel,24 had Pell treated the equation in a
widely read work, ь e., in his notes to Brancker's*3 English translation of
J. H. Rahn's algebra. After examining three copies of this translation,
G. Enestrom*4 stated that there is nothing relating to this equation. How-
However, x = \2y2 — zq is treated in Itahn's*8 Algebra, p. 143.
Euler62 noted that if az2 -j- bz + с is a square J2 for г = р, it is a square
for
z=±(~b + bR)+pR + \l,
so that the problem is to make 1 + a\2 a square.
Euler*6 again noted that, if / ^ ax2 -j- bx -j- с is a square m? for x = n,
it is the square of m' = apn + pb[2 -j- qm for x = qn 4- pm -j- (bq — b)/Ba),
provided that g3 = ap1 -j- 1. In the latter expression for x we replace n
by this x and replace m by mf and get
4- 2pqm -h^fg3 - 1) - n,
which makes / = Q. If А, В are consecutive terms of the series я, ху x\
* *., the next term is 2qB — A + b{q — l)/a. In the case / =- ax2 -j- 1>
whence b = 0, с - 1, the series becomes 0, p, 2pq, ±pq2 - <p, • • •, A, B>
« Bull. Bibl. Storia 8c. Mat. Fie., 12, 1879, 8&3. Attributed incorrectly to Marquie <k
THdpital io Comptes Rendua FarUr 88t 1879, 76-7, 223.
a Algebra, Oxford, 1685, Ch. 99; Opera, 2, ШЗ, 427-8. Reproduced by Konen,' 43-6.
e Letter to Goldbach, Aug. 10, 1730, Correspoodance Math, et Physique (ed., R H. Fusa)>
St. Petersburg, lr 1843, 37. Aleo, ЕШет.»^ Cf. Euler» of Ch. XIII. Cf. P. Tan-
Tannery, Bull, dea Sc. Math., B), 27, I, 47-9.
» An mtroduction to algebra, trandated out of the High Dutch into English by Т. Впшскег,
Much altered and augmented by I). P. London, 1668. On Rahn's algebra of 1659, eeo
Bibliothcca Math., C), 3r 1902, 125.
« BibUothcca Math., C), 3, 1902, 204; d. G. Wertbeim, 2,1Ш1, 360-L
« Comm. Acad. Pe^., 6, 1732-3, 175; Comm. Arith. CoU., 1, I849r 4] Op, Om., A), IIr ft

Pell Equation, ах2+Ьаг4-с = П. 355
2qB — Л, > -. Hence if one solution ap2 + 1 = q2 is known, we get an
infinitude of solutions p' = 2pqt etc» Euler noted special forms of numbers
a for which a solution of ap2 + 1 ^ g3 may be given at once, viz», (a, pt q):
e2 - 1, 1, e; # -j- 1, 2et 2e? -f 1; «V* d= 2ae^\ e,
=b ae^\ he, ±ак*е™ -ь L
If о is not of one of these forms, apply the method explained by Wallis,
which is here illustrated for 31p* -j- 1 = q\ Euler gave a table showing,
for each a ^ 68 not a square, the least positive integer p and the corre-
corresponding q satisfying ap2 -f 1 = g*. From лб = -J^^T/p, Euler noted
that, if q is sufficiently large, qfp is a close approximation to ^; let i*
be the ith term of the above series 0, p, 2pq7 * • • and Q the ith term of the
series 1, q, 2q* — I, • • • such that aF1 + 1 = Q2; then the successive
values of #/P are closer and closer approximations to ^a.
Euler* noted that the least integral solution x of ax* -j- 1 = □ is
226153980 for a « 61, and 15140424455100 for a - 109, and stated he
could shorten very much the work necessary by "Pell's method/' If
z2 — ey* =z N has the solution a, b, it has also the solution
Making use of the existence of integral solutions of p5 — eq2 = 1 for e not a
square, we can assign an infinitude of integral solutions of x2 — ey2 = N>
since
A1) JV = (a2 - eb2)^ - eff2) - (ap ± efy)* - e(bp ± a^J.
This formula of composition was known** by Brahmegupta.28
R. Simpson" noted that if we are given a and a fraction b/c such that
(Ъ* ЯР l)/c* « a, the series of fractions, converging to ^j,
' ^ ^_"^acc / _ bd H- осе & _ b/ -j- e*ff
с' e 2bc ' p cd + be * к qf ± bg '
are such that the numerator of any fraction (as hjk) is the sum of the prod-
products of the numerators and the denominators of ЬЦас) and the preceding
fraction (then f[g)y while the denominator (then A) is the sum of the prod-
products of the numerators and denommators of c[b and that preceding fraction
(ffg). By A1), every fraction N/D in the series has the property
№ — I = aD* if b* - 1 = oc2; but if b2 -j- 1 = ac* that property holds
only for alternate fractions, while N* -j- 1 « aD* for the others. He cited
the " obscure passage " where A. Girard**1 gave the approximations 577/408
p Math. Phy* (e<L, FUee)r 1, 1843, 616-7, 620^631; lettcra to Goldbacb, Aug- 4,
1753, Aug. 23, 1755,
«7 a. M. Chaalee, Jour, de Math,, 2r 1837, 37-50. Reprinted, Sphinx-Oedipe, 5t 1910r 65-75.
•■ РЫ. Тгашк London, 48,1,1763, 370-7; аЬг. ed,, 10, I80&, 430-4.
**° Lee Oeuvrea math, de Simon Stevin de Bruges ... par A. Girard, Leyde, I634? I, 170.

356 History or the Theokt of Numbers. echap. xn
and 1393/985 to V2 and an approximation to VlO. Jean Plana™ gave
reasons to show that Girard there in effect reduced VA to a continued
fraction*
A solution* of 440&0*' + 1 =* D is x = 40482981221781.
Euler71 published his formula A1), and treated ax2 + hx -j- с *= у*,
given the solution x = n, у =* то. Set x = л + pz, у ^ т -\- vz. Then
(И — a»*)z — 2ацп — 2wi + Ьд. If a is positive and not a square, we
can make v2 — atf *= 1 and obtain integral solutions, and then a third
set, etc.; if the general set is (xh yx)} we have
But we may obtain solutions not having v2 — ар? = 1; setting
we obtain the first formulas in Eulerb** earlier paper. Euler proved that
if an odd prime, not dividing or, is of the form b2 — oca2, it is of one of the
linear forms 4a» -J- г2, 4<*л -J- r8 — a, where г ranges over the odd and even
numbers < a and prime to a, respectively. He conjeetured> conversely,
that if Л is a prime or product of primes of these linear forms, then
A = x* — ay* is solvable in integers £not always true, Lagrange7*].
Euler72 again repeated his initial formulas and added that, if F, Q, R
are the values of у in three successive seta of solutions, R - 2^Q — P, while
the general set of solutions is said to be Qafter correction of signs]
where ^ is an integer. The method published by Wallis to find integral
solutions of x2 ^ ly2 + 1, where I is positive and not a square, can be more
conveniently exhibited by means of the continued fraction for ift. If
x =* p, у =* q is a solution, it is stated that pjq > Viand that pjq gives so
close an approximation to VI that a closer one cannot be found without
using larger numbers* After developing V? into a continued fraction for
z = 13, 61,67, he took a general г and set
г 111
where v is the largest integer < Vs, and a, b, c, ■ ■ ■ are found as follows.
In V5^ v+ 1/s, x = 1/(л6 — t>) = (лБ-j-v)/a, where * = г — я2; hence let
« Rfiflezionfl noiivellefl eur deux шбдншев de I^gmnge74 . . , ? Turin, 1859,24 pp; Memorie
R. Accad. Torino, B), 20,1863,87-108.
'° Ladies' ГКшу, 1769, pp. 39-41, Quest. 443. The DiarUn Repository, or Math. Register
... by a Society of Mathematicians, London, 1774, 677-9, С Hutton'e Dianan
МЬсеПлпу, 3,1776, 81-83. Т. Leybourn'e Math. Quest, propoeed in Ladies' Diary,
2,1817,1824.
"Novi Oomm. Acad. Petrop., 9, 1762-3 A760), 3; Comm. АтНЬ. СЫ1.. I, 297-315; Op.
Om., A), II, 576.
n Novi Comm. Acad. Petrop., llr 1765 A759), 28; Comm. Arith. СоП,, Г 316-336: Op.
O(l)III73

Pell Equation, ox*-j-kc-j-c=CL 357
a be the largest integer ^ ( V* -j- »)/«* 1д х = a -j- l/#,
— da z — (v — acrJ jff
where В = <ux — v} 0^1 + 2av — a^a. Hence let Ъ be the largest integer
^ (V^ 4- .B)//3. Taking у = 6 + 1/*, and proceeding similarly, we obtain
Euler's table:
1 A = vy ct — z — A*=z — &y a ^
ш
IV
7, p + (), ,
У v
etc., where in the last column the equality sign is taken only when the
fraction is an integer. It follows that A, B} C, - - - are ^ vt and the indices
a, b, cf • • ■ are ^ 2v. Euler observed in many examples that when the
value 2f is reached, the values at b, c, ■ ■ ■ repeat; but до proof73 is given
that the index 2v exists Q>roof by Lagrange74]. For each z ^ 120 and
not a square, he gave the values of v, а>Ъ,с> ■ • ■ (at least as far as a period),
and underneath them the values of 1, a, 0, yf -»-. Such values are given
also for certain types of numbers, viz., г = n2 + kt к *= 1, 2, «, 2n — 1, 2n,
and г = 4n* -j- 4, 9n2 -j- 3, 9»? -j- 6.
The successive convergents v, (va + l)/a, - — to лб are found by the law:
v, а, Ьу с, -•-, mt n, ••-,
1 t; on+ 1 (дЬ 4- l)v 4- ^ ^ iV пЛМ-Jf
б* Г a ' ob + l ' '"' Pf Q' nQ-j-P' '"'
These convergents are given the symbolic notation
1 (if) (^a) (vt a, b) (», a, 6, c)
0' 1 ' (a)'1 (a,b) ' (а,Ъ)С) ' '"'
where
{») = », (», a) = t>(a) 4- 1, (», «> Ь) = г(а, Ь) 4- Ъ,
(v9 а, Ъ, с) = v(a, Ь, с) -j- (Ь, с),
Не stated that
(v, а, bf ^, d, e) = р(а, 6, c} d, e) -j- (b, c, rf, e) = (», a) F, c, rf, e) -j- v(c, dy e)
= (v, a, b)fa Д, e) 4- fa g? W e) = fa g> ^, c)(d, e) + (v, a, b)(e)t
UM remarked by H. J. S. Smith, Britieb Aseoc. Report. 1861, %Щ pp. 313-5; CoU. Math.
Рлрегн,!, 1891,194, Euler'e p*per contain* all the clemcnte neoeteaiy to.give a rigorous
proof of tbie foci and hcnoe tbftt the ргосеж alwajH leada to & solution, other than
x - 1, у = 0, of д« «у* ^ I. Fkna,M voted that Eukr'e proof becomes пдогоая if
ali^htly modified Mby r^rmlre» Forfol, ■■-), Garee" of Ch. II wrote!>rA .-1
" MlBceUaaea Tatmnena», 4, 1766^9,41; Oeovns, 1,186T, «71-731.

358 History of the Theory of N timbers. [Chap, xn
and proved that
(py ~ z-i2 ~ - <*, (t/, aJ - z(ay = ft (■>, a, bJ - e(e, bJ = - y,
(v, a, b, cJ - *(«, 6, cJ = 5, (p, a, h, c, <tJ - a (a, &, c, dI = - <,
so that, for example, x2 — zy* =- — у has the solution x = (v, a, 6),
у = (a, b). No one of ft 7, 5, - - equals =b 1 unless the corresponding
index is 2v. Hence if any period contains the index 2v and if xjy is the
convergent defined by this period, we have x2 — zyp = — 1 or + 1, accord-
according аэ the number of indices in the period is odd or even. In the first case,
£ *= 2a* + 1, 7) = 2xy give a solution of £* — zy2 — -j- 1; or we may take
two successive periods and apply the second case. He applied this theory
to eight special types of periods, such as vt at Ъ, Ь, a, 2vt at • • ♦. He recog-
recognized that we need only use a half period. Thus, for the period just cited,
we employ the half period v, а, Ъ of indices and convergents 1/0, i?/l, Д/ft
C/7. Then x* - zy2 = - 1 for
x = (f, а, Ъ, ht a) = (a, b)(v, a, b) + {a)(vt a) = yC + pB,
у - (a, Ь, Ъ, a) = (a, Ъ)(а, Ъ) + (a)(a) = 72 + F-
But if z has the indices vt at Ъ, с, 6, а, 2у, with an even number of terms in
the period, we use the half period v, а, Ь, с and the additional convergent
Djb and find that x2 - zy2 = -j- 1 for
x = {a, &)(», a, fr, c) + (a)(v, a, b) - yD + /30,
у - (a, b)(a, Ь> с) + {a)(at Ъ) = уд 4- ftr-
As equivalent formulas were restated by Tenner,lia they are often attributed
to him rather than to Euler. The formulas are stated in general form by
Muir460* and Konen,3 pp. 55-6.
Finally, he tabulated the least solutions of p2 - If = 1 For each I < 100
which is not a square, and for I = 103, 109, Ш, 157, 367 [errata for I = 33,
83, 85, Cunningham309].
J. L. Lagranger< gave the first proof that x2 — ay2 ^ 1 has integral
solutions with у ф 0, if a is any integer not a square. He noted that
Wallis61 committed a petitio principii in attempting a proof, while the
method of solution explained by Wallis49 is tentative and not shown to
succeed. Lagrange started with the continued fraction
and its successive convergent^ m/w, AfjN} m'fn't M'jN't •■•. Taking
(xt y) = (M7 N)t (ЛГ, ЛГО, ■ • ■> we always obtain positive values < 2M/N
for x2 — ay*. Hence an infinitude of these values are identical. Let
(x, y)y (z', y')} (x"t yff), - - - be an infinitude of pairs of integers for which
x2 — ay2 has the same value R, First, let Rt a be relatively prime. By
multiplication and by elimination of a,
(A) Я* = (x^ ± щпП* - a(xy>
(B) R<y" - tfl -

Chap.xiij Pell Equation, ox*-{-fcc-j-c^O. 359
if R is a prime, (B) gives xyf =b yx* = qRy whence, by (A), xx' =b ayy' — pRt
where q and p are integers. Thus, by (A), p* — aq2 = 1. Next, let
Д = .4,5, where A and £ are primes. By (B), one of xy* -J- j/з', st^ — yxf
is divisible by Л£, or one by A and the other by B. In the first case we
have the same result as when R was a prime. In the second ease,
xy* ± yxf = qBt where q is an integer not divisible by A. Then (A) gives
xz' =b ayyr = pB, whence
(C) v2 - aq2 = A\
Arguing similarly with a third equation x — ay = R of our set, in con-
conjunction with хг — ay* — Rt we get p\ — aq* = A2- Treating this and
(C) as we did our first pair, we get a solution of r* — a& = L A similar
treatment is made for the case in which R is a product of several primes or
is an arbitrary number.
Second, let R = $Tt a = $b be not relatively prime. To treat the first
of two analogous eases, let В be not divisible by a square. Then x = $u and
T - &u* — by2. Hence T* = (ви? -j- by*)* - a{2uy)\ Since T2 and a are
relatively prime, we may employ this equation in place of the former
x2 — ay* = R. Hence there exist solutions of z2 — ay% = 1 and we have
a process to find them»
If p* «— aq* = I, then x? — ay2 — 1 for
x + y^a = E = (p -j- gVa)w, x - y^la = F = (p - q^a)mt
and
A2) x - \{E + П V- ^j-(n - F)
are expressed as polynomials in pt qt a. If p, q is the least positive solution,
then A2) gives all the solutions, m being an integer» All solutions occur
among the sets (Mf NO (AT, N')r ■•■ given by the convergents M/N,
M'jN', * * * to ли, and each is > ли» If m is a prime, and if x, у are given
by A2), x — p and у — qa{m~1]f2 arc divisible by m; hence, if r is the residue
@ or ± 1) of а(я>~шг modulo m, and if p\ q' are given by A2) with m replaced
by m - r, then pf* — aq'2 = 1, and q' is divisible by m, and either p' — p
or pf — 1 is divisible by m according as r — 0 or r 4= 0» Likewise when in
A2) m is replaced by M = n(m — r)(mf — r') • • *, where mf m\ »* • are
odd primes and r' is the residue of аAЛ'~1)'2 modulo m\ etc», « being any
positive integer, x2 — ay2 = 1 and у is divisible by N = mm' • • •, and either
x — p or a: — 1 is divisible by N according as Af is odd or even» After
giving numerical examples illustrating what precedes, La-grange stated that,
if a is not a sum of two squares, no number is simultaneously of the forms
z* — ay2, ayl — x*; but was not certain of the converse [cf. Legendre88}
If x2 — ay2 = R and x\ — ajfi = — Rt and if R is a prime, we can solve
p* — aq2 = — 1, and conclude that every number of the form x2 — ay2 is
also of the form ayl — x\. By squaring £ — аи2 = — 1, we get solutions
A2) of z2 — ay2 = 1; hence p =b дли must be the square of a quantity
у ± в ли, whence p = r2 -j- <w*f £ = 2ra. Hence Я — at*1 — — 1 is impossible

360 Hibtort of the Theory or Numbers. qchap. xn
unless p, q are of this form; and if they are, the resulting ty и give the least
solutions.
Lagrange** gave a direct method to solve a + Ы2 = u7 in integers.
Removing the factors common to t and w, it suffices to treat
A3) A - p* - Bq\
where p, q are relatively prime. If В is negative, we may assume that
| Л | > — B} since otherwise pq *= 0. If В is positive, we here assume that
A} > B, treating later the contrary case. Choose integers pi, qi such
that pqi — qpi =l ± 1, and multiply A3) by At = p\ — Bqt. Thus
Л^! = cc* — Bf where <x = ppi — £g£i. Since a2 — В is divisible by A,
(|*Л =t aJ — £ is divisible by A, and pA ± « can be made numerically
< | Л |/2 by choice of p. Hence U a2 — £ is divisible by Л for no value of
ct < | Л ]/2, A3) is not solvable. If such an a exists, the problem reduces
to the solution of
A4) A,=ri-Brf, \Al\<\A\.
If solutions of the latter are found, we deduce solutions
Л1 Л1
of A3) from ppi - Bqgi *= а, щх - qpx = ± 1. If, m A4), В < О or if
В > 0, A| > Bt we proceed as before and see that A4) reduces to the
solution of
At = p\-Bqlt «1 < i 1 At I , 1 A21 < I Ax | .
The case £ > 0, A^ < By falis under that treated later. Thus, unless such
a postponed case arises at some stage, we shall finally reach, if В is negative
(B = — b), a term An such that | An | — Ь or < Ъ. If |ЛЯ) = bt we
have Ь = pi -j- bgj, whence дл = 0 or 1 and A3) is solved. If | An | < bT
then qn = 0. But, if В is positive, we reach a term <*n *= e, where e < V§,
and AnAn+1 = e2 - B. Thus A* = =b^ An+1]= ^2>, where 2> and E
are positive and ДЕ = В — e2» Moreover,
=F 2> = p2 - B<r*, d= ^ = r2 - Bs\
the solution of one of which implies that of the other. Since DE < B,
one of the equations is of the next type.
The postponed type is =b E = r2 — Bs\ where E < VS, В > 0.
We first seek (§ 34, p. 435) an integer t} VS > * > V5 — Я, such that
В — с2 is divisible by E< If no such t exists, the equation is impossible in
integers» In the contrary case, take a particular «, and determine uniquely
integers E>t «^ X< by means of the equations
EEi - В - €» ЕхЕг - В - €ги ЕгЕг =* В - *и - - ^
— t\t tz =5 Х3-Б3 — «2, # * '»
. Acad. Berlin, 23, агшбе 1767,1769,242; Oeuvree, 2r 1868r 406-495. German transL
Ef Netto, OfltwaJd'B Klaseiker, No. 146, Leipzig, 1«M.

Chai\XII] Pell Equation, ox2+6x-j-c—П. 361
where the effect of the inequalities is to insure that the X's shall be positive
integers making 0 < e< < V5- It is proved at length that, if the proposed
equation is solvable, we will finally reach a least positive integer ft such
that the term Ещ is identical with E and such that E^i = Eu whence
E»+, => Е„ and also that Em = =b 1 for a certain m, 0 ^ m ^ ^. Then
tw^i equals the greatest integer & which is < V5. For brevity, set
j
Since Em = d= 1> /*/*» gives Я2 — ££* = ± 1?, and the general solution is
given by
By actually multiplying together the factors in/*, it is shown that
where the Vs are derived from the relations (p. 448)
The notation is at fault if m = 0, when we have R = lt S = 0r and if
m = 1, when we have R = t = p> S = 1.
Application is made (pp. 454r-94) to various numerical equations A3)»
For Pell's equation (pp. 494-5), we have E = 1, whence 0 — «, w = 0r
where n is a positive integer such that nti is even or odd according as
r* — Bs2 ~ + 1 or — 1. For the former, n is arbitrary if p is even, but
n must be even if p m odd, Heuce if В is any positive number not a square,
r2 — Be1 = -j- 1 has positive integral solutions. Lagrange noted (pp» 457-
461) that Euler's*»71 method to derive an infinitude of integral solutions
of ax1 -j- bx -j- с = у* from a given solution does not always lead to all
integral solutions unless fractional values of the parameters be used or
unless, in у* — Вт? =* А, Л Is a prime.
Lagrange**1 investigated the approximation of roots of algebraic equa-
equations by continued fractions and proved that the real roots of any quad-
quadratic equation with rational coefficients can be developed into a periodic
continued fraction, and conversely*
Lagrange7* derived his preceding formulas for the solution of
Acad. Bertie, 23, annee 1767, 1769; 24, алшбе 1768, 1770: Oeuviee, II, 560-652
(especially 603-15). TraiWde la resolution dee 6quAtiana nuKienquee, 179S; ed. 2,1808,
Cb, VI; Oeuviee, VHI, 41-50, 73-131,
e. Acad. Berlin, 24, annee 1768, 1770, 236; Oeuvra, II, 662-726. Fox simplification,
aee L*»

362 History of the Theory of Numbers. [Cbap, xii
by a method first applied to equations of any degree n (see Lagrange1 of
Ch. XXIII)» His method for t2 — Аи* = A, where Д is positive and not a
square, is as follows» First, consider solutions with и prime to A. Then
we can determine integers $ and у such that t = (to — Ay, $ < £A. For
this value of t, the initial equation becomes, after division by A,
where @2 - A)jA = Ex is an integer. Employ in turn each value of в for
which 0s = Д (mod A) and solve the new equation by developing into a
continued fraction either root of the corresponding quadratic
Second, for solutions with и = ru\ A — i*A\ whence t = rt\ with u\ Ar
relatively prime, we have only to treat tf* — An'1 = A' as before»
The same method applies to £t* + Otu + Du2 = A, C* > 4BD. By
the same substitution we now get Exu* — Quy -j- ABy2 — 1, where
Bt = (B6* + Св-\- D)[A, Q ^ 2B6 -j- C»
He noted that a conjecture made by EulerVl is false since 101 = x1 — 79У2
has no integral solutions, although 101 = - 4-4*79 -j- 382 - 79.
He applied (p. 719-723) the method of his former paper to deduce the
solution и — 34, t = 123, of 101 = I2 — lSu2, chosen probably in view of
his correspondence with Euler next mentioned.
Euler77 stated he found trouble in applying Lagrange's75 method of
solvng A3) to the case 101 = p2 — 135s» By that method we seek an
integer a < 101/2 such that «2 — 13 is divisible by 101. This is true for
a = 35. Then A4) becomes Al - 12 = pj - 13^. Since 12 is divisible
by the square 4, set the quotient 3 equal to tz — lZu2* Then t = 4; и — 1,
whence pi = 8, gx = 2. By Lagrange's method,
_apl=FBql_35*SzF13-2 „«gi^Pi^35-2^8
V~ Aj " 12 ' q~ Ai 12
As these are not integers, one should conclude that the problem is impossible.
However, p = 123, q = 34 are solutions, which fact led Euler to believe
that Lagrange's method is not sufficient» He noted that this solution
123, 34 is given by Pl = 47, ql = 13:
But what reason leads us to suppose that px = 47, qi ^ 13?
To test whether A - p* :fc Bf is possible or not, Euler gave for the
case A a prime the following rule, of which he had no proof: Subtract
from A any multiple of 45; if A — 4nB is of the form ab*} where a is a
prime or unity, and if a = p2 =fc Bq* is solvable, then the proposed equation is
solvable. Thus, 101 = p^ — 13g* is solvable since 101 — 4-13 = T and
1 - p* — 13g2 is solvable.
"Letter to Lagranger Jan., 177П; ЕиЬгЪ Opera po«tumar 1, 1862, 671-3; Lagrange'a
Оеи™*гХ1У, 214-8. Sec Lagrangc," ecd.

Chap. XII] PELb EQUATION, пХг+Ъх+С=П. 363
Lagrange's reply has not been preserved, but it convinced Euler78 of
the correctness of Lagrange's treatment of 101 — рг — 13g*, though, being
then bund, Euler confessed he did not follow the real meaning of all the
deductions, nor the significance of all the letters introduced.
Euler7' noted that ar2 — 4 = a2 implies that ax2 + 1 = у* holds for
Thus, if a = 61, we may take r = 5, s = 39 and deduce the large numbers
x, у in his table.
E. Waring80 quoted results due to Brouncker and Euler»
Euler*1 treated, essentially as had Brouncker,49 an2 -j- 1 = y*t where a is
positive and not a square» Thus, for a = 5, у is > 2n and Euler set
у = 2n -j- pt whence n2 = 4up + p2 — 1, n = 2» -j- V5p2 - 1- The radical
exceeds 2p, whence n > 4p. Set w = 4p -j- 5, whence p3 = 4p^ + <f -j- 1,
p = 2g 4- VS^-l-i» Having now the initial radical, we may set q = 0
and obtain p — 1, n = 4, i/ = 9. For a = e2 d= 2 or e* ± 17 we can give
explicit solutions n, y:
He repeated82 his table72 of the least positive solutions of an2 -j- 1 = m2,
a< 100»
Euler8* treated /^a-j-bx-J-cx2^ Das had Diophantus when a or с
is a square; also the case in which / is a product of two linear functions,
J, m of x, by equating / to the square of Jky as well as the case in which /
equals X1 -j- mn. In Ch» V, Euler noted certain forms which are never equal
to rational squares, as Зх2 + 2, Zt2 -j- (Зл 4- 2)u\ Ы1 + En ± 2)iA In
Ch» VI, he noted that, ^ven of2 -j- bf + с = д*, we can find new solutions of
ax* + bx + с = у2* Subtract and factor each new member; thus ws may
set
Multiply the first by p and the second by q and subtract» Hence
To obtain integral solutions take p2 = aq2 + 1 and change the sign of g.
Thus
The method for ax2 -j- с — у2 is similar, but simpler, giving x = qg -j- pf7
У =•= P9 4- ^/, and is derived a second way (§ 86) given earlier by Euler.w
я Opera poatmna, 1,574; letter, March, 1770, to Lagrange, Ocuvree, XIV, 219.
79 /Wrf4J 585; letter, Sept. 24,1773, to Lagrangc, Oeuvrea, XIV, 239-40.
» Meditation** Algebraicae, 1770,180-1&9; «1,, 3,1782, 308-337»
л Algebra» St. Petersburg, 2,1770, Ch. 7, tf 9&-Ш; French trausl., Lyon, 2» 1774, pp. 116-
134; Opera Omnia, (l)r I, 379^87.
"Also in Nova Act* Acad, Petron.. 10, adamjum 1792,1797 A777), 27; Comm, Arith,, II,
185.
"Algebra» U, СЬя. 4-6, ft 38-95, Fraoch transl., 2,1774, pp. 50-115; Opera Omnia, A), I,
349-78»

364 History of the Theoky of Numbers. [Ou*.xii
Euler81 solved ax1 + 1 ~ У* for special types of numbers a. Given
p* rr b2 + c2, determine g, f so that &g — tf = it 1 and take g = bf -f cg>
a=p+g*) then ajf-l^q*, x=2pq, y=2f+l. Next, if ap2zF2=^t the
divisor p* of gl±2 must be of the form b*±2c*; hence take a=f*dz2g*t
ef—bg—l or -1,д=ЬДь2ф. II ap'±4=4*, the divisor p* of g*=F4 must be
of the form b*Fc2; hence take a=f*^g2, cf—bg=2 or —2, q^bf^cg.
Lagrange" simplified his76 method applicable to equations of any degree.
Of two methods to solve F=Cy%—Znyz+Bz2^! in integers, one is to
render F a minimum, and the other consists in applying transformations
which replace F=l by Lp-2N№+M1? = l, where 2 | N | exceeds neither
I L | nor | ЛГ [ , while the determinants N2—LM and n*—CB=A are equal.
By multiplication by Jlf, we get t?—A$?=M where v=M$~N£. If A = — a,
where a>0, it is proved that f^O, Af-I. If Л>0, vj\ is a convergent of
the continued fraction for лЙ\ Euler's77 example, lOl^x2—13yJ is now
{pp. 6144>20) transformed into z* - lZv? = -1 which is solved by use of the
continued fraction for Vl3.
Euler10* of Ch. XXII deduced an infinitude of solutions of «*-X^=4
from one solution.
Petri PaoUM treated a+cV=y\ Since a is a difference of two squares,
set 1/-СЖ+1, cx+2, ••■, inturn. Then а=2ся:+1, 4сж+4, 6сх-Ь^> ■•■•
For a odd, use the firsts third, ■ • ■ terms, so that z will be an integer chosen
from the series (a—l)/Bc), (o-9)/Fc), * -. Similarly for a even. If a is
positive, the terms of the series decrease and there is a finite number of
trials. The case in which a is negative can be reduced to the preceding,
A. M. Legendre87 obtained important conditions for the solvability of
equations of degree 2 by use of Lagrange's" method for x2—Byz=A,
where A and В are integers with no square factor and A >£>0. By that
method,
A5) c?~B=AAtkt>«'2-B=A'A''k'\->>,azAl2,a'~ItA'±ct±A'!2, >--,
where A\ ■ • • have no square factors, and Лы < В, so that the proposed
equation depends upon
A6) x*-By*=A', x2-By*=A", —, x*-By*~A<»\
Legendre proved that, if for x2—By1=A and the first transformed equation
A6) there exist integers a, a', 0, &' such that
о*=В(тск1Л), с*^В (modi!'), P=A> 0'2=A'(mod B)t
the like conditions hold for the second transformed equation A6)» Since
a"*==B {mod A"), by A5), it remains only to prove the existence of an
integer 0" for which 0"Я=Л" (mod B)> If б is a prime factor of B, we
"Opure. Anal., 1,1783 A773), 310; Comm. Arith. Coll., Ц, 35-43~
« Additions to Eukrb AJgsbra, Lyon, 2,1774, pp. 464-^516, 661-баб; Oeuvree de Lagrange,
VII, 57-S9, IIWM; Euler1» Opera Omnia, A), Ir 548-573, 5Й8-637.
« OpusculA analytica, libumi, 178Q, 122.
^ Mem. Acad. So. Paris, 1785, 507-513. a. Legeadie, Th6orie dee Dombree, 1798, 43-50;
ed. 2,1806, 35-41; ed. 3, l, дао, 41-48; German tranel. by Maeer, I, 41-49. Ь his
texte, Legendre introduced the factor & in the right members of (]6).

Chap. XII] P-LL EQUATION, ОХг+Ъх+ - П. 365
seek an integer X for which \z=Aff (mod B). First, let В divide A\ Then
by A5), в divides a. Since kf has no divisor in common with B, which has
no square factor, and hence is prime to B, we can find integers n, p such that
kp=nk'-p$. Hence
*={nt-Afl)ka> пг^Л" (mod В).
Second, let В be not a divisor of A1 and hence not of 0'. We may set
а'=гф'к'-рВ. Then
0=A'W№-A4=A''F1V*-a?*=pflV\A''-n*) (mod 0).
The preceding result leads to the theorem: The equation x-—By*=A is
solvable in integers if Л and В are quadratic residues of each other, and if,
in the first transformed equation x*—Byz=A't A' is a quadratic residue of B.
We readily deduce the more elegant theorem: If each of the positive
numbers а, Ъ, с has no square factor and if no two have a common factor and
if there exist integers Л, д, v such that
cp2—Ь с?*—а
are all integers, then пх*+Ъу*=а? has integral solutions not all zero; if the
three conditions are not all satisfied there are no integral solutions. Apply-
Applying to (сг)г— bcy*=QCX* the earlier theorem, we have the conditions ctz=bc
(mod ac)y F-ac (mod be), /S'a=_4' (mod be). Set <x=cp, 0=c». Then
the first two give Ср*=Ъ (mod a), c^=a (mod b). By A5i), c^—Ъ=аА'к\
while afc2 is prime to be. Hence the third condition becomes ak1^'sc^-b
(mod be). This will hold if a\2-b&^0 (mod c) is solvable. For, it is solvabb
for 0' modulo Ь since сЛ10/8^ся* (mod b) is solvable for £'.
Legendre88 proved that x4—ауг= —1 has integral solutions if a is a prime
4n+l. Lagrange74 had stated that he was not certain of a converse that,
if a is a sum of two squares, every number x2—ayl is also of the form
ay*— x\\ Legendre noted that this is true if a is a prime, but fails for
a=2-17, 5^41,13 17. If a is а prime 8n-b3, ax*-tf^2 is solvabe. If a
is a prime 8n—1, y3—ax*=2\s solvable. While each of the preceding three
theorems was here treated separately, Legendre, in ed. 2, 1808, 5^-60, first
gave a preliminary discussion applicable to all the cases. Although he
took Л to be a prime, it suffices [Dirichlet108] to assume that A is positive
and has no square factor. Let p, q he the least positive integral solutions
of p2-Aq*=l> The g.e.d. of p—1 and p-H is/=l or 2. Hence
where MN=A,fgh=q. By subtraction, 2=jMg*-}Nh\ We must take
for M, N the various pairs of factors (including unity) of Л, Let Л be
a prime. The case 2=2^—2Ah2 is excluded since h<qt g<p. Let A be
a prime 4n+l. Then in 2=Ag*-h* and 2=д*-АЬ?> д and h are not both
■■ Mem. Acad. So. Paria, 1785, 649-551; Theorie dee nombree, 1798, 65-47; ed. 3P 1, 1830
64-71; M*»r, I, 6572

366 History of the Theory of Numbers. CChap. xii
even (since the right members would be multiples of 4), and hence both
are odd, whence ^=/i2sl (mod 8), and the right members would be
multiples of 4. Hence the only possibility is the case 2=2Agz—2h*f so
that k2-Ag*= — 1 is solvable. Besides the remaining two theorems for
primes 8та-ЬЗ, $n—1, cited above, Legendre proved that one of
is solvable if M and N are primes of the form 4n+3* Given a positive
integer A not a square, it is always possible to decompose it into two factors
My N, such that one of Mzz-Ny*=±l, Mx2-Nyi=^z2 is solvable when
the signs are suitably chosen. When z*—Ay*=— 1 is solvable, Л is a sum
of two squares. Cf. Arndt. In Table XII, he gave the least positive
solutions of m2—<m2= — 1, when it is solvable, and of m*—an*= -flin the
contrary case, for 2^a^l003, a not a square [errata, Cunningham,26*-309
Richaud,1* Whitford4 (p. 97), Gerardin811], but with no indication as to
which equation has the solution listed. It was reprinted (with fewer errata)
as Table X in ed. 3, 1, 1830, and abridged to a < 135 in ed. 2, 1808.
J. Tessanek*9 considered (a2+&)n2+l = [J, say (an-Ьр)*. Set n=p+q.
Then p satisfies a quadratic. Write b—a^fc, 2a-H — Ъ=д. Then
Replace p by q+r and solve for q in terms of r. Thus
tfq
where
g
Replace q by r+s and solve for r in terms of s. Thus
where
h"~g'-k'=Gb-7a-2,
Replace r by s+t. Then
According to the method of Pell,82 one ultimately obtains an equation in
which the number g under the radical is +1. To find values of n for various
,g0 , ; , ,g, p ,
n=3; etc. The terms free of a, b in 1, g, g*\ g{lY\ - - • are 1, 1, 4, 25, • ■ >,
i. e.t the squares of 1, 1, 2, 5, 13, 34, ■ • ■, whose differences of second order
give the same series. Thus the scale of relation is u1^.1=3ut%—ия-и so
that the general term is expressible in terms of the roots of 1 — 3z+z2=Q;
likewise for the coefficients of 6, a.
и Abh. ВдЬшшсЬеп Geeell. Wia»., Pr&g, 2,1786,160-171.

лр. хп] Pell Equation, а^-\-Ьх-\-с— □ . 367
John Leslie*0 treated х?+уг-[-Ьху=а2 by factoring a2 — y*t solved
if A, Cor B*—4AC is a square, and derived a second solution of ах*-\-Ъ = уг
from one solution.
P. PaoU91 noted that, if t=h, u=k give one set of rational solutions of
AP-['B=u*> all are given by
hr*-2kr+Ah _ . .
*- H_/ , u=k+r(t-k).
P. Cossali9* discussed Euler's and Lagrange's methods to solve A3).
C. F. Gauss0* showed how to find all solutions of P—Du2~7n2t given
two linear substitutions which transform any reduced form AX*-[-2BXY
-\-CY* of determinant D into the same quadratic form (see quadratic
forms in Vol. Ill),
J. С L. Hellwig*4 gave an exposition of Pell's and other equations of
degree 2.
R. Adrain95 reproduced the simpler proofs from EulerV3 Algebra, II,
Chs. 4-5-
F. Pezzi5* employed the continued fraction
X = fl4J =
^ui + at-l + an-i +^n ZnN^+N^-i'
where M*/Nn is the convergent derived by deleting ljxn. Take x=
a?i = l/( Va—a), etc. Then xn = (^A-\-bn)lcnt where
By substituting this value of xn and the corresponding value of #*+i in
хп=ая-^-1/х1Ч.1 and equating rationals and irrationals, and changing n to
n — 1, we get
Since the a?s do not exceed 2a, the a's repeat after a certain number n of
terms. Then Ml = ANl -h (— l)n. Hence x2 — Ay* = 1 is solvable in
an infinitude of ways, likewise x%—Ay*= — 1 if and only if the period
length n is odd. Consider any solutions of Mll=ANll-\-{—l)m. If Nm is
even, Mm is odd and m even. If A is even and Nm odd, Mm is odd and
(-1)« = (_1)*\ If A and Nm are odd, Nm is even and (-!)» = (—I)»*1.
t9 Тише. Roy. Soc. Edinburgh, 2,1790,193-209. Keprinted iq the Math. Repofiitory (ed,,
Leyboum), Loadon, 1, 1799, 364; 2, 1801, 17; EncycL Britannica, Cf. Berkhan.18*
n Elementi d'aIgebraT Pisar ly 1794, 165-6.
и Origuic, trasporto in Italia . . ♦ Algebra, Parma, 1,1797,146-155.
" Disquisitionee Arithmeticae, 1801, wta. 182,198-202j Werke, 1, 1863,129y 187; Crermao
transl» by Maeer, 1889, 120, 177-87. Cf, Dirichlet.1»
M Anfau^griindc dcr Unbeet. Ajaalytik, Braunechweig, 1803, 80-184.
" The Math» Correspondent, New York, 1, 1804, 212-222 (first American matb. periodical).
n Memorie di Mat» e di Fisica Soc. Ital. Sc.y Modena, 13, 1807, I, 342-365.

368 History of the Theory of Numbers. [Chap, xii
С Kramp" treated periodic continued fractions and application to
Лу2-^1 = П. The error (p. 283) on 11^-^49—x2 was corrected in the second
note.
P. Tedenat98 stated that, if у3—Ax2—В is solvable in integers, its
solution reduces to the integration of the equation у^^Ъту^+у^О in
finite differences, the integral being y=(r-\-s)/2l x = (r—8)lB^A)t where
Y, X being the least integral solutions of Yf—AX2=*B, and m, n being
integral solutions of m2—An*=l. This is Euler's7*result in changed nota-
notation.
P. Barloww gave 15 theorems on x?—Nyz — \ and the fundamental
solution for JY^102. He100 gavfc general formulas for the solution of
x2~Ny* = ±A or 23.
C. F. Degen101 gave in his introduction an account of у%=ахг-\-\ by
the development of Va into a continued fraction, and its solution by an
artifice for certain a's, as a^p2±l, p2±2. His table I (pp. 3-109) gives,
for a^lOOO and not a square, the solutions of у*=ах*+1 and the continued
fraction for Va [errata, Cunningham259-m*]* For example, in the entry
209 j> a]
14 2 5 3 B)
1 13 5 8 A1)
3220 [=x~\
46551 L=y]
the first line gives the continued fraction
* 1 1 ! 1 1
The second line shows auxiliary numbers 1, 13, 5, 8, 11, 8, 5, 13, 1 arising
in the process. Thus,
Table II (pp. 109-112) gives the solutions of y2 — ax2—! when solvable
[pimtted when о is of the form i2+l, when y = t, x— lisa solution]. It is
said to he solvable only for those values (Ф2, 5) of a which correspond in
tab!e I to a period with an even number of terms. For extensions of
Degen's tables, see Bickmore,21* and Whitford, p. 398 below.
1ед de Math, (ed., Gergonne), 1,18КЫ1,261^285, 319-320, 351-2.
"/bid., p. 349.
M Theory of numbers. London, 1811, 294» In г1 — 56587s/1 - lr the figure 7 ia omitted;
<£. A. Mirtinr Bull. Phil. Soc. WaahingtOQ, 11, 1888, 592, and Martini
**° New Mathematical ТаЫее, London. 1814r 206.
110 Сашш Р^Шалив Give tabula ншфцеишшяш a^quationis celebiuttaeimae y1 — аз* +
■olutionem pro ongulu numeri <tati valoribus ab 1 usque &d 1000 in numeris lib
uedemque integria «ihibenfl, Havni^e [CopenhagenX 1817.

Chap. XII] PELL EQUATION, <Я?-\-Ъх-\-С= □ . 369
P. N. C. Egen102 gave the 121 valuesof^t < 1000 for which x2-Ay2= -1
is solvable.
J. К Wezel101 proved that if S is the denominator of & complete quotient
(VA-br)/£ for the continued fraction for лЯ, and if p/q is a convergent,
then p2—Лд* = ±& By the periodicity, we ultimately get an S — 1. Thus
x2—4y* = dbl is solvable for the plus sign, and for the minus sign only if
the length of the period is odd. Also x2—Ay*= ^C is solvable if there oc-
occurs in the continued fraction for Va a complete quotient of denominator С
In the chapter on biquadratic residues in Vol. П1 will be given reports
on the papar by G. L. Dirichlet (Jour, fur Math., 3,1828, 35-69) where he
discussed &±qu*=p$*, p and q being primes and p=l (mod 4), and the
related pamphlet of 1861 by H. R. Gotting.
J. Baines104 found values of n for which
Then 15m*-h45m-b25=(mr/s=fc5Jifm^55(95=F2r)/Z>, where
Z>=r'-15>*. As by Euler, Z>=1 if (s, r) = (l, 4), (8, 31), F3, 244), D96,
1924), ---, whence n=6, 86, 401, 5361, —.
F. T. Poselger105 treated rx*-H = П by continued fractions.
C. G. J. Jacobi1" stated that the solutions of х*—ау*=1 can be expressed
in terms ol the sine and cosine of 2mr/a, and stated that he possessed a
generalization to the case in which a is a product of several factors. If
a = bct we can find in an infinitude of ways four integers u, v3 u>, x such that
the product of the four factors «±i» Vb^fcto ^±x jbc is unity, where two
or four of the signs are plus. The resulting relation can easily be given
the three forms y2-bz% = lt #f—сг] = 1, j£—az\^\. Hence the solutions
У, ",*t depend on «, y, w, x. The latter can be expressed by trigonometric
functions.
T. L. Pistor107 gave an exposition, illustrated by examples, of the methods
of Gauss and Legendre to reduce the general quadratic equation in z, у
to v*—Dy*=Nt ita solution by continued fractions if D > 0 and by trial
if D=—d, using 2=0, ±1, ±2, ■--, up to VJV/d- Од p, 44 is given a
table of the least solution of Pell's equation x2—Ityl=l, Z>=2, ■■ -, 200.
G. L. Dirichlet10* recalled Legendre's88 result that if p, q are the least
positive integral solutions of p2—Aq*=\9 then 2=fMgi—fNhlJ where/=*=1
or 2, and MN=A is a decomposition of A. Dirichlet proved that at
most one of the latter equations, in addition to l**g*—Ah1, is solvable.
Besides Legendre's theorems for primes A=4n+1, 8n-b3, 8n-l, Dirichlet
'«H&ndbuch der allgemeinen Aritiu, Berlin, 181&-20; ed. 2f Ir 1833r 457; II, 1834, 467;
ed. 3,1,1846,456: П, 1S497 468. a. Seeling.1"
"» Aimales Acsd. Leodienflis, liege, 1821-2, 24-30.
'м ОЪе GenUenyui'» Dmry7 or Math. Repodtory, LondoDr 1831, 38, Queet. 12в8.
JfllAbh. Abd. Wiaa. Berlin (Math.), 1832,1.
l" Letter to Legendre, May 27,1832; Werke, 1,4S8; Jour, fur Math., 80,1875,27C; Ann.de
J'Eoole Nonnale Sup., в, 1809,176-7; Bull. вс. Math. Aatr.,ft, 1875,136. a. Koenig.»
ш Uber die Auflueung der uobeat. GL 2. Gmden in eansen Zablen, Progr., H&mm, 1833.
101 Abh. Akad. Win. Berlin, 1834, 640-664; Werke, I, 219-236.

370 History of the Theort of Numbers. [Chap, xii
proved that, when A = 2a, where a is a prime, 2&—av?= 4-1 is solvable for
a=8n+7, 2t2-av?=~ 1 fop а=8п-ЬЗ> and t2-2au*=-l for a=8n+5.
This method of exclusion yields no result when a^Sn+l- But using
also the quadratic reciprocity law, he proved that P—2au^= — 1 is solvable
if a is a prime 16n-h9such that 2<e~1)/4^ —1 (mode), though the conditions
are not necessary. If a and Ь are both primes 4n+3, at*—fru*={a/6) is
solvable.* If a and bare both primes 4n+l and if {ajb) = — 1, &—abuz= — l
is solvable; but if (a/6) —1, And* (a/b)< = —1, (b/aL=—1, F—abu*=— 1 is
solvable, though the conditions are not necessary. He gave criteria for
the solvability of F—aicu*— — 1, where ay ht с are primes 4n-bl. Finally,
Dirichlet removed the initial hypothesis that p, q give the least solution of
p*-Aq* = l.
M. A. Stern108 developed the theory of continued fractions and in the
final article (pp. 327-341) made application to x2—Ayl=D) in particular
when Z>=±1, ±2. He tabulated 42 forms for Л, like w2n24-2wi and
Fn±l)*-|-(8n±2J, such that there is a small number of explicitly given
partial denominators in the continued fraction for VA, whence one finds
at once the least solution of x2—Ay2 = it 1.
B. Peirce and T Strong110 solved 376sHH4z+34=y* by setting
376s-b&7=x' and treating 376^-x/2=9535 by the theory of binary quad-
quadratic forms.
C. Gill111 noted the solution A364S57)*-369G1036J = 25 and that in
the least solution of t2*-940751u*= 1, и has 55 digits and t has 58 digits.
С G. J. Jacobim stated that, if p is a prime 4n+l, and x*—py*= —4,
then
where a ranges over the quadratic residues, between 0 and p/2, of p. If q
is a prime 8гс-ЬЗ, and xz—qy2= —2t then
If q and q* are primes 4л+3, and q is a quadratic residue of £', then
where qx2—§V^4, Cubing i( л[£х+ Vifa), we get solutions of gul—$V=«L
* Legeudre'a symbol (a/b) denotes + 1 or — I according авх'ва (mod h) is solvable or rwt.
Let с be a prime 4n + l, and ft an integer not divisible by с for which (fc/c) » -f Ir
vi*fJ М*-^ * + x (mod c). According aa *t*^V* « + 1 or — 1 (mod c), Dirichlet wrote
(Jb/c)< — + J or — lr reepectively.
i°» Jour, fur Math.r 10, 1833y 1-22, 154-166, 241-274, 364^376; U, 1834, 33-66, 142-168,
277-306,311-360,
u> Math. Musceilany, 1, 1836, 362-5; French banal., Sphinx-Oedipe, 8,1913r 117-9.
i« /bid., 230.
"Mon&teber. Abd. Wi«. Berlin, 1837, 127; Jour, fur Mtth., 80t 1845, 166; Werke, VI,
263-4; Opuftcula Mathematica, 1,1846, 324-5. Proof by Genocehi.IH

Chap.XII] PELb EQUATION, aX2+&J+C= П. 371
G. L. Dirichlet11* solved P—pu2^ 1 by use of trigonometric functions and
remarked that the method is not so well adapted to numerical calculation
as that by continued fractions and does not give the least positive solutions.
Let ai, - -», a* be the s = (p~l)/2 quadratic residues of the odd prime pf
and let bi, ■ • ■, b9 be the quadratic non-residues. Write i= V^T. In
where the upper or lower sign is taken according as p=4p-\-l or 4^+3, Y
and Z are polynomials in x whose coefficients (as shown by Gauss, Disq.
Arith., art. 357) are integers. By multiplication, we get
Let p=4/i+l. For x— 1, let У, Z become the integers g, k. Then
g2—ph*=£p. Hence g=pk, h2—р№=— 4. It remains to evaluate g aud
ft» Since <*i, '-fa, have in some order the same remainders as I2, 22, < •«, s2
when divided by p, we have
since
(mod 2).
In terms of the trigonometric product a, we evidently have
To pass from these solutions of k2—pk2 = — 4 to solutions of t* — pu* = 1, let
first p=8v-hl; then A and к are both even, so that
For p=8v-h5t it is stated that h and A; are both odd, whence solutions t, и
are easily deduced. But R. Dedekind114 noted that both k and к can be
even, as for p=37, 101, etc. Finally, if p=4/i-h3, it is shown that, for
z=i, Y and Z become дA±г) and hA41i)t where g and h are real integers,
and the upper or lower sign holds according as p=l or 3 (mod 8). Evi-
Evidently X becomes г. Hence Yi-^-pZ2—4X takes a form equivalent to
Q2—pk* = ±2, From this solvable equation, we pass to F—pu? = l by
setting (g+hi!py=2t+2u<Jp. The expressions for g and k in terms of
trigonometric functions can be found as before by use of x=it but are
not given.
11J Jour, fur Mftth., 17, 1837, 285-290; Werke, IJ 543-360. Reproduced by P. B&chmann,
Die Lehre . . . KreiatheUung, 1872, 294-9.
ш Dirichlet'a Werke, II, 413.

372 History of the Theory of Numbers. [Chap, xii
Dirichlet11* noted that while k—g/p is positive, the determination of
the sign of h presents difficulties. He showed that h has the same sign as
where n ranges over the positive integers not divisible by the prime p, and
the symbol (w/p) is Legendre's.
C. d'Andrea116 proved by use of continued fractions that x*—Dv? = l
is solvable.
Dirichlet117 noted that, if P is an integer > 1 not necessarily a prime,
where Ъ ranges over the integers < P and prime to P for which (b}P) = — 1,
and Y, Z are polynomials in x with integral coefficients. For x — 1, let
Y and Z become the integers Yu Zx. Then, if e— 1 or V? according as the
number of prime factors of P is > 1 or = 1,
where A is the number of classes of binary quadratic forms of determinant
P, and T, U give the least positive solutions of £*—Pu2 = l. For example,
if P=17,
Y1=31i 2i-8, e-2, T^=33, tf^8, whence h = l.
G. W. Tenner118 gave a convenient method to convert Va into a continued
fraction. Let a2 be the largest square <a. Then proceed as for
r
10
1,
1,
1,
2,
II
X
7,
5,
4,
2,
III
10
3,
6,
6,
8,
rv
113
9,
25,
36,
64,
V
—
104,
88,
77,
49,
VI
13
8
11
7
7.
Divide 10+10 by 13 and write the quotient 1 in column I and the re-
remainder 7 in II in the second line. Subtract 7 from a = 10 and write the
remainder 3 in Ш, and its square 9 in IV. Write the difference a—9 = 104
in V. Divide 104 by 13 (under VI in the preceding line) and write the
quotient 8 in VI. Similarly, to form, the third row divide ос-^З C of III)
by 8 (of VI) and write the quotient 1 in I and the remainder 5 in II; sub-
subtract it from a and write the remainder 5 in III, its square in IV, a—25=88
in V, and its quotient 11 by 8 (of VI) in VI- Continue until we find in VI
ш Jour, fur Math-, 18,1838, 270; Werke, I, 371-2.
"• Trattato elemental* <tt aritmetica e d'&lgebia, II, 1&40, 671, Naples.
u* Jour, fur Muth^ 21, 1840, 153^5; Werkey I, 493-6.
"•Einige Bemerkungen Qberdie Gleicbimgax1 ±1 = ^ Progr., Mereeburg, 1841.

Chap. XII1 PeLL EQUATION, ax*+bz-[-C=\3. 373
or II a number equal to the one above it. Then column I gives the de-
denominators (quotients) and VI the complete quotients in the continued
fraction; if the repeated number G in our example) occurs in VI, the last
number 2 in I is the last term of the first half of the symmetrical period
with an even number of terms;* but if the repeated number occurs in II,
the last number in I is the middle term of the symmetrical period with an
odd number of terms. If y1^ax2=-~ 1 is solvable, let Lfl, M/m be the
last two convergents for л(а, the second corresponding to the last quotient
in the first half period; it is stated that х = Р-[-т2, y=Ll+Mm [cf. Euler,72
end]. For example, if a=113, a = 10, the half period is 10, 1, 1, 1, 2 and
the convergents are 10/1, 11/1, 21/2, 32/3, 85/8, whence s=3l-b82-73,
^=3.32+8-85. But if we use 1, 1, 1, 2 and the convergents 1/1, 1/2, 2/3,
5/8 to Va—a, we have the same xt while y—а(!г-Ь;п2)+&\-|-тр. If there
be an odd number of quotients at • • •, 2ot, let Kjk, Lfl, Mjm be the last
three convergents for -Л, the third corresponding to the middle quotient;
it is stated that z = (A:+»i)Z, y = (K-\-M)l±l = (k+m)LW-l [equivalent to
Euler's y=lM-[-kLt since AL~Kl=±l~}> Tenner continued Degen's table
from 1001 to 1020.
Dirichlet11* proved that, if D is a complex integer not a square,
t2—Du2 = l is solvable in complex integers and deduced all solutions. It
applies120 without change to the case of real numbers. The proof rests on
the lemma: if a is a given complex irrational number, we can find an infini-
infinitude of pairs of complex integers x, у {^#0)such that N(x—ay)<4:jN(y),
where Nft+bfy^kP-S-b2 for к and b real. Then, since the modulus of r-{~&
does not exceed the sum of the moduli of r and s,
(x - ay) +
Since N{y) =1, the lemma gives
Hence N{x*—ahf) remains less than a fixed limit for an infinitude of pairs
of complex integers. Now take a= V5. Hence X2—Dy2 takes the same
value 2Ф0 for an infinitude ofpairs s, y, and hence for an infinitude of
pairs for which the s's and y's differ by multiples of I:
x*-Dy2=x\-Dy\ = l, x=xh y~yx (mod I).
By multiplication, (xsi—Dyy^J— D{xyx—yxtf — I2. Since xyx—yxx is di-
divisible by I, also xx\—Dyyx is divisible by /. Hence J2—Du2— 1 is solvable
in complex integers tt и (и Ф 0). All solutions are shown to be given without
duplication by
^Щ (n=0, ±1, ±2, ...),
•Note that VHe-lO+I + i+{+| + i+f + l+i + l+ l-ifii5-«b
u'Jour. fOr Math., 24,1842,328; Werker I, 578-588.
Ui Abh. Akad. Wi«. Berlin, 1854, 113; Jour, de math», B), 2,1857,370; Werke, II, 155,17&
Cf. Dedekind.^

374 History or the Theory of Numbers. cchaf. xii
where T, U is a fundamental solution, i. e., one for which N{T-\-U^D) is
the minimum of all the N(t+u-{D)>l* If D is real and positive, t> и are
both real or both pure imagin&ries. Thus if the fundamental solution is
real, all solutions are real. But if it be imaginary, only even values of n
give real solutions. Since pure imaginary solutions give real solutions of
t2—Du*= —I, the fundamental solution is imaginary or real according as
the latter equation has real solutions or not, and the least positive solutions
are T/i, U/i in the former case.
Du Hays121 derived, for the case b=0, Euler's85 recursion formulae be-
between consecutive sets of solutions of azz+c = O, and gave the nth set.
Chabert122 treated nyt-\--py+q= D by equating it to n(y—0){y—Ю and
setting {y—fOnff={y—pr)f~ Use an irrational / if &y 0f are irrational.
While we cannot always get rational xt y, the process is said to be far simpler
than Legendre's.
G. Eisenstein12* proposed the problem to find a criterion to decide a
priori if p2—Dq*=4 is solvable in odd integers, given that D is a positive
integer 8n+5, i. e., if the number of improperly primitive classes of quad-
quadratic forms of determinant D equals the number of properly primitive
classes of determinant D or is three times the latter number.
F. Arndt1*4 extended the work of Legendre88 on p2—Aqz = lt who treated
only the cases in which Л is a prime or a product of two primes. Let p,
q be the least positive solutions. First, let A be odd. Let 0i be the g.c.d.
of p+1, q, and 62 that of p —1, q* Then
ifipi, p-l=bfip2, e&=2q> pipi-Л, l = (§0iJpi-G®p2 ipodd);
0?*!, p — 1= в\<гь 0A= q, <Ti<T2=Af 2 = el<Ti—(%<T2 (peven).
If Л —4w+l, only the first system of relations holds and p\7 p2 are both
— 1 (mod 4} if £fli is odd, while both are =3 (mod 4) if $eL is even. If
A = 4m-\-3t either system may hold; if the first holds, pi=l, рз=3 (mod 4).
If Л is an odd power of a prime 4т+1у then pi = A, pi = l, and
-1 = (^2J-(^JЛ,
whence — 1 is a quadratic residue of Af and the number к of terms in the
period of the continued fraction for VZ is odd. Let (^A-\-In)fBn beany
complete quotient; then if ft is odd A=l£+I)t & = (k+l)l2r If, for
Л — 4m+l, the number к of terms in the period is even, the denominator
of the middle complete quotient is odd. If A=2n, where n is odd and
>3, it is proved that
If Po, qo be the least solutions of the latter, then p, q give the least solutions
of p2—2nqz=lr Since Po=3, qo = l when A=8> we can find the solutions
step by step, Finally there is treated the case A = 2nA\ A* odd.
131 Jour, de Math., 7,1842, 325-30»
w Nouv. Ann. Math., 3, 1S44, 250-3.
ш Jour- fur Math., 27, 1844, 86.
m Disquisitionee nonnull&e de fractionibua oontinuis, Dies. Sundiae, 1845, 32 pp. Extract
in Jour, fur Math-, 31,1846, 343-358.

Pull Equation, ахг-\*Ъх+с=П. 375
F, Arndtlfl! simplified the solution of я*—л4у* = ±1 when A has a square
factor.
J. F. Koenig" stated that Jacob! had remarked to him that if
and C, D are derived from Л, В by changing the sign of ^lg, then ABCD,
AC BD, AD ВС equal, respectively
(a) m*~-fgn\ m'-fn", m'*-gn"\
where
Given values of m, - -, n" for which the expressions (a) are unity, Jacob]
desired solutions a, •* -, d of (p). Koenig employed
and noted that a7, • • -, <P are hnear functions of г, while, by computation,
He gave a table of values of a, by ct d for/^2, 3, 5, 6, 7, дШ20, and values
of a, - fd giving x2-/^/2--! for/, ^<100,/^<1000.
J. B. Luce127, to solve x2—щ?=г\ set n=a2±b, Vn=a±;\ whence
In the resulting successive convergent», take the numerators and de-
denominators as values of x, y. If m=2afb is integral, ^—^ = 1 for
p^am+l, q=?m. If not integral, seek a square whose product by n leads
to an integral value of 2a/b. He gave a table of such square multipliers
for n ^ 158.
F. Arndt128 was led by investigations on binary quadratic forms to
XV4 J»0 If D=0 (mod 4), its roots are z=2t, y=u, where
If D-2 or 3 (mod 4) or D=\ (mod 8), its roots are x=2t, y = 2u> where
P-Du*=±l. For the remaining ease Z>^5 (mod 8), he tabulated the
least solutions for those values <1005 of D for which the equation
x*~Dy1=±4: has relatively prime solutions, the solutions being for
xi-Dy2 = -4 if it is solvable (such a D being marked D*). If x, у give the
least posit ve solutions of the latter, Х=хг+2, Y=xy give the least positive
solutions of X*—DY2^ +4. If the last is solvable in relatively prime num-
numbers, its least solution is easily deduced from that for x2—Dyi= 1.
Math. Phye., 12y 1S49,239-Ш.
u"ZerIegung der Gleichung 2* —fmf = ± 1 in Factoren, Progr., Konigeberg, 1849, 23 pp.
Extract ш Archiv Maih, Phys», 33» 1859,1-13. Cf. Jwx>bi.10»
m Amer» Jour. Sc. Arte (ed., SUKman), <2), 8,1849r 55-60.
ш Archiv MatlL РЬуя., 15, 1850, 467-478.

376 HisxofcY о*1 the Theory of Numbers. ЕСялг. xii
C. Hermite12* proved that if D is positive, x*—Dy2~l has an infinitude
of integral solutions, and all are given by
х+уЛ)>=(а+ЪЩ< (г = 0, ±1, ±2, --)
where а, Ъ are solutions such that a+h^ is a minimum.
A. Genocchi180 proved the results stated by Jacobi11* by means of
r2=FpZ2^4X [cf. Dirichlet113]. For x=i= V^, let Y and Z become
y+y\i and z+Zii> According as the prime p is of the form 8&+3 or fc7
we have
where the product extends over the (p—1)/2 quadratic residues of p.
P. L. Tchebychef131 proved that if a, 0 give the least positive solutions of
x*~Dy*— 1, and if x*~Dy*^ dzN is solvable, one solution has
If а, Ь and a,\7 hi are solutions x, у within these limits of x^—Dy^ —
then (abi+aj>)(abi—aib) is a multiple of JV, while neither factor is. Hence
if x*—Dy*= ±iV has two distinct sets of solutions within these limits, N is
composite.
A. Gopel132 proved, by use of continued fractions, that if A is a prime
of the form 4A'+3 or the double of such a prime, x2—Ay2 = zh2 is solv-
solvable, the sign being + or — according as A {or |Л) is —7 or 3 (mod 8),
and related theorems as to the values of A for which хг—Ay2—2 is solvable,
to be given in Vol. Ill under binary quadratic forms.
G. L. Dirichlet1" noted that if /=ar3+2fcty+c£*has for its determinant
D=b*—ac a number not a square, and if <r is the g-c.d. of a, 26, c, and if
is a transformation with integral coefficients of determinant unity which
transforms / into itself, then
where t, и are integral solutions of t2—Du2 = <r2; and conversely, if t, и are
integral solutions, the values of X, • • ■, p are integers which determine a
transformation of / into itself. For the more difficult case in which D is
positive, and / is a reduced form, obtain from the period of reduced forms
defined by/ all the transformations of/ into itself and hence, by the above,
deduce aU solutions of &~Dv?=v*b This theory, which will be given under
binary quadratic forms in Vol. Ill, is closely connected with the continued
fraction for the positive root of a-h2bw+cuj1^0»
""Jour, far Math., 41,1851,200-211; Oeuvree, 1,185-7. "" ™
«о Мйт. Coiuonn& Acad. 8c- Belgique, 25,1851-3, IX, X.
** Jour, de Math., 16,1851, 257-265; Oeuvree, I, 7S-80; Sphiitt-4)edipe, 10,1915, 4,18.
** Jour, fur Math*, 45,1853,1-14» Summary, iWtf.j 35, 1847, 313-8; Jour» de Math., 15,
1850, 357-362. Cf. Smitb,1" ft 123, P< 783; Coll, Math. Papere, 1,284-8,
«• Abb. Akad. Wine. Berlin, 1854, Ш-4; French trwigL, Jour, de Math., B), 2,1857, 370-3;
Werke, II, 155-B, 175-8. Zahlentheorie, ft 62, 5 83,1863; ed. 2,1871; edv 3,1879; ed.
4, 1894, Cf. H. MiDkow&ki, Georoetrie der Zahlen, 1, 1896,164-170,

. xirj Pell Equation, ах*+Ъх+с^П. 377
Dirichlet134 recalled the fact that if T, U are the least positive solutions
of P~Duz—l, where D is positive and not a square, all positive solutions
are given by
Ад infinitude of values of un are divisible by any positive integer 8. He
proved that, if N is the least n for which u* is divisible by S, the remaining
n'0 are the successive multiples of JV. If Df = DS* and if 7", V* give the
leflst positive solutions of F—Dfuz= 1, then N is determined by
For any prime factor p of S, let v be the least index for which u¥ is divisible
by p and let jf be the highest power of p dividing uv. Then, if e is arbitrary,
the exponent of the highest power of p dividing ирё is $+*, where t is the
exponent of the highest power of p dividing e. Let v{f 5* be the values
corresponding to the general prime factor pi of 8=Пр? and let N be the
Lc-m. of Vip?"*1 (i ^ 1,2, - •). Whenai, a2, •• ■ increase indefinitely, S/tf
approaches a Kmit. The application to quadratic forms will be given under
that topic.
(X A. W, Berkhan135 gave an exposition of the theory of or^+l^y*
and a table of solutions for a^lGO.
M. A. Stern136 applied new theorems on continued fractions to shorten
the work of forming an extended table of least solutions of х*—Ауг=1.
Given the period for one number, we can find an infinitude of numbers the
continued fraction for whose square root has a known period. He gave a
table showing the manner in which the continued fractions for the square
roots of 163 of the numbers < 1000 can be derived from that for 2.
A. Cayley1*7 gave for 1X1000, D-5 (mod 8), a table showing the least
odd solutions of x2—Dy*=— 4, when it is solvable, or, if not, of x2—&y* = -\-4,
when the latter is solvable. The computation was made by means of
Degen's101 table; if in the second line of the entry for D the number 4 does
not occur, there is no solution of x2—Dy2^4; if the rank of the place in
which 4 occurs is even, this equation and also x*—Dy*— —4 is solvable; if
of odd rank, only x2— Х>уг=4 is solvable. Also the least solution can be
found by means of the series of quotients (in the first line of the entry) by
stopping at the number preceding that above 4 and computing the con-
continued fraction determined by this series» From the least solution of
t2—IV=— 4 we get the least solution х^т*-Ь2, y-rv, of x2— Ву2 = +±,
and the least solution X=(t*+3t)/2, y^(r*+l>/2, of X*-DY*=-1.
From the least solution of T2—DIP=4 we get the least solution
m Monateber. Akad. Wise, Berlin, 1855, Ш-5; Jour, de Math., B), 1,1856, 76-9; Jour-fur
Math., 53,1857,127-fl; Werke, II, J83-194.
w Lehibuch der Unbeetimmten Analytik, ШЛе, 2,1856, J21-193.
m Jour» fiir Math., 53, 1857,1-102.
117 Jour, fur Math., 53, 1857, 369-371; Coll. Math. Papers, IV, 40. Reprinted. Sphinx-
Oedipe, 5, 1910, 61-3. Ertnta, Cunningham,»» p. 59. Extenmon by Whitford.1*

378 History or the Theory or Numbees. [Chap, xii
G. С Gerono1* proved, following Lagrange,8* that x2—ny1^! has an
infinitude of integral solutions, if n is positive and not a square. If x, у
are positive integral solutions, x/y is a convergent of even rank of the con-
continued fraction for Vn and corresponds to the next to the last incomplete
quotient of one of the periods.
H. J. S. Smith1" stated the principal theorems relating to P—Du* = l or
4 by use of Euler's72 notation (qlr • • -, qn). He noted, as had Lagrange™
and Gauss** (Art. 222), that the methods used by Euler*6»71-*3 are incomplete
because he always assumed that a first solution is known and merely
deduced from it those solutions which belong to the same set, whereas there
may exist solutions belonging to a different set, and lastly because he gave
no method to distinguish between the integral and fractional values con-
contained in his formulas for x, y.
L. Kronecker140 noted that if Г, V are the least solutions of T*-PIP=: 1,
log (T+l/VP) can be expressed in terms of special theta functions or elliptic
functions, and the number of classes of binary quadratic forms of deter-
determinant P. He deduced approximate values for T, U; likewise, for the least
solutions 4, 1 of P—17tta — ~1, 4+ VI7 has the two approximations
R. Dedekind*41 proved the existence of integral solutions t, и )
of F—Du? = \ by the method used by Dirichlet120 for complex integers,
but replacing his lemma by the following: There exist infinitely many pairs
of integers z, у such that z*—Dy* is numerically <1+2*{Б.
С. Richaud1" stated that x*—Ny*= — 1 is solvable for various types of
values of N\ If At * > >, L are primes of the form 8га-Ь5 and N—2A*t
2Au+ig»+i ог 2А^В^.. .L\ If Bt • •., L are not included among the
linear d visors of P-2Au\ and N=2A*&> 2Л%^+|С^+1 or 2Au*l£* - > Vх,
If a, b, * —, I are primes of the form Bn+1, and are not included among the
linear d'visors of fi*-2Au\ and N=2A2**1aT, 2A2m+1au+lb2tt+t or 2A'»+li^
* • >P\ If A, * - *, L are primes not included among the linear divisors of
Р-<*иг where a is a prime 4n+l, and JV= «Mfa+1, *Р»+1А\ «*lll+liifc+lB"l+1f
v*m*lA - "La* Also for 8 more s\ich sets of N's* He145 proved these
results and similar ones by use of the continued fraction for Vtf and the
reciprocity law for quadratic residues.
Richaud144 gave minimum integral values of x, у satisfying z2—Лу*=1
for A=a*±d (d a divisor >1 of 2a) and for many values of A such аз
(9a+3Jdb9, (9a+0)l=h9, B5a+5J-25. Likewise for x*-Ay*= -L
u" Nouv, Ann. Matb4 13,1S594 122-5,153-8»
"' Report British А&юс., 1861, \\ 96, 97, pp. 313-9; Coll. Math. Papers, 1, 195-202.
i" Monateber. Akad. Wto. Berlin, 1863, 44; French tranal. in Aitnalee ec. de P6cole Dormale
eup.t 3,1866,302-8. Cf. Smith, Report British Адэос., 1865, \ 138, p. 372; Coll. Math.
Papera, It ШЧ&.
1Л Dirichlet'fl Zablentbeorie, 55 141-2,1863: ed. 2, 1871; ed, 3,1879; ed. 4,18W.
i« Jour, de Math., B), 9, 1864, 3S4^8»
1Л1Ш., B), 10, 1865, 236-280; B), 11,1866,146-17B.
"« Atti Aocad. Pont. Nuovi lined, 19,1866,177-182.

Pell Equation, ах*-\-Ъх+с=П. 379
M. A. Stem1*1 proved (p. 27) that хг—Ay2=d* has one and but one
solution in integers if dn F<dn< VZ) is the denominator of a complete
quotient which belongs to a partial denominator a»+i of the u negative "
periodic continued fraction
г n 1 x
[a,alfa2, .-]=a-- --
<*i — a* — • • •
for Vil, and xt у are the numerator and denominator of the convergent
[a, au • • •, an% The first dn which is unity leads to the solution of
x2—Ay* = l in least integers; this d« is the denominator belonging to the
final term of the first period. He found (pp. 30-43) the conditions for
dv% = 2. Finally there is atabie giving for JV<100 the partial denominators
of the half period and the complete quotients for the negative continued
fraction for VAT. Lagrange86 had shown by an example that Pell's equation
cannot be solved by use of a continued fraction in which the partial de-
denominators have signs chosen at will.
X Frischaiu™ noted that Gaussw (Arts. 197-202) obtained the least
solutions Tt U of P—Du*= <r* by use of a reduced quadratic form of deter-
determinant Z>; It is here shown that Г, V are independent of the particular
reduced form used.
N. de Khanikof147 used a table showing the last two digits of the root
of a square ending in 01, 04, ►• •, 96 to find the endings of possible integral
solutions of A+BP=u\
P. Seeling1*8 treated the form of numbers the continued fractions for
whose square roots have periods with a given number g of the terms,
treated the cases g = 2, * —, 7 in detail, and tabulated the period of the
continued fraction for VJ, 2 ^A ^602. He noted that Egen102 omitted
from his table all numbers of the form n*+l, though they belong there.
Egen stated that x*—Ay*= — 1 is solvable only when the period of the con-
continued fraction for VA has an odd number of quotients. Seeling stated
that it is possible in relatively prime integers x, у only when A =4m+l or
4to+2. Hence if the period for VZ has an odd number g of quotients,
A=4m+1 or 4яъ+2; this is proved for g=l> 3, 5, 7.
L. OttingexMi gave tables showing several solutions of хг~Ау*=±Ъ
forA=2, —,20; Ь=1, —, 10,3*,5*,7* (fc = lf 2,3,4). If we have found
by continued fractions the least solution of p*—Лд^ = ±Ь and know a
solution of F—Au*=l or —1, another solution of x*—Ay1 = tbb is given by
A
q, ypq
A. Meyer1*0 proved by use of ternary forms that if D is a positive integer,
2* the highest power of 2 dividing Dt o"=4, S* the greatest odd square
dividing Dt and Z>=2'jS>1Z>i, then there exist integers £, ijj relatively prime
w Abhand. Geeeil, Wi». Gettingcn (Math*)» 12,1866, 48 pp.
ш Sitmmpiber. Akad. Wise, Wien (Math.), 55, II» 1867,131.
w Comptce Rendus Paris, 69,1869, 185^8.
Ml АгсЫу Math. Phya., 49,18G9, 4-44.
"■ Ihid>t 193-222»
"♦Dies., Zurich, 1$71; Viertdjahwecbrift Naturf. Geeell. Zurich, 32, 1887, 363-382, a.
Got.*»

380 History of the Theory of Numbers. [Chap, xii
to 2 Д such that for all primes p and q satisfying
P«f, Я=ч (mod8Si>i)f
the equation &— pqDu2— 1 has a fundamental solution T, V for which
neither Г+l nor T—1 is divisible by pq-
L. Lorenz1*1 found the number of integral solutions of m*-\-en*=N>
where e = l, 2, 3,4 or — 1, and N is a given positive integer, by transforming
the series
into a series of another form and finding the term qN of the latter. For
details when 6=1 see Lorenz94 of Ch. VI.
P. Seeling152 noted that, if Л is positive and not a square, and the con-
continued fraction for VZ has the symmetric period n\ a, b, •••,&>», 2-й,
solutions x7 у of x2—Ay* = ±l are given by the numerator and denominator
of the convergent belonging to the quotient 2n. The sign is plus if the
number of quotients in the period is even; while if it be odd, the sign is
plus after 2, 4, 6, — >, periods, minus after 1, 3, 5, — periods. If
хг—Ay2 — — 1 and the number of quotients in the period is odd, then
A^4»i+1 or 4m+2and A has no factor 4?n+3; if A is a prime 4m+l, the
number of terms in the period for VJT is odd; if A is a product of two or
more primes 4?ra-f-l or the double of such a product, no general rule has been
found* Finally, he tabulated all numbers A <7000 for which the period of
VZ has an odd number of terms, so tnat xx—Ауг= — 1 is solvable.
A. B. Evans and A. Martin1*3 found the least solution of rz*+l= □,
where r=940751, and noted that ra^-|-38 = O has no integral solution.
Moret-Blanc161 noted that if x = h, y=kisa. solution of 2x2 — 1 = y2t then
x=hu+kv, y = 2hv+ku give a second solution, provided u2—2^ = 1, as for
u-3, v=2.
F. Didon stated and С. Могеапш proved that, if D= Dn+2J+I, where
n is a positive integer, t2—Lh£=4 has no solution in odd integers, and the
least positive solution is *=16Bn+lJ+2, и=8Bп+1).
О. Schlomilch16e discussed the continued fraction for
L. Matthieflflen16* noted that if x=/, y=g give the least solution of
* 1, all solutions are given by
» Tldfflkrift for Math., C), 1,1871, ft7. a. *J. Petersen, Aid,, p, 76.
« Anshiv Math. Phyii., 52,1871, 40-9.
i" M&th, Quest, Educ. Times, 16,1871, 34^6.
>" Nouv. Ann, Matb., B), 11, 1872,173-7.
«JW*,48; B), 1^1873,330-1.
«• Zeitechrift Math. РЬув,, 17,1872, 7D-71.
1W Ibid., 18,1873,426.

Chap. XII] . Peix Еотатюк, ах?-Ъ-Ьх+с=П. 381
If x=jy у —Я give the Least solution of ая?~у2 = —1? all solutions are given
by the preceding and a similar formula.
D. S. HartJM stated that, if the fundamental set of solutions p<>, q0 of
p3_JVg- = ^l has been found, so that we have a set in addition to 1 0, the
simplest method to find successively all further sets of solutions is to use
the relations p^2p0r+r', q^2pas^$'r where r, s are the last found va'ues
of p, q> and r\ $' the next preceding values»
B. Minnigerode1" modified the theory as presented by Dirichlet133 by
using a different definition of reduced forms and using the continued fraction
11 11
OJ—Go— — —»
<*1—U2— * ■ " -^dp—i— GJ/
with negative quotients (see the chapter on Ыпагу quadratic forms in
Vol. III).
W. Schmidt160 showed that all positive solutions of P—Du2=±4: are
given by the development into a continued fraction of a root of a certain
reduced binary quadratic form of determinant D.
T. Muir1Wa treated the development into a continued fraction of the
square root of any positive integer or fraction. In particular, he obtained
(p. 19) in general form the results of Euler,7* calling Killer's (a, * -. I) a
continuant K{af - - •, I).
D. 8. Hart and W. J. C. Miller16' proved by use of p2-103n* = l that
lQ3Cx—2)г+1 — □ has no integral solution x and that 22421/3 is the least
positive solution.
M. Collins and A. M. NashM1 proved that &+D" = (N*+&)y2 js soivablc
in rational numbers if ^—271+1 by taking
Several163 solved x1—953y*= ±1 by the continued fraction for V953,
S. Tebay164 poted that, if p, q are the least solutions of x2—ny*^ i then
Let na*±k=m\ To solve пР±к=Пу set t=a+r. Then
уг if ^
S. Bills185 illustrated a "new, practical" method of solving a?—Ay2^±l
by taking Л— 953. Then jS—30 is the root of the square just <^^ JVom
>" Math. Quest, Educ. Times, 20, I874r 64.
"• Guttingen NAchrichten, 1873, 619-652. (X A, Hurwit».**
Mi Zeitschrift ШЬ. РЬун.г 19,1S74, 92-94.
w^The Expression of a Quadratic Surd as a Continued fY&ctioD, Glaaeow, 1874 32 pp,
Cf. R. E, Morita, Ueber Contmuanten und gewisse ihrer Auwendungen iin?Za.lilen-
theoretiachen Gebiet^, Dies. Strasaburg, Gottiiigen, 1902.
^ Math. Quest. Educ. Times, 20, 1874, 66-7; 28, 1878, 65-66.
i* /Ш., 22, 1875, 23-24.
l» Ibid.t 78-80; 23,1875,107.
mIbidrj 23, 1875,30.
ie Ibid., &

382 History of the Theory of Numbers. [Chap, xn
0,1, S as the initial triple and AfT iV, P as any triple, derive the next triple by
where [k] is the largest integer ^k. When we reach a triple with P=2St
we get & solution. Application is made to solve nF±A = D.
Several writers1* proved that, if т is the least integer for which Лг*— 1 = D
and if Л1Р+1 = П, then R is a multiple of r.
D. S. Hartier showed that 560 is the least z making 953z2+87z+l = П,
A. Martin10 noted that x=1284S36351 gives the least solution of
x2—5668^=1, whereas Barlow" gave erroneously a number of 48 digits,
[Barlow solved я1—56587y*=l; the omission of 7 was a misprint.]
H, J. S. Smith1* proved that, if T, U are the least integral solutions of
T*—DTP={~I)*, then T+U-fi) equals the product of the i complete
quotients in a period in the development of VZ) as a continued fraction. Also
theorems on die number of different periods of complete quotients.
D. S. Hart170 gave a "new "method to solve x*—Ay*=l.
Then (z+ry)(x-ry) = l±my\ Setx-n/=l. Then
But the solutions are not in general integers. He and A. Martin1*1 found
positive integral solutions of 94х*-Ь57з+34 = □.
A. Kunerth172 required rational values of p for which x=N/D is an
integer, N and D being given quadratic functions of p with integral coeffi-
coefficients. Replacing p by a suitable linear function of qt we get*
where S=p—Q<№ is known. Any common factor of dt f may be removed
from each member of the second equation. Write vjw for the rational
number q and equate each, positive or negative factor of S in turn to
v"*—gufi. Hence for g negative, there is only a finite number of trials. To
apply to y*—ax2+bx+c with the given solution x\t y\t set y=p(x-~
in if=a(a*-*I)+&(s-*0+ri. We get
The case b=0 is treated at length» The method is applied (pp. 24-32) to
Pell's equation з/*=ах*+1; as y=px+1, x= —2pf(j?—a)> He reproduced
(pp. 56^8) Tenner's rule.1M
» Math. Quest. Bduc. Times, 23,1875,109-110; 24,1876,109-111.
»"/«rf.r25, 1876,97.
"■Tbe AnaJyHt, D« Moin«, 2, 1875, 140-2; Math. Quest. Educ. Times, 26, 1876, 87;
Math. Magazine, 2, 1890, 59.
"■F*oc. London Math. Soc,T 7, 1875-G, 199-208; Collectanea Mathematics, Milan, 1881,
117; ColL Math, Papera, 2, 1894, 148.
"' Math. Quiet. Educ. Tim*, 28f 1878, 29-30.
™ 1Ш., 101-2; 24,1876,39-Ш.
^fflnmpber. Akad. Wiffl. Wieu (Math.), 75, П, 1877,7-58.
* Ibere are five errors of eigue on pp. 15-16. Li the examples the eignfl are correct.

Chap, XUJ PeIX EQUATION, ОХ*+Ъх+С= □- 383
A. Martin171 noted that, in the least solution of s2- 9S17y* -1, x has 97
digits.
D. S. Hart noted that (r4-*V-l = D for y=m*+n*, if
ms-rnzb
where one of r, s is odd and the other even, while n is to be found by trial.
Martin1^ found the least solution of x*—9781y*^I.
S. Roberts1* noted that if P—Jhf=— 1 is solvable in integers, where
£>=2V*' - -, /t=0 or 1 and *, /5, . •. odd, then *-.D'u2= -1 is solvable,
where />'^2V**yS***- • >. Since any prime 4»+l is а Д any odd power
of it is a Z>'. If Z>=AJ, the solvability of P—du*=— lisa necessary, but not
sufficient, condition for the solvability of &—Ih£= — \*
Roberts17* proved that, if f, и are the least solutions of f*—Au*=l, there
are values tu ub less than I, u, for which either Af^J—Nu\ = ±l, MN=A,
or Af/J--^iiJ=d=2, MN=A, unless Af=L If the first of these equations
is solvable and M<N> then M is the middle denominator of the penod of
the continued fraction for VJ; but if the second holds, and not the first,
2M is the middle denominator,
H. Brocard177 gave a bibliography and historical notes on PelPs equation.
K. E. Hoffmann178 recalled that Lagrange proved that xb y0 is a solution
of x'—Ay*^\ if Zofyo ш the convergent corresponding to the first or first
two periods of the continued fraction for VX Other solutions follow from
While it is usually merely stated that x*{yn is a convergent to a later com-
complete period, a direct proof is here given by use of the 4i closed form " of
a periodic continued fraction {ibid., 62, 1878, 310-6),
A. Kunerth17* gave ft " practical " method of solving
A7) у*=ах*+Ъх+с.
If a rational solution is known, we may transform A7) into
Hence every such transformation yields two values —8/7 and — f/« of x
giving rational solutions. If x^mfn, y=r{n is a solution of A7), take
y-n, &=—m. Then г=та+п&} from which we may determine a, 0.
Then c, f may be found from A8), Tb proceed without a known solution,
subtract (ctx+p)* from A7) and employ the condition that the difference
be a product of two linear functions:
A9) (b
■* The AnaJyHt, Dee МоЫ*, 4, 1877, 164-5.
"/ШЧ5,1878,118-9.
Math. Vbritorr 1,1878, 26-7.
71 Froc. London Math. Soc., 9,1877-8,1M.
»Ibid., \0t 1878-9, 30^32.
in Nouv. Сопедр. Math., 4,1878,161^9,193-200, 228-232, 337-343-
l" Arcniv Math. Phya., ft4r 1879» l-«.
lTf SitBUngeber. Akad. Wiaa. Wien (МаШ,)» 78, П, 1878,327-37.

384 History of the Theory of Numbers. [Снл*. xii
Set D = V—iac, fl = (K+ha)jBa). Then Ю=аА*+В(*2-а)* Hence we
have to ass:gn to A and a such values that the latter sum is a square.
To apply (pp. 338-346) this method to the congruence yz=c (mod Ъ),
where bis a prime, we have A7) fora=0. Then A9) holds for A=b-\-2pail
The first denominator may he made equal to zbb if ±b is a quadratic
residue of с Then a— ^wa(v^-\-0wd).
Kunerth180 continued the same subject. Let alt /3i be a solution of
r=7na-\-nfi. Then а^^—пр, /3=0i+mp. Substitute these in A8), with
7 = n, d— —m. After several reductions, we get
Then A7) has an integral solution if and only if p can be chosen to make the
value of x for which the preceding vanishes an integer.
A. B. Evans and others1*1 proved that, if рп/я* is the last convergent in
the first period of the continued fraction for t[A, and r is the largest integer
= VA, then pn=rqn—qn-i- Hence we can derive x from у in a solution
of &-Ау*=1.
J. de Virieu1*2 used the final digits to show that xy is divisible by 5 in
B0) 24x*+l=y\
E. Lionnet1*3 stated and M. Rocchetti and F. Pisani1*8 proved easily that
three successive seta (x<, у*) of solutions of B0) or 2х2±1 = 3у* satisfy
xn+i = 10Xn-x^iJ y*+i = XOyn~ Уп-i, with (xuyi) = @,l) or A, 1), (z2, y2)
= A, 5) or A1, 9) у respectively. For solutions x of the second equation, 3x*
+2 is of the form 360n-|-5and is simultaneously a sum of three consecutive
squares and a sum of two consecutive squares. Рог х*-\-1^2у*7 хп^&хп-х
-*„_*, у»~Ьуп-\-у»-2, (xi, lfi) = (l, 1), (x2j Vt) = Gt 5).
S. ReaUsi" used х*~ку* = {о?-кр*){А*~кВ*)\ where
to derive a new solution of 2?—ку* = Н from a given solution a} 0 and a
solution of A*—kB*= 1.
H. Ротсагб185 noted that, if m is odd, and а, Ъ give the least integral
solutions of o2—rrib*=tl and c, d give the least odd integral solutions of
*—4, then
Several18* proved easily that хл+р=2х&п—Xn-V, ул+Р=2х1>ул—уп^,> if
xn, у * be the nth set of positive integral solutions of ж2—Nyt = 1 [xa = 1, jfo=0].
in SiUungaber. Akad. WiM. Wien (MathOr 82, II, 1880r 342-75.
1Ь Math. Quest. Educ. Times, 30,1879, 49.
i« Nouv. Ann. Math., <2), 17,1878> 476.
i" /Ш., B)J 18r 1879, 479, 528; B), 20r 1881, 426-7, 373-4. Cf. Pisani^ of Ch. VII.
«* Nout. Correap. Math., G, 1880, 306-312, 343-350.
i* Comptea Rendus Parie, 91, 1880,846.
i~ Math. Queet. Educ. Times, 34, 1880,114.

Сяар. xiij Рвы. Equation, ax*-\-bz+c=O. 385
W. P. Durfee1*7 stated that, if **, y0; xt, y%} ■•- be the integral solutions
of ах?-у* = — 1, arranged according to magnitude, then
S. Giinther188 noted that the solution of 2z*—l=y2 was apparently
known to Plato. Ita complete solution implies that of 2x*-\-l=y2, and
conversely. To solve (а*-\-Ъ*)х*—l^y*, seek the integral solutions of
{«-(аЧ-Ь1)**=a*£ and test whether or not a2f divides 2(сР+ЪР№±Щп;
if so we have a solution of the initial equation.
E. de Jonquieres18* found the period of the continued fraction for VA
for special types of numbers A, and treated periodic continued fractions
whose numerators differ from unity.
D. S. Hart190 stated that a process, simpler than Euler's and Lagrange's,
to find integral solutions of ах*-\-Ъх+с= □ is to subtract such a square
(ix-hmI that the difference will factor into two linear functions with integral
coefficients. Then H+MN** D = (L—Mr/&y gives x; equate its denomi-
denominator to unity,
E. Catalan191 discussed Лз*=у*+1. Thus A is of the form a*+63. If
P, 4 g^ve the least solution, x is divisible by p. Set z=pz; then
Hence consider (а*-\-1)х*—у*+1. For ita solutions,
It is shown that хл is a sum of three squares if «^3. If Ъ>1 in the initial
equation, xn is a sum of four squares. Every integer y>l, for which
(a4-l)a*=y*-l, is a sum of three squares. Cf. Catalan51 of Ch. VII.
S. RobertB1" proved that g2—2)^=1 can be solved by using the nearest
integral limits exclusively or superior limits exclusively as the partial
quotiente belonging to the continued fraction for Vo, instead of using the
customary inferior limite exclusivaly. But he admitted his results are due
to Stern and Minnigerode»168
G. de Longchamps191 gave a bibliography of PelPe equation.
J* Perott194 proved that there exists a positive integer X such that, in
ux is divisible by a given odd prime, where tu «i give the least positive
solutions of P-du*=l. He repeated (pp. 342-3) Pomcartfs185 remark.
M. Weilllto noted that х*-Ау*^№ has the solution x=Atf+P, y=2tu,
ill, «gives solution of P—Au*=N. Taking N = 1, consider a, ai, a2, •• -,
*« Johiw Hopkins Univeneity Circular, 1,1882,178,
'» Blfitter fur Bayer. GynmaeialecbJweeen, 18r 1882,19-24.
"' Comptee Rendus Рапе, Щ 1883, 568, 694, 832,1020. 1129, 1210, 1297, 1361, 1420, 1490,
1571, 1721.
*" Math. MagaaiDe, 1,1882-4, 40-1.
lfl А*юс. fran^j, av. BC.f 12,1883, 101; Atti Accad. Font. Nuovi Unca, 37,1883-4,
"■ Proc, London Math. 8oc.p 16,18834, 247-268.
m Jour, de math. «Зёпц 1884r 15 A885,171r on continued fractione).
ш Jour, fur Math., «, 1884, 335-7.
*» Nouv. Ann. Math., C), 4,1885,189-193.

386 History of the Theory of Numbers.
ak = 2aL1-l, obtained from а2-Ли2 = 1, ух^2аи1 а1^Аи2-\-а*=2аг-1%
♦ >>. He gave an explicit expression for ak and noted the connection
with the formula for cos m<j> in terms of sin ф and cos ф.
H. van Aubel1M proved the statement by Brocard177 that
give the relations between three consecutive sets of solutions of x*—f
where p, q give the least solutions. Also theorems giving p and q in terms
of the convergents found near the middle of the period of the continued
fraction for VA- If the period has an odd number of terms, A is a sum
of two relatively prime squares, but not conversely. He treated values
of Af b for which the solution x^by-\-lf у=2ЪЦА—Ь?) of x^-Ay1^! fe
integral. He noted cases when integral solutions can be derived from two
sets of fractional solutions.
Several197 solved the problem to find the polygons the number x(x—3)/2
of whose diagonals is a square, by treating Bv— lJ—8гг2= 1.
H. Ricbaud"8 found the least solution of я2-ЛУ=* -1 for ЛГ = 1549.
He noted corrections to Legendre's** table for N=823 and 809.
J. Vivanteiw treated Dx2—3=yi (cf. binary quadratic forms),
E. Lucas300 gave periods of the continued fraction for V», when n is a
quadratic function»
Several*01 solved x*-19y*=8h
J. Perott202 reviewed various classic papers on B-Du1= — 1 and proved
that, if q is a prime of the form 16n+9, F—2qu*= —1 is solvable if and only
if 2<«^>'4--l (modtf); while,if gisaprime I671+I, the condition 2<«-1>/<=l
(modg) is necessary, but not sufficient. If gisaprime8nr+5, ii—2qzu*=s— X
is always solvable; but, if q is a prime 8/1+1, a necessary condition is that,
in the decomposition ^=c2-h2d2, d be divisible by 8. This condition is
sufficient if q is of the form 16m+9, but not if g = 16wi-|-L
F. Tano203 proved that x*—Ay*= -1 is solvable if A=ala2- * -a^wheren
is odd and a1} - -, a* are distinct primes =1 (mod 4) and if at most one of
Legendre's symbols {aijaj) is + 1 for i<j. He gave theorems on the
case A = 2a^ ■ >ал.
J. Knur201 gave in detail the Indian30 cyclic method to solve г1—cs*^l,
claiming a simplification. This method is said to be much shorter than
that by continued fractions. He tabulated the least solutions for с ^152,
A. Hurwitz205 developed any real number xQ into a continued fraction
by use of Хъ = йо— 1/tfi, xt=a1—1/x^ - • •, where an is chosen so that x*—a»
lies between —1/2 and +1/2. Minnigerodelw had shown that the de-
1M Ahboc, franc. *v. ъс., Н, II, 1885,135-151,
1W Matheeie, 6, 1886, 162.
^Jour. de Math. E16m., C), 1,1887, 181-3, Cf> Whitford/ p» 97.
1M Zcitechr. Math. Phy»<r 32, 1887, 287-300.
JM Jour, de math, specifies, 1887, 1.
*n Math. Quest. Edue. Times, 48, 1888t 48.
'M Jour, fur Math<r 102, 1888, 185-223.
™ Jour, fur Math.. 105, 1889, 160-9.
*« Die Auflflsung der Gleichung *« - ttf = 1, 18. Jahrwiberich Oberrtalschule, 18в9,
>MActa Math., 12, I8S9, 367-405.

Equation, ааР+Ъх+с^ □ . 3S7
velopment is periodic if Xo is a root of a quadratic equation with integral
coefficients. The necessary and sufficient condition for the solvability
of*2--*V=-lis that
Vp = (a0; oil <**, — ■, «r, — fli, —a2, • ■ ■, —<ъ; <ii, a*, * - ■)■
G- Chrystal206 gave an exposition of x'—Cfy*=:fc tf convenient for English
readers.
F. Tano*17 proved by developing Va*±4 into a continued fraction that
x*—(а*+4)з/* = —1 is solvable in integers when a is any odd integer, while
-^-(a2—4)y2= —1 i« impossible except when a=3. There are infinitely
many integral solutions of x2—fo2—±a if a is any odd integer and к a sum
of two squares. To prove that there are infinitely many integral solutions of
where N is any integer, we add the two equations
x*-(а*+*)у*=a, x\-(a*
ifJVisodd; but, if N is even, we first change the second members to — a, 4.
By multiplying x2-aV-^=±a by u\—аЧ\-\-4г% = 1 for г = 1, 2, •-•,
in turn, we find that there is an infinitude of integral solutions of
2
G. Frattini208 noted that, if #©, уй is the fundamental solution of
7?— (a?-\-l)y*= — JV, viz., a solution with Q<y0^ VJV, then all its solutions
are given by
where n ranges over the values 0, 2, 4, ■••; while all solutions of
x*—(a2-hl)^2^-|-JVftre given by the same formula where n ranges over the
positive odd integers.
Frattini209 proved that, if К, Я (Н< ViV) form a solution of
x*-~ (a2—l)y2—N, every solution in positive integers is given by
m=Q, 1, 2, .-.
I Multiplying x*-Dy*=N by 02, we get
whose solutions are derived from one by the preceding formula, vis.,
'When N is changed to — N, the same formulas hold if we replace К by ±Kt
where, in the final formula, Нш i№(«+l)/2D. Tchebychef's131 first result
^ & corollary.
*» Algebra, 2,1889,450-60; ed.2,2, 1900,478-86.
";BtJl. dee 8c. Math,, B), 14,1,1890, 215^8.
ш Periodioo di Mat., 6,1891, 85^90.
/Ш, 169-180.

388 History of the Theory op Numbers. ГСнар.хп
Frattini*1* reduced the solution of x*—Dy*=N to the solution of one
of the equations x2 - Dy2=Np*} where p = 2m+1 - n > 0, m* being the largest
square <D=m2+n. Let x, у be a solution of the given equation such that
y^VN/p. Then x±(m+l)y. Let z=(m-\-l)y-h, whence h^O. Then
our equation becomes a quadratic for y; the radical in the root у must be
an integer k. Thus
P
The sign before ft must be plus. Hence if y& ^N/p, and if k, h give positive
integral solutions of s2 - Dy*=JV>, positive integral solutions of x5 - Ity*=N
are given by
Applying this result to the new equation, we conclude that, if h^ ^Ny posi-
positive integral solutions of the proposed equation are given by
h\ bf being positive integral solutions of z*^-Dy*=Np** The reciprocal of/
is w+1— лA><1. Thus we finally reach an equation x*—Dy2=Npx with
a solution у exceeding ^Np^/p, and hence a solution of the proposed equa-
equation.
Frattini21! шес1 similarly x*-Dy*=N(-n)\ X^l, 2, ■-, to solve
x*—ZV=N, and applied the two methods to x2 - Dy2 --N. He412 deduced
the theorem of Tchebychef.
Frattlni*13 supplemented and interpreted geometrically the theorem of
Tchebychef. From Frattini2* we derive the result: If 0, $i, q%, - - * are
values of #in successive positive integral solutions of x*-Dy2—l, the series
0, qi^lN, qz^N, - - • separate the positive integral solutions of x^—Dy^^N
m such a way that the number of solutions, in which у equals or exceeds
any number of that series and is less than the following, is constant. The
geometric interpretation is that the vectors of the successive solutions of
x*—$yz=\ divide the angle between the positive x-axis and the lino of
slope 1/ VD into consecutive angles each of which contains an equal number
of points with integral coordinates satisfymg x*—Dy2 *= N. Again, if 1,
Pn V*t ''' are the values of x, the series 0, VW(pi+l)/2D, jNfa+tylZD,
• - * separate the solutions of x?—Dy2=—N as before; for interpretation,
use the y-axis instead of the з-axis.
C. A. Roberts gave only the denominators of the continued fractions
for Vp, where p is a prime 4n+l ^10501 (thus giving what corresponds
only to the first line of each entry in the table by Degen,wl and not the least
solution of Xй—py2 = ±1)» The introduction to the table is by A. Martin*
»• Periodico di Mat., 7,1892,7-15. ~ "
i» Ibid., 49-54r 88^92r 119-22.
«*IbuL, 123-124, 172-7-
n»Atti Reale Aeead. Iinoei» Eendicontir E), 1,1892, Sem. lt 51-7; Sem. 2,
** M&th. Magaxme, 2,1892, 105-120.

Chap. XIIJ FELL EQUATION^ ОХ*4- Ъх~\-С= □■ 380
E. Lemoine21* proved that all positive solutions of x*-\-I = 2y* are given
Nii^-i+Nzniy^NinjVherbNm^ia+Nnbte the nth term of the series
= а, щ = b, ■ • •, Un = 21*^1+4^, во that
л. =2--
If ar, у is a solution of x2 Н-1 = 2^2, then x+2y, x+y is a solution of я2 —1 = 2#2
and the same holds if the equations aie interchanged.
G. B. Mathews,21* employing the fundamental solution (T, IT) of
t2—Dv?=vl, and tlie notation of hyperbolic functions, put
Then the general solution is T« = <r cosh пф, C/"M = {<rj Vl)) sinh пф.
К. Schwermg*17 started with Jacobi's elliptic function x=sin amu, the
function inverse to
u= \
and an "oddJ> integral complex number rj—a-\-bi, where a is odd and b
even, so that g=a*+b* is odd. Then
\-ajc хф{хА) q—1
If 17 is a complex prime of the form 4k+3-\-Dk'+2)i, then ф{х*), on which
depends the division of the lemniscate by ?;, is factorable into
Let g be a primitive root of the prime g, so that g*=i (mod 17). Taking
x=l, we get odd complex integral solutions /, и of t*~i^=^2i{ — lI*^
By squaring ^-|- V^1^ we Se^ complex integral solutions of Г2— ^Z72 = l.
H. Weber218 employed the modular equation (an algebraic equation in и
and v of degree 24 in each) which holds between the two elliptic functions
u=f((S), v=fB3u>), to deduce the identity X2M-Y2N=1,
Squaring Х*Ш+ Y Vtf, we getx+yvfD, where x2~Dy2^l, D=MN.
С E» BickmoreJ1* computed (for a committee of which A. Cayley was
chairman) a table, extending Degen's101 and showing, for 1001 ^a^ 1500,
the least solutions of yz—ax2—1 when a is not of the form J2-|-l (in the
contrary case, y=tfx=lt give a solution), and, when the latter is not solv-
*u Joraal de Sc. Mfttb. в Aiitr. (ed., Teixeira), II, 1892, 68-76, 115.
119 Theory of Numbers, 1892, 95»
m Jour, fflr Matb., 110,18S2, 63-4 A15,1893, 37-8)»
*« Math. Anmden, 43,1893,185-196.
111 Heport Britifih Аяяос» for 1893,1894,73-120; Csylcy'a CoU. Math. Papera, 13,1897, 430-
467» Errata by Cunningham.»*-w

390 History or the Theory of Numbers. [Chap, xii
able, the least solutions of y*=ea?-\-la From a solution of the former we
get the solution yi = 2y*-\-l, xY = 2xy of if[=ax\+\*
A. Huxwitz*20 proved that the positive relatively prime solutions of
v?—Dt?=m, where [m\<2jD, are given by the fractions u/v approxi-
approximating to jbt where ufv and r/з are said to form a pair of fractions approxi-
approximating to x if 2 lies between them and if su—rv= 1.
A. H. Bell221 found a special solution of x*=Nyz+l by setting
x^-1+Nymln,
whence у—2тп!(т?№—п2) and asked when the denominator is unity»
He treated the ease JV=94 and x*-Qly*= — L
Emma Bortolotti222 noted that a root of a quadratic equation with dis-
discriminant A and having as coefficients polynomials in x can be developed
into a periodic continued fraction whose elements are linear functions
of x if and only if Аи?— v* = l is solvable in polynomials w, v in x. If A is
of odd degree in x, the latter equation is evidently impossible.
A- Meyer223 noted that if &—Du*^ l has a fundamental solution 7\ U,
in which U is relatively prime to a divisor Dt of D, it has solutions in which
и is congruent modulo Dx to an arbitrarily given number.
G. Speckmann22* employed the identity
' ixy=m2,
for m = l, and called the resulting solutions of Pell's equation regular if
x = I and irregular if x2 is a divisor >1 of nW±2nm. To solved—Dy2—M
{M+square), he sought a square т? such that M+r? is a square $?; then
a solution is x = £+kyz, у =* ij, if D = 1 +2kJ-h^V*
O. Frattini225 discussed the solution of x2—Ay* — 1, where Л is a poly-
polynomial in uf especially when A is of degree 2 or 4.
Ch. de la Vallee Poussin32* indicated the advantage in using continued
fractions in which all but the first quotient are negative integers,
O. Speckmann527 noted that the fundamental solutions T, Uott2-+Du2~l
are T=x+2, 17=1, if Z>=s2 +4z+3; T=2x-\-Zt U=2, if D=x*+3x-\-2;
etc. He noted identities like
(na?+my - (п^а'Ч-'Зтп^Н-Зт^о3 = яг3.
A. Palmstrom228 gave many recursion formulas and relations between
sets of solutions of (а+2);с2— {a—2)i/2=4. If xlf yx are the least positive
*" Math. AnnfiJen, 44, 1894, 425-7»
** Amei. Matb, Monthly, 1, 1894, 5M, 92-4,169, 239-240.
**» Rendiconti Circolo Mat. Palermo, 9, 1895,136-149,
** Jour, fur Math., 114, 1895, 240.
**♦ Uebcr unbeat. Glcicbungenj Leipzig and Dresden, 1895-
** Gioraale di Mat., 33, 1895, 371^8; 34r 1896, 98-100.
5» Annalcs Soc, Sc. BmxeUes, 19, 1895, 111,
»7 Archiv Math. Phys,, {2), 13, 1895, 327-333; 1^ 1896, i4S-5.
»» Bergena Museums Aarhog for 1896r Бетвеп> 1897r No. 14,11 pp. (Frencb).

Сндр.хн] Pell Equation, ах2-\-Ъх-\-с=П. 391
solutions of x*—Ay*=— 1, then
so that y/yiy x/xi have the same properties as the above x, y, where now
a=4^J+2. If X\ is the least positive integer for which x2—Ayi=l, we see
that, by taking a—4x*—2r the solutions of odd rank have the same proper-
properties as the solutions of x*—Ay* =—1.
C. Stormer22* quoted the known result that, if a, b are the least positive
solutions of x*—Ayt= — 1, other solutions are given by
and solutions of x*~Ay2=-\-l are given by xZn-\-y2~'fA = (а-\-Ьт/АJя. Не
proved that
+02tan-itan^^tan*L
Stormer230 noted that if я2—Ву*=±\ (Z»0) bas positive integral solu-
solutions and уг is the least y, either there is no solution у such that every
prime divisor of у divides also D, or there is only one such solution, viz., y\.
A. Thue^1 proved that in x*—Dy*^m the least positive у is —flVm,
where v is a positive number for which u2—Dv?=l, provided D is not a
square and u>l.
A. Boutin232 tabulated the periods of continued fractions for Vn,
n ^ 200, and when n is one of 30 special quadratic functions of a parameter
£ef. Stern109]. He2** gave the complete solution of y*—(m2 — ljx2 = 1;, with
details when m=2.
EL Brocanl2" gave references to problems depending on x2—2y* = ±l>
E. de Jonquidres28* proved by the use of binary quadratic forms that
(а2-4)**~4у2=±1 is not solvable if a*3, that (а2-1)^-4^2 = ^1 is
not solvable [error for —l]f that (o+l)u*— ay*^l (a>0) has the least
solutions *^4а-Ы, у^4а+3, that (ma*±l)»B-»iy* = ±l has the least
solutions z=4ma2±l, 2/=47naJ±3a, and gave long expressions for solutions
of (mcPzktyx2—fny*^zhl (a and m odd). The method employed is similar
to that of Gauss (Disq. Arith., art. 195), but with the variation (inspired
by Legendre) that he omitted from the period of neighboring reduced
forms those having the middle term zero. He applied (pp. 1077-81, 1177)
Gauss' method of reduction to (ww»2+4)x2—my**=t. He gave (p. 1837}
values of D for which P—Du7=— 1 is solvable in integers: D=a2(n2-\-l),
D=4n2+4n+5, where a is a divisor of any term of odd rank in the recurr-
recurring series having 0, 1, 2n as initial terms and having 2n, 1 as the scale of
relation. It is not solvable if D=a2(n2+1), n a multiple of a.
m tfyt Tkkatrift for Math., 7, B, 1896, 49.
»»Vkfenekabe-SelabibetB Skrifter, Quntunia, 1897, No. % 48 pp. Cf Sto
ш Arehiv for Mftttu OS Nattnrvidenakab, 19, 1897, No. 4.
- MfaaU!, B)t 7t 1897,8-13.
B) 8 I8»8 1Я11в1
-Ibid., 113-3.
ш Comptee Re
Comptee Rendi» Рят, 120,1898, 863-8П, 991-7 (correction, 132, 1901,750, and I'ioter-

392 History or the Theory op Numbers. [Chap, xil
De Jonqui&res'23* noted that a solution of P~Du2 = — 1 or — m2 can be
found from two similar transformations of a quadratic form {A, B, C) into
(a, b, c) or ite inverse (— а,Ъ, —с).
R. W. IX Christie417 found the least solution of x2-103^ = 1.
G, Bicalde23* stated that if z^l, хъ з?«, - '} y=0, yt, Уь ♦ ■ ■ are the
integral solutions of x*—Ay2—1 (A not a square), 2(x2»-bl) is a square P
and уг* is a multiple of t; if 2(^«+i — 1) is a square for one value of n, it is
a square k2 for every nf and Угя-и is л multiple of Ary and one has the solutions
of v?-~Av*= — L A. Palmstrom (pp. 210-11) noted that the first statement
follows from
As to the second statement, PalmstrSm proved that (a^n+i^lj/fri^l) are
squares, whence 2(z2tI+,—1) is a square for every n if fbr one n. If «i, V\
are the least solutions of u*—At?2^ — 1,
so that 2(a:in+1—1) is a square. But the latter may be true when
is impossible.
A. Goulard2*9 proved that, if m is odd, 2(^—1) is a square if and only
if 2(x,—1) is a square. The latter is not a square if p is even, while, for
p odd, it is a square if and only if u2— A\?= —1 is solvable.
A. Cunningham and R. W. D. Christie210 each noted that X2-pY2=^l
becomes я2—py2— ^2 under the transformation X—я*±1, Y—xy. Then
if p is a prime, it is of the form 8те+3 or 8n —1 according as the upper or
lower sign holds. By choosing values of x, y, we get solutions of the
proposed equation.
C. de Polignac241 proved that if tu ux are the least positive solutions of
t*—Dui = l1 where D is positive and not a square, and tn, un any other
solutions, there exists a linear substitution Xi=(Q1x+jSi)/(P1a;+i?i) whose
nth power xn = {Qnx+S1%)i(Pnx-\-Rn) gives tn=Qn, ил=Рл}щ.
G. Ricalde*42 gave the identities solving хг—Ау* = 1:
as well as those due to Euler,65 He and others241 made minor remarks on
the linear relations between three successive solutions of x2—ay2 — ±1.
*» Comples Hendue Pane, 127,1898,596-601, 694-700. Slightly different from Gauss»11
"> Math. Queet. Educ. Times» 70,1899, 5L
»» UintennSdiaire dee math., 6, 1899, 75-
™ /Wd., 7, 1900, Ш.
«e Math. Quest. Educj. Times, 73, 1900,115-7.
•л JWrf.,75, 1901r67-S,
*" L^ntermediaire dea math., 8, 1901 r 256. The third identity lacked the flirt exponent 2.

Chap. XII] PELL EQUATION, OZ*-\-bx+C= D. 393
A. Boutin241 noted that, if A is a properly chosen quadratic function of m,
xi^Ay2 — ±1 are solved completely by an infinitude of polynomials in mf
which satisfy certain differential equations of order two» Thus for
у»-(«Ч1)*-1, у'-(т*+1)*>= -1,
the recurring series
for even indices solve the first equation, and for odd indices the second.
As functions of m, xn and ул satisfy the differential equations
Similar remarks are made for A = 25m2—Ыт-\-2 and for я2—Ay2— 1,
J. Romero noted that (л^±ж)я-(пУ±2«г+А)^=±1 if
A. S. Werebrusow noted that in я2—Ay*~±l, A may have the form
a2m?-\-2bm+c if Ь*-вЪ = ±1.
A. Holm5** employed the (n+l)th divisor Dn when Vc is converted
into a continued fraction the length of whose period is c. bet pc, q0 be the
fundamental solution of xl—Cy*=l. From one solution pn, qn of
x2 — Суг=( — 1)*DA we get all the solutions by use of
where, if с is even, m ranges over all integers, positive, negative or zero;
while, if с is odd, m ranges over only the even integers.
H. Weber147 treated /*—Z>u2=±4 from the standpoint of quadratic
numbers i(t-\-u^D)f where t and и are integers.
Necessary or sufficient conditions that я2 — Dy*= —1 be solvable have
been noted»34*
E. B. Escott asked and A. S. Werebrusow249 replied for what values of
a, h, > ■ • is [a, &, ■ **, Oft*, c, ■■-,&] integral (cf. Dirichlet's Zahlerxtheoric
p. 49).
P F. Teilhet250 stated and several proved that if 0 is a root of 72~3/3*= 1,
and /3#0, then 60*+! is not a square. Hence n(n+l)(n+2) =3A2 is
impossible.
P. von Schaewen2*1 made/= Ax*-\-Bx-\-C a square in the following cases
(их which D^B2--LiC): (i) A^n*Aj, 1)=т2А, AiH-D^ □ =^, since
•" L'iuterm^diaire dee math-. 9,1902, 60-62.
*■/«:, P. 182.
ш Ргос. Edinburgh Math. Soc., 21, 1902-3, 16&-180,
^АгсЫт Math, Pbye,, C), 4f 1903, 201; Algebra, I, 18fl5, 395-400; ed. 2, 1898, 438-443.
*• Lintermfidiaire dee math., 10, 1903r 102, 224; 11, 1904, 156-8, 242; 12, 1905, 53-6,
249-250; 13,1006,243-7 (Werebnwow'e result* ate егготеотм).
■«ЛИ., 10, 1903, Й8; 11,1904,154-6.
»■ Ibid., 11, 1904. 63-9, 182-4.
ш Zeitechr. Math. Naturw. Unterricht, 34,1903, 325-34. Frogr. Gym. GbgAU, 1906.

394 History or the Theory of Numbers*. [Chap, xn
is of Eider's form P*+QR. (ii) C+D=D. (Ш) -AD=\3. (iv)
-CD = H. (v) One of ЛA-Я), C(l-D), D(l-A), D(l-C) a square,
and the generalizations to b=w1Dll ЛA—Dl) = \J=q2 (etc.), aince then
R. W. D. Christie** noted that, if od-bc^=fcl, а*+Ъ*^Р,
is satisfied by
The problem is now to choose a, b, c, d to шаке у=П. Не and others
{p. 87) solved x*—149y2=l without using continued fractions. He and
E. B. Escott (p. 119) gave the identity
Christie** noted that if pjqn is a convergent to VD, where D is a prime
4m-\-lt then ?*«+i=g!!4-^+i Cc^ Euler,^ end].
O. Frattmi,264 employing a positive integer D and positive rational
numbers E, Fy defined the index of E-\-F-4d to be the maximum number
of such factors into which it can be decomposed. If one solution a, & of
я*—Dy* = l is known, all solutions of x2—-Dy1^N are given by
where the index of the particular solution x\ yf does not exceed half the
index of the solution of the Pell equation. But we may regard as known
the solutions whose indices do not exceed a given limit (depending only on
a finite number of trials),
Frattmi265 extended the preceding results to the algebraic case in which
D, N, x, у are polynomials in a parameter a. Finally, he proved that, if
D'is a positive integer or a polynomial of even degree in a, x2—Dy7=l is
solvable if and only if -4d is developable into a simple periodic continued
fraction such that
where the a's and с are integers if D is integral, otherwise polynomials in a.
A. Cunningham^ gave the least solutions of both 7*—IV = :tl, D<100,
from DegenV01 table, but checked by Legendre's88; also further (multiple)
solutions for D^20; also the least odd solutions of t*—Dv2=±2, ±8, ±16
for D<500, and D = ±4 for D<1000 (computed from data in Degen's
table). He noted three errors in the table by Bickmore.51*
Cunningham and Christie** showed how to find an infinitude of integers
*» Math. Quest, Educ. Tunee, B), 6,1904, 98-101*
*»Educ. Time», 67,1904, 41.
* Periodic* <ii Mat», 19, 1904,1-16.
*■ Ibid.. 57-73, Cf. FrattJni,'» H. E. Heine, Jour» fur Math., 48,1854, 256-8.
" Quadratic Partitions, 1904, 260-6.
»' Math. Quest. Educ. Timeo, B), 7,1905, 79-80.

Снар.хщ Pell Equation, ах*+Ъх-\-с= П. 395
Хп having the same Fin Xl-P+Y2^ -1. They and A. H. Bell** solved
x*—\§уг= —3 without using the usual convergent^
Cunningham^* used known solutions of y1—Dz?= — 1 to factor numbers
of the form gl+l-
A. Aubry*0 give a history and exposition of the Pell equation.
J. Schroder*1 noted that if PJQa (a -1, 2, - ) are the convergents to
1+
holds only for k=2. P. Epstein (p. 310) noted that this result for * = 2 is
a case of the known relation between the general solution of z*—Dy*=zbl
and its least solution. It is also a case of the following theorem. If
t>^a'+b, and Ъ is a divisor of 2a, while Zk/Nk are the convergent^ to VZ>,
then
Several writers262 discussed the р'в for which xT—(y*—I)/?2 = 1 is solvable.
A. H. Hohnes^ noted that 41 is the least prime у for which
7s*-111-if.
A. Holm2" noted that, if p,q give a particular solution of x2—Су2 = ±Д
and r, s one of x2—Cy5^ 1, all positive solutions of the former are given by
x-y^C~±{p~q<C)(r-&<C)nt n^O, ±1, ±2, --
R. W. D. Chrfetie266 noted that if we set ^;=cos в, #=sin Bt
which give the successive sets of solutions of Xl—PYl^l if Xx=xf Y
is the first set [cf. Wallis,*8 Euler**]» This was verified for any n.
Christie2" proved that, if pnj qH are any convergents of pl—2ql =
2 tan- ^ ^
£ Р * P*+l
ChristieMr noted that successive solutions of X2—pY2^! are given by
the initial solutions being 1, 0; x, y. From a solution of x2—601у*= —1,
one of JF-eOlF^l is found (pp. 545)
** Math. Quest. Educ. Times, B), 8, 1905, 28-30,58. — —
*' Ibid., 83; Мзд Math4 35, l9Q5-67 166-185. He noted (p, 183) eight errata in Degen'a"»
table and various erraU in Legendre'a»* tables of 1798 And 1830, including 4 =397
(cf. A. Geraidifk} I'intenneU dee math., 24,1917, 57-8).
thCM1905233
,(),l,
« Aicbiv Math. РЬун., C), 9r 1905, 206^-7.
m L*interme<iiaire dee math.r 13, 1906, 93, 22^-230; 14,1907,136.
"• Amer. Math. Monthly, 13,1906, 191 A48-9 for erroneous solution).
** Math. Quest. Educ. Times, B), 10» 1906, 29.
"•ЛИ., B), 9r 1906,111.
-Ш*52^
11,1907,39. Cf.p.96.

396 History of the Theory of Numbers* [Chap, xn
A. Auric1" developed into a continued fraction the root of any quadratic
equation of discriminant Д; it is a question of factoring f±2, where t, и
gLve the least eolution of **—Au*=4.
B. Niewenglowski** noted that x1—оу*=—1 is solvable if and only
if the least positive integral solutions of a?—a^=-hl are of the form
x=l+2u*, y=2w> The latter represents an hyperbola; if P and Pi are
points on it with integral coordinates, the line through P parallel to the
tangent at PY cuts the hyperbola in a new point with integral co6rdinates.
A* Cunningham™ gave tests for the divisibility of solutions of
by a prime.
The existence of a fundamental solution of Pell's equation is a corollary
to Dirichlet's theorem on the units in any algebraic field. For the case of
a quadratic field, reference may be made to J. SommerV71 text.
" E. A. Majol "ш gave eight values, 75,78, 321, ■ < ■, of Д for which there
is a common prime divisor 4m+3 of A and у in the fundamental solution of
A* Boutin57' gave the period of the continued fraction for V3 for many
forms of A, chiefly quadratic functions of a, and for various such ЛЪ
listed the least solutions of x%—Ay2=±l. He listed the values of N,
0<JV<1023, for which x*—Ny*= — 1 is solvable, a necessary and sufficient
condition for which is that there be an odd number of terms in the period
of incomplete quotients in the development of -JN-
*C. Stermer274 gave a simple proof of his*10 theorem and applied it to
solve the following problem: Given the primes рг • ■ •, p*y find all positive
integers N for which N(N+h) is divisible by no prime other than pi9 ■ ■ ■, pn
when k=l or 2. This is solved by the theorem that, if c=l or 4, all posi-
positive integral solutions x of x2—1 = ap? - ■ p? occur among the fundamental
solutions of the equations х*—0фг=1{1=1, ••, v)y where Du * -, D,
are all the values of aj>\1 * • -p'~ when tu > ■ >, en take independently the
values 1, 2v
G. Fontene*" proved that, if a, b give the least positive solutions of
x2—fcy*=l, all solutions are given by x+yVfc~(a-hbVfc)"; the proof is
essentially the classic proof, but follows the proof by Mile. J. Bony (ibid.,
13, 1907, 316).
A. Chatelet2* proved by an elementary formulation of the classic method
of solution by continued fractions that, if A? is not a square, s2— ky*=l is
always solvable.
~~~ Вий. Soc. Mat*, de France, 35, 1907^21^ ~"
»»/№., 126-131; Wiadomoeci Mat. Warsaw, 12,1908,1-26 (Polish).
™ Report; British Аййос. for 1907, 462-3. Cf. Cunjungbun.**1
*"Vori«mg«i tlber Zahlenthewie, 1907, 98-107, ИЗ, 33&Ч5, 356-8; Fr^ncb tranal. of
revieed text by A. L6vyp 1911,103-113,11», 351-7, 370-3.
™ LJintenn6di&ire dee math., 15,1908,142-3.
» Aeoc ftan9. av. *c., 37,1908,18-26,
»* Nyt Tidffikrift for M*t4 19, b, 1908,1-7; Fortechrittc der Math., 39,1908, 246.
» BuH math, elementairee, 14,1908^9» 209-212.
™Ibid.t 307-331.

Pell Equation, ах*+Ъх+с=П. 397
К. W. D. Christie277 expressed the solutions of z*-5i/2 = ib4 in terms of
fifth roots of unity. He and others278 obtained a double infinitude of frac-
fractional solutions of x*—руг=1 from one integral solution;
£. B. Escott and A. Cunningham27» factored u* in &
Christie and Cunningham** proved that, if pi—2д* =
Cunningham281 applied his270 method to factor vM in rl—2pl=l, and
gave further examples of the factorization of solutions of Pell equations.
Cunningham281 noted relations between the solutions of a*—3j/2 = — 2,
2*—3u>*=*l, in connection with the factorization of (ae+27U*)/(a*+3$2).
G. Frattini*3 proved that if D and N are polynomials in a, and D is
of even degree, and if z2—Dy*=l has a known solution in polynomials
in d, then all solutions of x^—Dy^^N can be found from one.
G. Fontene*28* poted that, if а, Ъ give the least positive solution of
A. Levy28* gave another proof of the result proved by Fontene.275
A. Gerardin** noted that, if g-Aii-l,
Each и^ is a composite integer. For/^—^й= —1,
G» Ascoli487 gave an elementary treatment of atf+y
F. Ferrari288 cited known results leading to a practical method to find
all integral solutions of x2—Dy*=±l in the solvable cases.
W. Kluge288 proved that for the integral solutions of
577 Math. Queet. Educ. Times, B), 13,1908, S5-6.
"""*■. B), Г
■••/Wd.r B), 16,1909,74-75.
тП ЛЙ., 95-6; B), 17,1910, 54-5.
'"tod., B), 17,1910,110-2.
MAtM dd IV cjongreeeo internal dei mat., 2,1909, 178-182. Cf.
«** BuU. math, dlemefltairee, 15, 1909-10, 66.
^Sphinx^edipe, 5,1910,17-29.
£ S«ppl. al Pteriodico dl Mat., 14,1910-И, 33-8.
g der VerBammlung deutecher PbiJologen and SchulmfinneTj Leip»cp 51,
1911, 135-7. Unterricbteblitter Matb. Naturwiss., Berlin, 19,1913, 9-U.

398 History of the Theory of Numbers. [Chap. ХП
2kxM+x*-iho\d. To apply to £—D%£=(—1)%
when D=fc*-|-1, make the substitution t*=kun+vn; then uny v» satisfy the
initial equation with p= — 1. Hence Un+i^^feu^-l-w*-!, /я+^Й^^в-ь
A* Cunningham discussed the values of D for which
(аЬ=ЫУ-(а=ьЬУ^О (mod B4DI),
where a, & are of the form 2Dn+l and 1№£|г'=<1&-
H. B. Mathieu*1 asked if (m*—lJs'H-i^i/2 has solutions not given by
E. Dubouis*" stated that there are no others in view of Fontene1,*1* the
exposition by Legendre being insufficient. All the solutions can be found2*9
by applying Gauss, Disq. Arith., art. 200.
R. Fueter*4 noted that Dirichlet1Oi gave sufficient, but not necessary,
conditions that 2* ~* my*= —4 be solvable for certain positive integers mnot
squares* Whenm^l (mod H)yx and у are even and the problem reduces to
z3—tny*= — l; a necessary, but not sufficient, condition that it be solvable is
that in the domain defined by V—m there be an even number of classes in
every genus.
A. Cunningham1'5 wrote r%t vl and rT, v* for the xth solutions of
i*—2J*=~l, 1^-2^=1, and noted that E. Lucas (Ch. XVII of Vol. I of
this History) proved that every prime p divides some t>,, where x=(p—l)/n
when p=8wrbl9 s=(p+l)/n when p=8wi3, and n=2m. It is here
proved that, if n=4m, &т, 16m or 32m, then p=8w-J-l— a'^
with b = 4p> d=28> and the number of factors 2 of n is given. If
either р^8й>=ь1=3«/+1, Р^^+бЯ1, Ог p=8a±3=3w'-l, р
St. Bohnicek^* proved that if т is a semiprimary prime in the domain
R defined by a fourth root of unity and if the norm of t is =1 (mod 8),
£*—xi|f=2> ^—iriji ^ 1 have the solutions
so that £, ij are odd, d and щ integral numbers in R. Here T=USU,
Ti^TlSu+i, where jSr is the lemnbcate function defined (p. 680) in terms of
Jacobi's theta fimctione. But £*—xi^=i or 2i is not solvable in integral
numbers with £, q odd in the second case* If т is semiprimary^ £*—ti^=4,
Й—nif^^haveoddsohitions^ijinijonryif r=l (modX1),»^! (mod Xs),
where X = l-(-i. ТЪеге are similar theorems for £*—xi^=l, 2, i or 2^,
when the norm of t is not ^ 1 (mod 8). Application is made (pp. 719-725)
to x*—py1— ±1, —2, dzl, where p is a rational odd prime, use being made
of eyclotomic functions.
Б. Е* Whitford4 gave an extended history of Pell's equation and (pp.
98-112) extended the tables of Degen1" and Bickmore*11 by listing for
^I/interm6diaire dee math., 18,1911,166-7.
ш Ibid., 22Q.
"i/N£,19,1912,47.
M L'mtermediaife d^ matb, 19,1912,72.
>n Jftlneiber. d. Deutecben Mmth.-Verebiganc, 20,1911, 454».
** ВгШвЬ Амос. Report for 1913,412-3.
"• ЭДппфЬег. AkAd. Wise. Wien. (MaUl), 121, Ш, 1912, 701-7.

Pell Equation, лг*Н-!кс+с=П. 399
1500<A^1700 the least solutions of x2-Ay2^-lt when solvable, and
always those of x2—Ay2^ +1. He noted (pp. 154-5) that the former is
solvable for 38 of the 110 composite numbers А=аг+Ъ* between 1501 and
2000. Finally (pp. 162-190) he tabulated for 1500<A^2012 the period
and auxiliary numbers for the continued fraction for Va [corresponding to
the first two lines in Degen's table].
R. Remakm modified Dedekind's141 proof of the existence of solutions
1=l and obtained upper limits on the least positive solutions:
Known methods of solving y5—2^^= —1 have been recalled.2*
Th. Got2" simplified the proofs by A. Meyer1".
M. Simon800 noted that Brahmegupta's first rule shows that he knew
how to Bolve all equations (a) 4(X2=F2)x2-hl=j/2 and (b) (X*±2)«4-*ey*.
The identity (X2=fc2)X'= (A2±l)*-1 gives x=\, y=\*±l for (b) and s=X/2,
y=-X2:Fl for (a). But if X is odd, and a solution a, 0 of (a) is found, it
becomes @*—1)я*/а*-|-1~y2y which is satisfied if x—2otpf whence the solu-
solution is x=\(\2^l), y=2(X2=FlJ-l.
G. Metrod™ noted that in u2-2f = lt v+2% «>lt and v±Ba)\ In
v?—3t? — l, v=2* only for t=0, 2; v is not a power of an odd prime, and
рфBа)*, where a is an odd prime* In u2—pv*=1, where p is an odd prime,
cases are noted in which f—2' or a*} where a is an odd prime ^p.
E. E. Whitford*2 extended Cayley's13^ table from D=1000 to D-1997,
but gave the solution of both x2—Dy*= —4 and x2—Dy*^ -J-4 when they
are solvable. He noted the application to finding the fundamental unit t
(least unit >1) of the domain defined by VZ>; the least positive solutions
of x7—Dy2—1 do not determine « when one of the equations x2—Dy*=~ 1,
4 or —4 is solvable.
0. Perron*3 obtains by use of continued fractions tbe limits
for the least positive solutions of x*—Dyz^ 1. Kemak2*7 had given larger
limits. Cf. Schmits,3* Schur."*
T. Ono804 stated that, if x2-5j/2=4,
-- •>-*-* -*-*■■■■
Infinite series involving successive solutions of this and x*~Dy*=p*
have been treated.306
"V. G- Tariste"** noted relations between successive s'a or y's for
m Jour, for Math, 143,1913,260-4, Cf. Knmecker,'" Pferron.**1
SB* L'mtermediaire da matbv 20,1913,254-6,
m AonaJce Fac. 8e. Toulouee, (S), 5,1913, 94-8.
»• АгеЫу Math. Fhy*., C), 20,1913, 280-1.
"*8pbmxrOedipe,8, 1913, 137^L
«я Annale of Matt, 15,1913-4,157-160.
"• Jour, fur Math., 144, 1914,71-73.
"■ LiQtfimeduire dee math., 20,1913,224.
ш Ibid., 21,1914, 37-38, 47-48; 22,1915, 21-23,277-8.
** Ibid., 22,1915,125-6L

400 History or твой Theory of Numbers. [Сна*, xii
A. S. Werebrusow** stated erroneous conditions involving JV=a*-|-&J
for the solvability of x2 — Ny* = — 1.
Thekla Schmitz*8 proved that, for the least positive solutions of
z1—Dyz^\t х-\-у^О<2е^г where eis the base of natural logarithms.
A. Cunningham** described and noted errata in various tables on the
Pell equation: Euler,71 Legendre,*8 Degen,101 Cayley,117 and Bickmore.*1'
Eveliovitchi110 gave an elementary method of solving (ух2-{-1=у*л
We may take x=2u, y=5u—v, 2v=w. Then s=5w-J-2r, y = \2w-\-&rt
r*=6tp4-I. Hence if {xt=Q, y^l), ■■*, (xif yt), --are the solutions
arranged in order of increasing magnitude, £tfi=5e<+2pij y%+i = 1
The same method is said to apply to лг*+1=у* if а=4Л-|-2, 4a-hl^
A. G^rardin111 applied the remark of Hart17* on Ay*-1 = П,А =г
To treat similarly xl— Ay2=2, set A~a*-2b*, ±y=a*—2/32, and solve the
system of equations
Thus, if А=Ш~12?-23г, we get p=7, <*=59, y=2S$$y which leads to
Legendre's solution of x2—151^=1. For s2—Ay2=— 4, set А=а*+Ь*,
y~zf+P and solve the system
Thus, if A=3*-|-10l, we get the least solution *=3, z^4* An error for
д=397 in Legendre's88 table is noted. He announced an extension in MS.
to 3000 of the table by Whitford,**
M* Cftssin1" gave relations between successive solutions of x2=3y*+l.
Several111 gave relations between successive solutions of г1—Ds*=±l
or c, and of их2—vy***w.
•J. Schui414 obtained closer limits than had Remak^ Perron,101 and
Scbmitz.**
On ^-З^^!, see papers 100 of Ch. Ij 12, 24, 29, 33, 51 of Ch. V; 94
of Ch.VH; 230 of Ch.XXI. On 2*4=1-П, see papers 112-129 of Ch.
IV; 92 of Ch. XXIII. For аа?+Ъу*=с or ох*+Ъху+суг=К see Ch. XIIL
On 5я^±4= D, see Wasteels72 of Vol. I, p. 405. On the application to fac-
factoring, see Vol. I, p. 368. For "Fell equations of higher order/' see papers
313-23 of Ch. XXI, 1&-25 of Ch. XXIII, and Ch. XXVI. Pell equations
occur incidentally in the following papers: 56, 70,107, 152, 178, 185, li^J,
189, J96, 202, 204, 210, 219, 223, 227 of Ch. I; 135 of Ch. IV; 41, 109 of
Ch. V; 138,193 of Ch. VI; 66 of Ch. XV; 55 of Ch. XVI; 21 of Ch. XVII;
27(M of Ch. XXI; Ш, 250 of Ch. XXII; 95,99, 163 of Ch. ХХП1.
*" L'mterm£diaire dee math., 22,1915, 202-3; 23,1919, 56 for admuaum of error*.
«" Archiv Math. РЬул, C), 24, 1016, 87-9» Cf. Perron^
"» Мею. Math., 46, 1916, 4^69.
"•SocMath. de F^ance^ Comptee Rendua Stances, 1916, ЖЫ.
ш Sphinx-Oodipe, 12, June 16, 1917, 1-3; TcnacignemeDt math», 19, 1917, 316-8; Tinter-
u^diaire dee math., 24,1917V 67-6S.
ш L'intermMiAire dee math., 25,1918, 28, 93.
ш Ibid., 83-87; 26,1919, 51-54.
'" Oftttingen Nachrichten, 1918, 30-6

CHAPTER XI1L
FURTHER SINGLE EQUATIONS OF THE SECOND DEGREE.
Equation ъшеав in one unknown.
Brahmegupta1 (bom 598 A,D.) solved axy=bx+cy+d. Let e be an
arbitrary aumber and set q = (ad+bc)le. To the greatest and least of e, q
add the least and greatest of b, c, and divide the sums by a. We get the
values of z, у (that of x on adding to с and vice versa). Thus, if
zy=3x+4y+90, take e=17, whence q=6y 0 = 17+3, z^6+4. Another
method is to give a special value to one of the unknowns.
Bhaecara' (bom 1114) gave a like rule for a=l, but added в and q
to (or subtracted them from) Ь and с in either order, and gave both geom-
geometric and algebraic proofs of the rule. Thus for xy=4z+3y+2> take
e^ 1, whence q** 14; adding 4, 3 to 1,14 in both orders, we get 17,5 and 4,
18 assets of values of xty; taking e=2, we get 5, II and 10, 6. The same
example was treated in § 209, p. 269, by assigning any value as 5 to у
and deducing я ^17.
On axy+bx+cy+d=0 see WereP*, and papers 121-141 (on optic for-
formula) of Ch. XXIII; also, Bervt* of Vol. I, p. 451; and *P. von Schaewen>
L. Euler2 noted that 4ят—т—п is never a square since
is impossible; also iprrm—m—n is not a square if m is of the form 4»*$~n.
Euler4 proved that no number of the form 4ш&—т—п or 8wm—3wi—3n
can be a 'square, and many such propositions.
Euler* stated without proof that 4шт—т—n= D is impossible. This
arises from the fact that the divisors of mx^+y* are of the form 4?nz+ly so
that d=4mz—l is not a divisor, whence йпфт-hj/3. He* treated similarly
the case *n=I, and proved that 4*nn—m—п"фП.
P. Bedce7 erred in his proof that 4mn—m—1ф П.
Several* proved that 4mn—m—n is never a square or triangular number.
S. Gunthet*solved у*—<и?=*Ъг by nse of the continued fraction
K ^
^7 (^)J |$ 61-64, AJgebra, with arith. and mensura-
mensuration, from the Sanscrit of Bnltm^ppta and Bh&rara, tranal, by Colebrooke, 1S17,
pp. 881-2.
Vfj*^£u, H21*4i ОЫжо***pp.270-2,
Bettedirift fur d. B«tedrfw4№i, 38,1913,141-6»
О M»th, Phy^, (ed^ Fuse}, 1.1SI3,191, 202 A80, 259, 200); lettera to Goldbaeh,
19 ri 1^1743
* Сошии And. Petn^u, 14,1744^ 151- Corom. Arith,, I, 48-49; Op. Om., A), II, 220.
' Opem partcm^ 1, Щ 220 (ftboat 1778).
« Oonwp. Math. Fh^., («A, Fiien), lp 1843,114-7; letter to Goldbtch, Маг, б, 1742.
i N п. Hath,, 11, 1S5Z, 378 (Bufer'i correct proof, p. 279).
' Jour. <fc Mt^ <S), «, 1871,3S1-340.
401

402 History of the Theory of Nukbebs. £Chaf. xin
Let Qi be the denominator of its ith convergent. Then
Hence a Bobxtioni&y=Q»tx^Qn^llhz=Q*+,the last being used to determine
и and n:
If b is odd, set &^Bp~l)b; then (J) is divisible by Ъ if p<h, and we may
take 2n=k—1. If 6 is even, divide я and # by a power of 2.
P. Mansion10 gave a short proof of the preceding Qi—<*Qi_i =Q2*.
S. Realis11 noted that, if a, 0, у is one solution of оаЧ-ЬхуН-сз^Аг, *
second is given by z = (h+a—c)a+(&+2cH, y-BaH-b)a-h№—a
since
If, forc=l, we solve the initial equation for y, the radical will be a rational
number и if иъ—Dx2=*±hzy D^b*-4a, which was treated (ibid., p. Ill)
and if D>0 by Gunther.*
A. H. Holmes17 proved that 96x—96i/-h21^n is impossible in integers»
On axI-hbx-hc=Kt/ see Desmarest**
Solution of x?~y*=g>
Diophantue, II, П, took 0-60, x=#-\-Z, 3 being a number ^^60,
wheace у-17/2.
Leonardo Pieano11 took a square a?<g and set (х+аУ=х*-\-д, which
determines x. He gave a second method- Let g be odd, 0=2n+l. Since
l_(-3-(-...-(_Bn-l)^n2, we may take y=n, whence n*-\-g=(n-\-l)\ He
treated separately the cases д=*2к> д=4к.
R. Descartes14 noted that 6*-32 = S*, ll^-lO*^^3; {ax¥-x*~x* if
J. L. Lagrange1* concluded from his general theory of binary quadratic
forms / that every integer is of the form у1—г*. This1* is not true of the
double of an odd prime, and Lagrange'a argument is conclusive only when
the discriminant of/ is not a square.
S. Canterzani17 treated s2 -\-A ^ D, by deciding whether or not Л is a
sum of differences of consecutive squares. First, let А Ы even. The sum
of 2/ consecutive differences 2h+l, 2Л+3, - • • ю 4/ft-H/1 and hence +Л
"Jour, de Mfttk, C), 2,1876,341. ~ ""
u Nouv. Corresp- Math., 6,188Or 348^60.
« Amer. Math. Monthly, 18, Iflll» 70.
" 1л F^actica Geometric, 1220. Seritti di L. tfaano, Поле, 2, 1862, 216-8,
i<Oeuvns,X,302,poetbmc>wMS. Cf. рсфегв 23-26 of Ch. XX.
w Nouv. тбш. AcAd. Sc. Bertm, *ni^e 1773; Oeuvre, Ш, 714.
Ji L'intennediaire dee math., 18, 1911, 33.
" Memorie ddt'Ietituto Naiionftle Italxauo, Oaaee di Fb. e Mat., Bolo^oa, 2, II, 1810,445-76,

Solution of x?~>y*=g. 403
if A is not a multiple of 4. For Л*=4В, the sum equab A if h>=Blf—f;
iben**-hA = (A-h2.0* for x=h. Next, let A^2B+1. The sum of 2/+1
consecutive differences 2A+1, ---is B/+l)Bfc+2/+l), which can be
made equal to A by choice of h, whence z2—A^-k* if
B+f+l
*- 2f+l +/>
T. Clowes" noted that the difference of tbe squares of x+1 and x—1
equab the difference of the squares of a+o and а—Ъ if x—ab.
L. Poiasot" stated that any integer N, not the double of an odd integer,
can be represented ae a difference of two squares and in as many ways n
as N can be expressed аз a product of two factors both odd and relatively
prime or both even and with no common factor > 2. If N has k distinct
prime factors, rt"=2M.
P. Volpkeflie took g=2rh*r< }?k> where the Л'з are distinct primes.
As known, tbe number of decompositions of g into two factors is
or у-И accor<ubgas at least one of the exponents /i, a, • ■ •, т Is odd or all
are even. Hence, in the respective cases, the number of decompositions
into two distinct even factors, i. e., the number of solutions of x-—y1 = g7 is
or ri—i, if м>0. For д=0, the number of solutions is v or v—£, respec-
respectively.
R. P. L. Oaude21 noted that any odd integer +1 is a difference of
two squares since db is the difference of the squares of (a±6)/2, while the
double of an odd integer is not* Every integer which is a difference of two
squares is such as many times ae there are different combinations 2,3, • ,n
at a time of its n prime factors.
G. C» Gerono" stated only known results.
L. Lorens11 concluded from
that the number of solutions of mz—n*=N is double the number of divi-
divisors of N or JV/4 according as N is odd or is divisible by 4; none if Nj2
is odd,
G. EL Hopkins14 noted that in я*—yi=Ba1- "О*)*, where alT — -, an
«e primes, z or у has Ce—1)/2 integral values.
«Tbe Ledierf aod Gentlemen's Di^ry (ed^ M, Naeb), New York, 3P1822, 53-4.
11 Comptea ВакЫ Pbria, 28, iS4Dv 582.
"Alii Axxad. Pcmt. Nnovi lino^i^ 6t 1&52-3, 01-103; АшшД di Sc. Mat. e Fis., 6, 1855t
120-8; ОищХев RenduB Faiie, 40,1855,1160; Nouv. Ana. Math p 14,1S55T ЗИ.
я Nouv. Алл. M*Ul, B), 2,1863, 88-90.
tt/UAT 90412,
M Tiebskrift for ОДЬ-, 0), lp 1S71,113-4.
11 MatL QoeeL БЬе. Hrnee, l6p 1872, 40^7.

404 History of the Thkoby of №швввв.
A. Sykora*5 repeated Claude's11 firet remark,
L. P. da Motta Pegado,» A. Z. Candido,* T. H. Miller,21 G. Biaconcim,*
and H. E. Hansen* stated known results.
"H. Rifoctitlee "M noted that every integer N is the quotient of two
differences of two squares. For, N=2{a*~Ъ*) or a2—6s according as
iV= 2 (mod 4) or not. Then apply formula A1) of Euler,wCh. XII, for e^l.
W. Sierpinski31 proved that the number r(n) of distinct representations
of a positive integer л as a difference of two squares is twice the difference
between the number of even and odd divisors of n. Also
where [t] is the greatest integer Ш. If 6(n) is tbe number of divisors of n,
*(*)-ai;*Cb)-2Se
S. Guael1* proved that
*A.L. ^i
For sohitions of x*-l=g, see Stormer574 of Ch. Х1Г. Cf. Gffl.M
SoltjtioN OF аз?+Ъху-\-су*=<1г\
DiophantuB, IV, 10, desired two cubes the ratio of whose sum to the
sum of their sides is a square. Taking г and 2—s as the sides, we must
have 4-6в+3^= П, say B-4e)\ whence «=10/13.
Diophantue,IV, 11,12>eorvedx*=b0*=s=b^ Takex=rz,y=sz. Then
(т*±&I(г±8) is to be a square. For the upper signs he found (as in IV,
10) that r**5, a~8, s = l/7. For the lower signs, take r»H-l* so that
3^_l_3e_l_l= П, say (l-2sJ, whence s~7> »»1/13.
In these three problems, Diophantus made no use of the fact that
(&±y*)l(x±y) «a:1=Fxjr4-^- But, in V, 7, he made хЧ-s-H the square of
x-2 for s=3/5, whence 3i+3-5+51= П.
C. G. Bachet in his comments solved similarly f=vt or Zp*, where
/=x*±xjr+^ Farmat (Oeuvres, III, 249) remarked that we can solve
/=a, where a is the product of a square by one or more primes of the
form 3n-bl or 3,
L, Euler11 proved that if fx?+gxy+hy*=tz* is solvable for t=kt it is
solvable for t=kl, where l=p*+gpq+fhg*. We have only to multiply the
*ArehivMAtb. Phye., 61,1877,446-7. "~ "^
» Jornbl de So. MAtb. e Avt.v 1,1878,15CHS, 171-2.
«Proc Edinbnr^h M&tlL 8oc.p 9,1800-1, 23-5.
» Periodico di Mat., 23,1908, 21.
*> L'eiiBttgnement math.» 18, Ifll6p 48^65.
"Lintermediairedtf math., H, 1«И, 26-6. Proof, S, 1901,23^40, by continued fn^tioti*.
n Wibdomoeci Matematyctne, Wareaw, 11,1907, Supply 89-110.
■ibid, 111-Q.
*- WiAtmdig Tijdflchrift, 13,1916-7, 207-9.
" Ope» pcrtuma, 1,1802, 209-211 (ftbotrt 1771).

. XIII] Soltjtion op ах*+Ъху+су*=с№. 405
given equation by I and note that the product of /х*+дху+Ну2 by I is of
that same form.
С Gill34 solved х*-у1=Ъс by setting х+у=Ъ cot A{2. Next,
is satisfied by
2-bx=y cot A/2, г—х = (ах+Ъу) tan A/2,
Eliminate 2. The resulting equation gives x/y, whence
y=t (sin A+a sin» A/2), x=t (cos' A/2-6 sin2 A/2).
Take *=m*+n*, sin A = 2mn/k Then
G. L. Dirichlet" proved that Аг1Ч-2В^1/Ч-Сг1/2=х* is solvable in integers,
with x prime to 2Z), if the left member is a form of determinant D of the
principal genus*
J. Neuberg8* noted that x*—xy-t-yt=z* holds if
T. Pepin" gave special methods to obtain a particular solution of
ах*+2Ъху+cy*=z\ Given one solution x=ay y=p, z=yy to find all,
eliminate D=b3—ac between
and write pfq for the irreducible fraction equal to (fi2—
Hence
Conversely, these imply the initial quadratic equation. Hence fix,
equal quadratic functions of py g. It is shown that p is a factor of
A. Desboves*8 noted that by specializing his1** formulas we find that the
complete solution in integers of X2+bY*+dXY= Z* is
where (as below) di is to be inserted before the second members. For the
case d=0, the ordinary method is to factor Z2—X* and get
X=ctf~№, Y=2pqt Z=ctf+№ (Ъ=а0).
For each pair of factors a, p of Ъ, the latter equations give all the solutions.
It is inexact to say with A. M. Legeudre*9 and others that the general
solution includes as many particular formulas as there are ways to decom-
decompose b into two relatively prime factors. The complete solution in integers
of X2-|-F2=cZa for c=m?+n2 (the only solvable case in view of a theorem
M Application of tbe angular analyme to the eolation of indeiei. ргоЫетв of the Beoond
degree, New York, 1848,15-17.
" Zahlentheorie, f 155> § 158,1863; ed. 2, 1871; *L 3,187B; ed. 4,1894»
" Notzv. Cont«p. Math., I, 1874-6, 197-8. Cf. papers П2&, 124,125 of Ch. V, and 72 of
Ch. IV. Cf. 3. Bertnuid, Traite eMm. d'algebre, 1851, 222-4.
^Atti Acc*d. Pont» Nuovi lined, 32, 1878-9, S&-97.
" Nouv» Ann. Matb., B), 18,1879r 260; proofe, C), 5,1886, 226-33.
" Theorie dee пошЬгев, ed. 2,1008, 29.

406 History of the Theory or Numbers. [Слар. хш
of Legendre) is
X = (cq*-p2)m, Y=p*n-2tpq+cnq\ Z=-f-2n<pq+c<f,
obtained from x=m,y = ntz = l. The complete mtegral solution of
aX*+bYi+dXY=cZi (c=
is found iToinx=y—z = l to be
By changing the notation of the parameters, this becomes
J. Neuberg and G. B. Mathews40 proved that the general rational
solution of x*+xy+y2=z* is x=pi—q*f y=2pq-\-q2, z^p*+pq+tf. A.
Cunningham41 deduced %x+y=P-3u*, \x^2tu from {\x+yy+Z(\xY=zx.
Ch. J. de la Vallee Poussm43 proved that a necessary and sufficient
condition for integral solutions of axt+Vbxy+qf^mz*, where m is prime
to 2(hz-ac), and the g. c- d. of a, 2bf с is unity, is that m be representable
by a form of determinant Ъ*—ас and of the same genus ag ах2+2Ъху+су*.
Б. Sos43 found the complete solution of
х*+Ъху+у* =z* or y(bx+y)=z*—x*
by Betting y=\(z—x),\(bx+y)=z+x. EUminating y, we get
where p/<^ is a fraction in its lowest terms. Hence
The same method applies to азР+Ъху+су2 =z\ a or о a square.
A. Gerardin44 found a general solution of аХ*+ЪХУ+с¥2=}&*, given
one solut on a, 0, y, by setting X=a+mx, У=^4-"Ч/, Z=y. Then m is
determined rationally and
Gerardin45 granted that ah?+bh-l-c=m*t replaced h by h-\-x, m by
m+fx, found x rationally, and hence obtained a solution of ау*+Ъуг-1-аР~ tJ*:
A. Aubry48 solved 2d?xz^2dx+l—<P=y* for d and made the radical
rational by means of a Pell equation. L. Valroff47 made the substitution
R±S JX_
X~ 2Y ' V~S>
« Math.Quest. Educ. Timee,46r 1887,97» See рарегоЗб, 171. О. рлреге 68,69of CbЛV.
«Ли., 75,1901, 33^4.
« Mem. сошхмшея et autnea шёш. acad, Belgique, 53, 1895-6, No. 5, 43-64.
« Zcit&chrift Mail]. Naturw. Untexricht, 37, 1906,186-190.
« BuU. Soc. Philomathique, A0), 3, 1911, 218.
«Sphinx-Oedipe, 1907-8,177-9.
« L'intermedi&ire dee math., 20, 1913,114.
*rSphini-Oedipe, 7,1912, 74-6.

Chap. XIII] SOLUTION OF 0^+^ = 0. 407
and noted that the resulting equation in d has real roots if
which is a consequence of 2X2+2Y2=R2+SZ.
rON of ахг-\-Ъу*=с.
L. Euler48 noted that
4» noted that, if m2=afm2+l, then ax*-byz=af*-~ Ъд* for
C. F. Kausler5* treated the solution of m'x*+nry1=N, where N=±A-\-l,
l, y=2Y, whence
Let B>4m-H and set B = {4m+1)D+E. Then
xcx+i)_ (
W 2 '
Since Bn-t-l)t—E'~ Dwi-|-l)z has the solutions
the question is whether t— О = Y*> If so, we test {10 by the table of pronic
numbers X(X+1) ш Nova Act-a, XIV, 253. A similar treatment is given
for the case mf = im—1, n/ = 4n4-l-
C. F. Gauss61 solved тх2Н-»у2=Л Ьу the method of exclusions.
F. Arndt52 noted that, if/, h are given relatively prime integers, the least
solutions of fp^—kqz^^k, k = l or 2, can be found, without using continued
fractions, by means of the least solutions of x?— /%2— 1, given in Table X
of Legendre's Theorie des nombres (errata noted, p. 246). We have only
to take x = 4^1 -\-2fp2/ht y = 2pq/k. He gave a table of the least roots of
S. ReaHs5* solved («H-4)sl—ny2—4 by formulas simpler than those
given by the usual method of employing a Pell equation. If a, 0 give а
solution, then
give a second solution. We thus get an infinitude of sets of solutions
A, 1), A+я, ЗЧ-и), - - ■, which are said to give all. Replacing x by 2u+l,
У by 2»+l, we get (nH-4)(t**+u) = nft^+p). Hence the above work solves
the problem to find an infinitude of pairs of triangular numbers whose
ratio is n : n+4.
«Opera poetuma, 1, 1862, 490 (about 1769).
" Ibid., 215 (about 1774)»
и Nova Acta Acad. Petrop., 15, ad annoe 1799-1802, 164-9.
* DjequifflOonce Arith., art. 323; Werke, 1,1863, 391; German traoal. by Maser, 377-38$.
■Arehiv Math, РЬув.р 12, 1849, 211-276,
u Kwxy. Ann. Math., C), 2,1883, 535^542.

408 Histobt of the Thboet of Numbebs. [Chat, xni
D. Hubert54 remarked that the proof that a proposed diophantine
equation is not eolvable ш rational numbers is often made by showing that
the corresponding congruence with respect to a prime or prime power
modulus is impossible. For the case of a quadratic equation in two variables
it follows conversely that the possibility of solving the congruence for
every prime power modulus implies the possibility of solving the equation.
For, the known cr tenon for the solvability of a ternary quadratic dio-
phantine equation leads to the result: If mf n are any integers, the equation
2^ 1 is solvable for rational numbers x, yt if the congruence
*^! (mod %/) is solvable in integer* x, у for every prime p and
positive integer e. There is no immediate extension to higher equations,
since
is irreducible and has no rational solution, while the corresponding con*
gruence modnlo p* is solvable whatever be the prime p and positive integer
e. Again, *М-13^4~81 ^ ^ irreducible function which becomes reducible
modulo p* for every prime p and integer e.
Several writers" found all solutions of х(х+1I2=у(у+1)/3 by means
of 2u'-3z*=-l.
On Mx*—Ny*=zkl or 4, see Legendre,*8 Jacobi,11* Weber,m Palm-
strom,328 and de Jonquieres2** of Ch. XII.
On zI+gy2=m see Cornacchia4 of Ch. XXIII.
On ax1-^cy2=n> see Euler51 and Nasimoff.48
Solution of ax*-t-hxy+qf*=k.
L. Euler55 noted that the problem to find the тшттцтп of Ax*+2Bxy
-t-Cy* for integral values +0 of x, у presents no difficulty if B1—AC^O
and hence is here treated for Bt—AC positive and not a square. Then
the proposed form may be reduced to maP—ny*, where m and n are positive
integers whose ratio is not a square. If m=l, it can be given the value
unity by Pell's theorem. If n—1, it can be given the vatue —1.
If mx2—ny*=k for x=a, y = b} it has an infinitude of solutions. For,
if p*—mnq*=l (in an infinitude of ways, since mn+П), then
mx2—ny*= (ям*2
This holds if
so that we get xt у as rational functions of а, Ь, р, q*
The problem to make mx2—ny* a minimum corresponds to finding the
rational fraction x/y giving the closest approximation to -Jnfm. Develop
the latter into a periodic continued fraction and take the convergent ob-
obtained by continuing to the largest quotient. Thus, for 7s1—13$/*, thecon-
u Gtttmgen Nachrichten (Math,), 1807, 52-44.
« L'mtenwediaire dtf m&th.t 22, 1Й15, 239, 255-260.
M Novi Comm. Acad. РЫгор., 18,1773, 218; Comm. Ahth. I, 570; Opera Опшп, <1), Ш,
310. Od the inoompletene» of Euler'o method^ see Smith1" of Ch. XSL

SoiajnoN о* ах*+Ъху+су*=к* 409
tinued fraction for V9I/7 has the quotients 1, i, 1, 3, 9, 3,1, 2, 2 (with the
period marked). Since
хт2+1+3 IV
£ = 15, y —11, give the ттттпцт 2 of 7x*—13y*.
Given (p. 577) the solution х=с^у—Ь<Л
/=Аз*-2Вху+Су*=с (h=B*-AC>Q, *+□),
to find an infinitude of solutions, use the solution of
corresponding to the Pell problem p— Bq+^k<?+\. Now AJ has the
factors Ax—By±y^i, and ф the factors p—Bq±q^k. Hence we use
A) Ax—By+y*fk=(Aa-Bb±b'Rc){p~ B9±qilk)*9
for any of the four combinations of signs. To find the minimum of / for
integral xf y, develop the root (В±^Х)/Л into a continued fraction and
proceed as above.
G* L. DLrichlet noted that in addition to the infinite set A) of solu-
solutions there may exist further similar sets of solutions. Given any positive
number <r, we can find one and only one solution x, у of set A) for which
where t, <[ give any positive solution of £—£^ = 1. All solutions of these
inequalities can be found by a finite number of trials. Hence we find the
initial solutions a, b defining the various sets A).
A. M. Legendne58 discussed the integral solutions of
B) Ly*+Myz+Nz*~ =ЬЯ.
After preliminary transformations, we may assume that z is prime to у
and Я. Distinguish the cases in which the roots of LP-\-Mt-\-N—0 are
imaginary, real or equal. First, let 4LN—Мг^В>0. Set x=2Ly-\-Mz*
Then x*+Bz*=C=4LH. Give to г the successive values 0,1, -■-, [VC/5]
and see whether the resulting value of C—Bz* is a square z* and then
whether the resulting x makes MzTx a multiple of 2L. Second, let
4LN~M*= —В, В positive and not a square. If Н<\^Ё, develop a root
of Lx*+Mx-\-N=0 into a continued fraction; if one of the complete
quotients (| jB-t-I)/Dha&D=Ht at least one of the equations B) is solvable.
But if H>i^Bt we may set y^nz+Hu where п^Щ. Thus if Ln2+Mn
+iV is not a multiple/Я of Я for some value of n between —§Я and \Н,
B) is impossible; while if such a multiple is found, the equation reduces to
fz*+gzu+hu*= dbl for g=2nL+M, h=LH. See Lagrange7*-ю of Ch. XII.
E. F. A* Minding59 noted that, if A^№—ac is positive and not a square,
and if Я<}Д we can decide whether or not ах*+2Ъху+су*=±Н is
17 Bericfat Akad. Wi*j. Berfm, 1841, 280; Wcrke, I, 628-9.
"Thfcrie dee nombra, 1T&8, 09-122 G7-98); «L 2, 1908, 88-110 F8^87); ecL 3, 1830,
1,104^129 (81-103); tianel. by Merer, 1,105-131 (81-106).
и Jour, fur Matb., 7t 1831,140-2,

410 Нштсшт or тай Тнш>вт of Numbers. [Cbajp. xm
solvable in integers by devekipmg a root of uv*-\-2bb+e=^0 into a continued
fraction, admitting negative terms»
H. Scheffler80 treated ax2-1Ebxy—Gf=k. We may take x7 у relatively
prime. Let D=№+gc be positive and not a square. Set a=Qe, b=Pn,
c—Q-i Develop the ioot z/y=К={tId-\-Pq)!Qq into a continued fraction
and let the quotients be aD, au • •• >. Set
D-Pl
«—i—* »—i, Qn ~ ~7i -
Take Qh = k and seek all integers PJ, numerically ~k/2, such that D—P9
в divisible by k. For each such existing P^ develop К' = (ч{о-1-Р/1)/к
into a continued fraction. There is no solution unless we «an assign a
common period Pr=P[f Qt=Q\ (r+* even) of the two developments. By
use of such a common period or a repetition of that period, he obtained a
process for finding atf relatively prime solutions xt y.
C. L, А. Кипте*1 treated z2zby*=x±y in four cases.
J. J. Nejedli62 assumed that D-bz-\-ac>0 in
Set z—a$f+yi. We get a similar equation, apart from the sign of &,
C) Qtif=2Pjyyi+aiJl-k, Pi-сЮф-о, Q^c-aai+tojb.
Taking «o to be the greatest integer in r — (b-\-^D)/a and repeating the
process on C), we can solve the given equation. The process is equivalent
to the development of r into a continued fraction.
S. Realis*3 noted the identity f(x, y)=f(<x, fl)f(A, B), where
x, у) =
с*В\ у= -арА2+2а*АВ+(Ъ*+с0)В*.
Given the solution /(a, p) = h, we get another solution of f(z, y) = h if (as
is not always tbe case) solutions of f(A, B)=±l can he found. la par-
particular, from solutions /(<*, 0) — ±1, f{A, #) = d=l, we get new solutions of
Д*,*)-±1.
J. J. Sylvester" proved that/^*-|-2£xy—2fz*=±l is solvable ш integers
if А =2/*Н-0* is a prime and / is odd- Since w2—A^=l is solvable, «et
1^4,1 = 0-^ u—l—Affq^, where p, q are relatively prime. Then
p*-4gr1=2/<7 = =Fl or zh2,
the upper signs being excluded by the form 8?И-3 of Л. If p*—Aff — 1,
v=2pq7 we writep, q forte, rand pu q\ forp, ^and see in like manner that
p\—AgJ = l or —2. Penally, we reach тх—Л^ = — 2, where xand фагеod<L
Since every prime divisor of **-\-2 is known to have the form
" Jour, fur Mftth», 45,1853,349-309.
« Ueber einiee Anig. Dioph. Analysis, Weimmr, 1862.
° Б^п Beitrag «дг Aafl6eung ttDbegt, quad, GL, Frogr. bubaeft, 1874.
« Nouv, Опор. MaIJl, 6,1880, 342-350.
и Math. QtntfL £duc Tnnee, 34, 1881, 21-2.

Solution of аа?-\-Ъху+ср*=к. 411
^2J. By the coefficients of лР2,
S. Roberta used reduced quadratic forme and results of A. Gopel.
E* Cesiro66 proved that the number of sets of positive integral solutions of
Ax*-\-Bxy+Ctf~n D>G\ C>0)
is я-/B*)— #/** in mean, where $*=44C—tf*
S. R6alisw noted that if o^ £ ia & solution of x*-\-nxy—nj^ = l then
s=(n+l)a—nfi7 y=(n-\-2)<x—(л+1)/3 is a solution. From the evident
solution I, 0, we get the solution n+1* n+2. Using y=n-\-2t and solving
the initial equation we get x=n-\-l and the new value x= — n7—3n—1»
Applying the formula to the latter we get a. fourth solution, etc. The ath
set ха, уа of solutions of this series is given, as well as recursion formulae.
RiaEs*7 noted that mx2— (trc+ndbl)x2/-|-ns/J=Aha&the solution
D) x~(m—п)а — (т—л±1)£, y = (m — n41l)oL — (m — nHJ
If a, p is one solution. Starting from this set D), we get again the first
set a, /3» Evidently D) hold also for an equation derived from the given
one by increasing m and n by the same number; also for
For хг—(п+2Уху-+пуг=^1, the solution 1, 0 gives the solution n— 1, n.
For у=щ *we have x=n—1, n*-\-n-$-lt and hence find an infinitude of
solutions. There is treated the equation obtained from the last by changing
the sign of the constant term, and
x2-2(n+l)xy-\r{2n~l)y2=l or -2.
Recursion forraole&are given for the integral solutions of x2—Axy+Byi=h
when A—2 is divisible by A—B—l.
*P. S. Nasimofi48 gave an exposition of Jacobt's series for elliptic func-
functions and application to the number of solutions of ах*-\-Ъху-\-су*=п, ш
particular for x2+l§y*=nJ 4x2-\-4xy-\-3tf*=n, ax*-\-cy2=n (а, с odd).
F. J. Studnicka" noted that if pb and qk are the numerator and de-
denominator of the kth convergent for the continued fraction
1 1 1 „
Using g»=ogn_i+gn-l, we get
and hence the solutions of axy+x2—y*=±l. Cf. Huge28» of Ch. XII.
• Mem. Soc. R. 8c. de Ьй«в, B), 10,1883, No. 6^ 157-9.
m Now. Ann. M*tb.p C), 2, 1883, «4r-7,
17 /Wrf^ C), 3,1884,306-15, ЕЫа, р. 44S.
•• AppHcatien Ы effiptic fmictiee* to the theory of Dumbere, Moecow, 1885, Ch. 1. French
гбншё ш Ambits &c. tie I'^rtifc оагилГе sop^., C), 5, 1888, 23-31.
" Fa* 95t«anejber. {Math, N*t.)> 1388» Ю-0&

* Ferval" дате an infinitude of solutions of each of the equations
A. HurwiW1 called rfs and ujv a pair of approximating fractions for a
number between them if t«—ot = L If 0<m<2V2> and if at least one of
А, С is positive, and D^-iC>0, every pair of integral solutions of
Av?+2Buv+C&=m is such that u/p b an approximating fraction to one
of the roots of Az?+2Bx-\-C^Q- If both Л and С are negative, we get
tbe ваше result by assuming also thai &>— AJ1&4D—m).
H. Scheffler*3 made successive additioos to get p, 2*py 3*p, - - - and then
a table of values for p*?-\-j)in]. ТЪе aim is to solve axM-uxjf-l-cjf— q.
R. W. D. Christie* sohraixM-zy^tf^^tl by use of continued fractions.
Cf. J. Wasteels7* of Vol. I, p. 405, of this History.
A. Cunningham aad Christie74 solved tf—шф—oH=L
A. Levy79 recalled the special case of Dirichlet's theorem on the unite
of an algebraic field, tbafc if (д, b) is the least positive solution фA, О) of
x*-\-xy—ку*—1,жЪ&е к is a positive integer, every solution (t*, r) is given by
Several writers76 solved x*+xy+l^ = 1.
С Ruggeri77 used the series with the recursion formula ;Vh =£»~b*»-i
to solve aj^—bxy^c^—ht when Ы—4ar=5m1^
See papers $8,89; also Leslie*9 of Ch. XII.
Solution or Ax2+2Bxy+Cy*+2Dx+ZEy+F*=Q.
L. Euler* noted that if А^Н-2В^+СУ+2Dx+VEy+F=Q has the set
of solutions x=*a, у=Ъ, and if А—1Р—АС>{), яо that p^=^+l
a second set of eokrfioas ш
J. L. LagrangB78 showed how to find the rational and integral eohitiotts of
Solving it algebraically lorx m terma of y, we get
M Jour, de шлШ, «рбс, 1S9Q, 94, 141.
" Math, Annaltm, 4A, 1894» 425-7.
л Vermbchte Mattu BiSicSteav ^^^ ^ ^^ Quadratiecbe ZerfUlImg 4er Zabten dorcb
11 Math. Qw*t Efluc. 35пм*ц 75, «90,71.
« /Ud, B), M, Iff», 21-25.
" BuQ. de mmib. йётч 16,1ЭД9-Ю, 113-& Cf. J. Зошгоег, Vodesungem «her Zabietbeorie,
1907, 100-7; РгамЫталЛ by Levy, 1911,103-113.
^Amer. Math. Monthly, 16,1906, 44.
» Periodico di Mat., 26,1Ш, 206-276.
« Novi Сотш, Amd. Bstra^ 13,1765 ^7B^, 8 j Сопшь ЛетА., 1, *17; Op. Опь, {l^Hl,?».
f» Mem. And. Berlin, 2Z, вшмё 1767, 1700, 272; Оеет*», П, 377-«1, ДО-ОВ. Cf. Ыв
mmplificftUcme in trie ftdditkrae io EoWfl Aledbvn, % I774r $51* ЙЙ5^ДО7; Oeuvree de
Lagranee,VH, 103,14ft^7; ВвЬЛОрвслОюпйа, a), I, 5»3P€L6-S3. Cf. Snatk"

wnere
tf_
В ' X~ 2a 2aB '
Hence the rational solutions of (X) follow from the rational solutions of
u*=A+B&. The latter depend on the integral solutions of AT*=p?—Bq*t
discussed by Lagrange.110
To obtain the integral solutions of A), it is necessary that not only
u and I be integers, but aleo that ±u—f be a multiple mB of B} and that
±£—S—/3m be a multiple of 2a. If В is negative, v?—BP—A has only a
finite number of integral solutions, which can he found by trial. This
is not true when В is positive, as will be assumed henceforth. We may set
u=^<rpf t—trq, where p, q are relatively prime» By Lagrange" of Ch. XII,
the solutions of pt~B<f=AI<r2 are given by
whence
Here a, b, X, Y are given integers for which X2—BF2=dbl. We may
restrict attention to the case X%—BY2—-l-l, to which the contrary case is
easily reduced. The problem is now to choose positive integral values of
the exponent n for which the resulting values of xy у are integers, viz., for
which ±jp—/is a multiple of B, and =t^g— d—&{±<?p~/)/#is a multiple
of 2a. These two questions are special cases of the general question of
the divisibility of
B) F+Gp+Hq=F-\-P(X+Y<B)«+Q(X-Y<B)«
by R=r"tr1?1* * -t where r9ti, • - - are distinct primes. It is easily shown that
(X±yVB)'—1 is divisible by r, where p=2rif В is divisible by r, p=r±l
if Б^да±1 is divisible by r> and p-r if r=2. Then (X±Y<{B)*-1 is
divisible by rw for e=*rm~1p. Hence if n—ke+N, B) is divisible by rm if
and only if r" divides the function obtained from B) by replacing n by Nt
во that we need only test the values <e of n. Similarly we need only test
the divisibility of B) by r™1 for n<rT1pi- Suppose that the test succeeds
for n=JV and for n—Nu etc., in the respective cases. Then determine n
so that it shall have the remainder N when divided by Tm^1pf the remainder
Ni when divided by if^pu e*c- We saw that also a second expression
Fi+Gip+Hiq had to he divisible by a certain number Ri. The conditions
on n are similar to those just stated. Hence the method leads to all the
(infinitude of) integral solutions of B) when it is solvable.
Lagrange* multiplied A) by 4ac end sefc
We get i&=*ay*+2by+c. Multiply by a and write t=ay+b, К=Ъ*~а
Hence P—аи*—R. Assume that it has a known solution t=P7 u=
M Mecellanea ТаиппашЕ, 4, 1766-9; Oeuvree, i, 725-31.

414 History op the Theory of Numbers. [Сиар. хш
Then A) has the solution
fv\ P~b Q~S &
Since we may change the sign of P or Q, we get four solutions. If
R=A**B*> - -, where At Bt - - - are expressible in a single way in the form
P*—cQ2, it is known that H is expressible in this form in exactly Jx
ways when T = (wi+l)(rt+l)- — is even, and in (т+1)/2 ways when ж
is odd- If a is negative there is only a finite number of solutions of
**—au*=Rt since P—au2^l is not solvable, so that the number of factors
Af Bt "• is limited. But if a is positive, let pt q be the least solution of
p*—ag2 — 1; then every solution is given by
I
for7?i=l,2,3, --- Then
R = (P>
if
D) P.-j
If we employ as P, Q the various sets corresponding to the factors > 1 of
the fonni2—au*of R and take m=lt 2,Zf - < -f we get by D) ail the rational
solutions of P\—oQ\=U. Returning to C), Lagrange proved that, if
the values C) of zfy which correspond to the case m—Q are integers, there is
an infinitude of values of rn (the multiples of an assigned number depending
only on a and a) for which the solutions xf у are integers,
L, Euler*1 gave two methods of finding the general rational solution of
given one solution x=af у=Ъ„ Inf(x, y)—f(a, h) =0t set
We get
Eliminating у by the second of the preceding p^r of equations, we get
)—2Dp*-2Epq,
t=Ap*- Cff>
and hence obtain, when p, q are rational, the most general rational solution
of the proposed equation. Integral solutions may be obtained from values
of p, q making « = ±1 Or ±2.
For the second method, set
k=Bi-AC) N=(BD-AE)/kt P
Then
« Kovi Comm. Acad. Petrop^ 18,1773,185; Сошт. Anth.T 1,549^65; Op. Оиц (l)f Ш, 207.

СИДР.ХШ] A&+ZBxv+Otf+2Dz+2By+F=0. 415
let G and H be the values of P and Q for x=a, y—Ъ* Then
q the first factor on the left to the second factor on the right and
vice versa. Thus
y=-b-2N, *=a
Or use the Pell equation &—kr2=l, having an infinitude of solutions if к
is neither negative nor a square, and set
By equating the terms free of Vfc, we get rational expressions for z, y.
Euler82 treated the solution in integers of
E)
given one solution z=*а, у—Ъ. Denote the roots of **=*2«—1 by
p=s+ tI**— 1, q=s — Vs1—1.
Make the substitution
Since pq=l, the members of E) equal respectively
These are equal if
G) 4^=£"|f+l
Por n~0, let x=a, y~b. Then F) gives
and the resulting value of (/+дУ—(/—дУ reduces to G) since E) holds for
x=a, y=6. % Hence the values of /F ^ from (8) lead to solutions F) of E)
provided *, in the expressions for p and q1 is such that the resulting x1 у
are rational. For n=l, the expressions fora:, у become, in view of (8),
+HH. B ^
t-afr*, a solvable Pell equation if af is positive and not a square»
Hence if the latter is solved and we set p, q—e±rV^ and define f, g by
(8), then, for any integer n, F) gives a solution, which is proved rational
as follows. Galls', y' the values obtained from F) by changing n ton+1;
*", y" those by changing n to n+2. Then
* М<пь Acad. Sc. St. Peteieb,, 4, 1811 A778), 3; Comm. ДгН1ц П, 263.

416 History of the Theory op Numbers. [Chap.xiii
Since the values given by n=0 and n—I are rational, those given by any n
are rational. Enler stated that if we employ only even values of n, we
obtain integral values for x, y. Cayley1*2 gave a generalization to several
variables.
A. M. Legendre83 reduced ay2-\-byz+cz*-\-dy-\-fz+g=O to
(9) citf+ty^H-czf-AjD, D=b*-4<ic>0, -
by setting y = (y1+<x)ID, z = (z1+P)jD) a=2ed-fbt 0=2af-bd. If (9)
has a solution, it has an infinitude of solutions given by
J5 J
A0) yiy
where <f>, $ give the least solution of ф^—Щщ=^\. It is a question of
the values of n for which у and z are integers. Since
р-фп9 а^пф*-1*, ф*^1 (modi)),
we see that the expressions for y} z are integers if and only if
=О (modi)),
(mod Л).
In either case the resulting values of n are said to be of the form V-\-Dk
[denied by Dujardin**! where к is arbitrary, so that there is an infinitude
of values n. It remains to solve the problem: if F and G are given by
(I0j) and if ф*-Вф2^1, find all values of n such that \F+t*G+v is divisible
by a prime not dividing Ity, For this, the method of Lagrange" is given.
Dujardin" agreed with the statements in the preceding paper down to
the erroneous one that the values of n are of the form V-\-Dk> But the
quantities S, £ are divisible by D and the conditions marked (»=2m) and
(л~2ти-И) are satisfied only if the coefficients of the unknowns are relatively
prime. Hence a+yt P-\-t must be divisible by D if n is even, and тФ+«,
«Ф+/3 if n is odd; then the conditions cited are satisfied for all values of m~
The correct conclusion is therefore that n varies according to an arithmetical
progression of difference 2 (not D)- The latter result is said to follow also
from the law of recurrence between three consecutive solutions of (9),
which leads also to the following rule» Given two consecutive solutions
y'i9 z\ (i-lt 2) of (9); then if no one of the systems у\+а,г\+& (t=l, 2) is
divisible by D, there is no solution in integers; but if one of the latter
systems is divisible by Dt then to every system of the same parity as it
there corresponds a solution of the proposed equation.
C. F. Gauss86 treated the integral solutions of
A1)
Set
а Ъ d
bee
d e f
<х=Ъ*-~ас,
«■ TWorie de* nombree, 17&8, 451-7; ed. 3, 1830, II, 105-112, No. 439.
« Comptee Rendufl Pane, 119,1894, 843, 934. ВергтЫ, Sphinx-Oedipe, 4» 1909, 45-7.
^DiequiaitkmeeArithmeticae, 1801, art», 21<b22l; Werke,!, 1863, 215. German trattal. by
EL Maeer, 1889, pp. 205-2X1.

Az*+2Bxy+Cjf+2Dz+2Ey+F=0. 417
By the substitution p=az-\-pt q—cty-\-y> we get ap2-\-2hpq-\-cq1=ocA. The
theory of binary quadratic forms leads to all representations of «Д by the
form (я, bt c). From the resulting sets of values of x, yt discard those which
are not integral [cf. Smith88].
To find (art. 300) the rational solutions of A1), set x = t/vt y=ujvt and
find the integral solutions of the resulting equation which is of the form
considered by Gauss.1"
J. L. WezelM reduced ax*-\-cxy-\-dx-\-ey-\-C=0 to xxyx^k by a linear
substitution, and treated equations solvable rationally for one variable.
For axz-^-by2-\-2cxy-\-C=0) we solve (p. 40) for x and find no trouble unless
В=с*—аЬ is positive and =#□- The latter case is treated elegantly by
continued fractions. Develop the root r~ (л[В—с)/а of a2?-\-2cz-\-h = Q.
jjet Q=(VB+fl')/Cf be the complete quotient with denominator Ct and
ot p!*I the convergente immediately preceding this. Then
qQ+qo
since л[В is irrational For ах*-\-Ьу*-\-сху-\-дх-\-еу+С=О, we set
x^{xt+2hd—ee)fDf y=(y'-\-2ae-cd)fD,
where D=c?~4abt and get an equation of the form last treated:
E. Desmarest87 noted that the substitution X=xja reduces the solution
of аХ2-\-ЪХ-\-с=Ку to the problem to find multiples x of a satisfying an
equation of typefx=z2-\-qx-\-r^Py> To solve a particular equation of the
latter type, he would employ two auxiliary doubly-entry tables, a com-
complicated method based upon the functions
and the fact that their products by fn are also of the form /*, where
x = fnN—n—q and /„АГ+n, respectively* One of the auxiliary tables has the
headings fA) nPlf oi\ - - • and in the body of the table are entered the values
for successive K*s of the roots R and remainders p defined, for example,
when N=2K-\-l, by use of
dP^-JP+p, R=BK+2)n+qK-q~lt p=A(K+lJ, A=4r-q\
Troublesome methods are indicated (pp. 42, 43) by means of which the
square R* nearest to the given P enables us to find the entry in the body of
the table which will yield the desired value of n such that the heading of the
column of the entry will for this n be the value of у (or a known multiple
of y). The example X2+31X+241=PF is treated (pp. 2Ф-25, 301-2)
for all primes P< 1000; but he knew (p. 104) that it can be transformed by
X=x—15 into x*-\-x-\-l = Pyf which is proved to be solvable if and only
if the prime P is 3 or 3g+l.
"Annaka Aead. LeodieneiB, li^ge, 1821-2, 1-48.
17 Th&uie dee nombree. Trsite de Гаол1уее indeterounde du second degre h deux inconnuea
. . ., Pane, 1852, 4-126.

418 History of the Theory of Numbers. [Chap, xiii
(рр)/,
Д^Ь2—ос+0, it is transformed as usual into F=M, which is treated as
usual by the theory of binary quadratic forms. К Д=*0, it is transformed
into и*+т=*Ру, which is of the type first treated. In each case there is a
discussion as to which of the solutions are integral*
H. J. S. Smith88 noted that Eider's81- ю methods are incomplete for the
reasons noted in Ch. XII, Smith.1** He modified Gauss'86 method by em-
employing the g,c,d. Ь of cet 0t 7, and employing the new variables X=p/5f
Y=qlb. Thus aX*+2bXY+cY*=*'A\ where a'-a/S, Д'-Д/S. Then if
Xni Y* is any representation of a'A' by (a, b, c)t we separate the integral
from the fractional solutions xt у by separating (by Lagrange's method)
those values of Хл, Yn which satisfy the congruences Хя—0/3=0,
F«—7/5^0 (mod a') from those which do not, and obtain a finite number
of formulas exhibiting all integral solutions.
G. Wertheim8* treated A) as had Lagrange,79 and by reducing it to
ax2-\-2bzy+cyz^M and then applying the theory of binary quadratic
forms.
C. de Comberousse* treated A) for the case 7 = 0, whence y=Q/L,
where Q is a quadratic and Ь a linear function of x~ Thus L must divide
a certain constant Nt whence set L=d,d any divisor of 2V.
Kautenberg91 reduced the solution of an equation of degree two in
two variables to Bx2-\-Cx-\-D = □ and gave other known results.
R. Marcolongo,52 G. B. Mathews,93 P. Bachmann,94 and E. Cahen»
treated A).
Focke" gave the usual application of quadratic forms to our problem.
E. de Jonquieres97 showed by detailed examples that the methods of
Lagrange (continued fractions) and Gauss (period of reduced forms) for
solving indeterminate equations of the second degree are less different
than they seem, since they employ the same auxiliary quantities, and rest
on the development of practically the same ideas»
G. Bisconcini1* noted that z=y=2 is the only positive integral solution
of xy=x+yt and z=0,1, y=0t 1, the only integral solutions of xz+y2 =x+y_
J. Westlund9* proved that x*+y*={2x—1)/3 is impossible in integers*
C. CiamberUni1" stated that (x+y)(x+y-\-l)+2y=a has a single posi-
positive integral solution if a is a positive integer.
T. Pepm101 used the method of Gauss.8* ^_^
" Britieh Amoc. Beport, 1861, 313; CoU. Math. F&peia, lv 1894, 200-2.
« Element* der Zatientheotie, 1Ю7, 226^236, 369^374.
м Algftre Bupfoieure, 1, 1887,185-191.
n Ueber diopt Gl. 2 Grp Progr K. GymnM Maricab^rg, 18S7.
»GiomalediMat^25l l887p 161; 26,1883,65.
» ТЪеогу rf Numb™, 1892, 257-261.
n Arith. der Quad. Fcrmen718»8t 224 231.
« Elem. de la tbeorie dee nombres, 1900, 286-299.
^Uberdie Auflflfflmg AdioplbGHeicb. mit HllfederZahlentheoric, Piogr. Magdeburg, 1895,
" Comptee Rendufl Par», 127V 1898, 694-700.
"Psiodko dl Mai., 22, 1907, 121 2.
"Amer. Math. Monthly, 14,1907, 61.
»»SuppL al Pteriodico di Mat, 11,1008, 104^-5.
ia Mem. Pont. Accad. Nttovi lincei, 29,1911, 319-327.

Cjtap. ХШ] ах*-\-Ъу*-\-сг*=О. 419
U. Fornari102 treated (z-l)(x-2)-\-yBz-\-y-l)=2m.
W. A. Wijthoff1*3 solved {x+y+lJ=Qzy.
M. Rignaux1Ma stated a complete solution of A1), with ac<b*, by recur-
recurring series»
For ix(x+l) = ly(y-\-l) see paper 55. For 3x(x-\-l) =y(y-\-l), see
Euler79 of Ch. I. T. L. Pistor107 of Ch. XII gave Gauss'85 method. On
ах2-а'х=Ьу*-Ъ'у} see Gill107 of Ch. I.
ax*-\-by2+cz2=Q (except х2-\-у2=2г1)*
For xt-\-y2^2z2t here excluded, see squares in arithmetical progression
(Ch. XIV).
Diophantus, II, 20, proposed to find three squares such that
A) y*-x* :zi-y2=a:b,
where a : b is a given ratio. He took a/6 = l/3, y = x-\-l} whence
Take 2=x+3, whence x=5/2. In IV, 45, he took а/Ь = 3, з*=4, */ = *
whence f22=3i24-12f-i-9= C-502, if «=21/1L
Alkarkhi104 (beginning of eleventh century) solved x*—y*~2(y*—z2) by
taking у=г+1, z=z-\-2t whence г=1/2,
Leonardo PisanoIW first treated A) for several special cases. For
5 = a_j_lj take x=2a-lt y=2a-\-l, z=2a-i-3; then y2—x2 = 8a, z2-y2=Sb.
In general, if integers А, к, п can be found such that
= ha,
then v2-^=8b, *2-fl2=8M> for
The conditions for the above sums are
h+l+h+a=2kt (h+a)n+n{n+l)/2=kbt
or
These fractions must equal integers, as in the case for the values a —11,
b=43, 7i=16, k — 8, h = 2, used by Leonardo» A. Genocchiiw remarked
that Leonardo^ method consists essentially in separating a progression
h-\-lf fe-i-2, * ♦ *, k+m+n into two parts such that the sum of the first m
terms is ka and the sum of the last n terms is kbt whence
mn(rn-\-n) , t _2amn-\-an*—bm*
2k —
от—an от—an
101 П Pitagora, ID, 1913, 57-60.
1W Wiakimdige Opgaven, XI. 1912-4,192-5»
""' LJintcrm6diwre dee math., 26,191ft, 9.
101 Extrait du Fakhri, Frtnch tranel. by F. Wocpcke, 1853, 116.
leTreScritti, 103-112. Scritti, 2,1862, 275-« (Opuscolf), <X Ch. XVI.
xn Ancali di 8c. Mat. й FiH,, 6t 1855. 351-2 (miapriat of sign before Ьлй in the fraction for
2*+l)

420 History of the Theory of Numbeks. [Chap, xiii
Since а, Ь are relatively prime, we can make bm—an= 1 in an infinitude of
ways. ТЬепх-2Л+1, y=2(h-\-m)+l) z=2(h+m+n)-\-l.
F. Woepcke107 gave an analogous interpretation of Leonardo's method
and wondered why Leonardo preferred this ingenious method to the more
natural one [of Diophantus] of substituting x=y-\-m, z = y—nf and thus
finding xty, z as rational functions of mf n, a, b. The last method and other
simple ones were used by С L. A. Kunze.108
For a different presentation of Leonardo's method and a proof of the
equivalence of the problem with that of concordant forms, see Genocchi*7
of Ch. XVI.
Mateunago,1" in the first half of the eighteenth century, noted that
rx*-\-y*=z* has the solution x=2mn, у=гт2~п*) г=*гт*-\-п\ If k—l*=t*
кх*-1у*=г* has the solution y^a+tff, z^W-<xt, provided x^cSl
which is of the preceding type* Again, (^H-P)^2—y2 = zz for
J. L. Lagrange11* treated the solution of
B) Ari=p*-Bq*
in integer The cases Л^П, В^П are easily treated (pp. 381-2) by the
methods of Diophantus. In B) let pt q, r be integers, p and q relatively
prime, while A and В are integers neither a square nor divisible by a square,
and (as may be assumed) j A j > | В \ . A necessary condition is that
there exist an integer a such that а1—В is divisible by A. This is shown
by multiplying B) by pi—Bq\, using
C) W-Bq*)(ri-B<fl) = (ppl±Bqqiy-B(pqi±<lPiY,
and taking pq1—qpi=^l> whence Ar>(j£—Bq*)=a*—B. We may .also
take | a | < [ A \/2, since also (цА±аУ-В is divisible by A. When such
an a existe, AA^ct1—B, set al=}i1Ai±<x, the integer /ii and the sign
being chosen so that | at \ < | 421/2. Then а\-В is divisible by Ail
call the quotient A2. In this гпагщег we get a series of decreasing integers
\A \ , | Ax | , \Ai\t - ^ and hence get | An \ ^ \B \ - It suffices to stop
when An is of the form a*C, where С has no square factor and | С \ ^ | В \ .
Multiply together the equations
anduseC). Hence AA\*> 4^iAn=P2-BQ2. Multiply by B). We get
Cq\~v\—Br% where qi=AA^-Am-lar. Hence
D) Bi\=V\-Cq\.
Conversely if D) is solvable, B) is solvable. Treating D) аз we did B),
we get Crl^pl—Dql, etc. Since | A \, \B\t \C\, • • • form a decreas-
decreasing series, we finally get a term ±1. If it be —1, we proceed and get
+ 1. The resulting equation F^=x2—y* is easily solved in integers. Let
wjour.de Math,, 20,1855, 59. ~~~
^ Tfeber aaigfi Aufg. Dioph. Analyam, Wanur, 1862,14-15.
i« Y. M&ami, Abb. Gescb. Math. Wies., 30,1912, 231-2,
utMem. Acad. Berlin, 23, шшёо 1767,1769, 385~Мв; Oeuvrefl, П,

Снар.ХШ] ffiC2+^-i-czs=0. 421
М be the g.c.cL of V and x+y and set V=MNf x-\-y=Mp. Then z*= p<r,
x—*y=N<r, where tr is an integer. If I is the g.c.d. of p and <r, we have
p^lm*, <r~ln2t whence
z=lmn, s
We may set l*=2t since we may multiply x> yt z by 2/Z.
L. Euler1" stated that the general solution of ах*-\-(}у*=уг? is given by
x^±y'f^=(f^±gJ^)(pMf*n±q4-pnyt
if one solution. aP-\-Pg*=y№ is given; the solution by taking n = l is not
general. Again, by taking x=fp-\-pgq, y=gp—afqtwe get axz-\-Py*=yh2Rt
where Е=рг-\-арц* is the square of r^H-a^s4 for р = гк—а&$2, q^2rs. Again,
if we multiply the initial equation by h2 and <хр-\-рд*=у№ by z2 and sub-
subtract, we get
=
gz—hy hx—fz
Set each fraction equal to p/g and equate the two values of г; we get yjx.
To obtain another solution, set F^a&q^—p2, G=2pq, H=afiq2-\'p2, whence
H2=Ft-\-ctfG*> Multiply the latter by <у№ = а.р+$д\ The product of the
right members leads to the solution
z-hff, x^fF+figG, y=gF-aJG.
A necessary condition for fx^+gy^^kz* is that —fg be a quadratic residue
of A.
Eulerllu made ca?-\-q? a square by use of
x
Eulerm considered the rational solutions of
E) fx>+gy^hz\
If, for/ and g fixed, the equation is solvable when ft = Ai, Л2 and Лэ, then it
is solvable when h=h1hzJh, He stated (p. 558) the elegant empirical
theorem that if E) is solvable when h=h\ it is solvable also when
h=hi±4nfg, provided the latter is a prime.113
If E) be solvable, then (p. 566) —fg is a quadratic residue of k> For,
since z, у may be taken relatively prime, we can determine p, q so that
PV—qx = l. Then
{fx2+gu2)(fp2-\-gq2)=P+fg (t=fp*+goy)
is divisible by h.
ш Opera poetuma, 1, 1862, 205-21X (about 1769-1771).
m* Algebia, St. Pteterebuig, 2, 1770, $| 181-7; Lyon, 2» 1774, pp. 219-26; Opera Omwa,
A), 1,425-9. Cf. Еи1и« and LegraUKe** of Ch. XX
шОршс. anal., I, 1783 A772)» 211; Comm. Arith., I, 55U-569.
ш A. M. Legendre, Мёш. Acad. So. Paris, 1785, 523, stated that this theorem is true, but
omitted the proof (not easy) ad it was necessary to separate cqs&. He stated the
generalisation: If №±щр=№ ie solvable then /x^inr1—a? is solvable if c = h-\-fgn
is a prime and if я is mich that the two members of the quadratic equation are congruent
modulo 8.

422 History of the Theory of Numb его. [Снар.хш
If E) be solvable, also fx*+gy*=hiZ2 is solvable when hx is a certain
integer <k. For, k—^-\-fg is divisible by h for some integer t<hj2; call
the quotient Ab Then
so that fX*+gY*=h1Z* is solvable. It is not shown that AX<A4 In case
a set of decreasing values Л, hlf hh - • - eventually contains / or gt we can
determine z, у (p. 569, § 62).
A. M. Legendre114 proved CLegendre* of Ch. XII] that, if each of the
positive integers а, Ъ, с has no square factor and if no two of them have a
common factor, then ax*-\-by2=czz has integral solutions not all zero if and
only if there exist three integers A, /i, v such that
ак2+Ъ су*-Ъ су2-а
с ' a ' ~Ь~
are all integers.
Legendre explained the method of Lagrange110 to solve B), modified
by use of a principle employed elsewhere by Lagrange76 of Ch. XII. The
present method is essentially due to Lagrange.115* We may take A and В
positive, since otherwise
x*-Ay*=-Bz2 or x*+Ay*=Bz* (Л>О,Я>0).
In the second write Bz-z\ AB=A', whence z'*—A'y*=Bx2. The second
is obtained from the first by the transpositions of two terms. Consider
therefore x*—By*=Az2, A>B>0, where у is prime to A and x, while A
and В bave no square factors. Set x^ay—Ay*. Then
The first coefficient must be an integer, say A'k2, where A' has no square
factor. By changing <x by a multiple of A, we may take a between —Aj2
and A/2. Multiply the resulting equation by A'k* and set kz=z\
A'k2y-arf=x'>> we get /-В/=ЛУ:, Af<A. If A'>B, we repeat the
process. Finally we get a similar equation with one coefficient unity and
hence easily solved. While this method is not the simplest one for solving
the proposed equation, it is a very luminous one.
C. F. Gauss116 proved by use of ternary quadratic forms the theorem of
Legendre114 that, if no two of а, Ъ, с have a common factor and if each is
neither zero nor divisible by a square, then ах*-\-Ьу*-\-сг*^О has integral
solutions not all zero if and only if —bct —ac} —oh are quadratic residues of
a, bt c, respectively, and at b, с are not all of the same sign. If а, Ь, с are
Aead. 8c. Paris, 1785, 512-3; Theorie dee nombres, 1798, 49; ed. 2, 1808» 41;
ed. 3,1830,1, 47; German tranal. by Млеет, I, 49.
"■ Theorie d«8 nombree, 17G8, 36-11; ed, 2, 1808, 28-32; ed. 3, 1830,1, 33-39. (Maacr, I,
Зв-39.) For hie remark on a^+by1 =& see Legendre,"
ii* Addition V to Eulerb Algebra, 2, 1774, 538-55; Eulerb Opera Onmis, A), I, 586^4;
Oeuvrea de Lagrange, VII, 102-14,
ш DiequieffioiMe Arith, arte. 2Mr-8; Werke.1,1863, 349. German traneL by Maeer, pp.
335-343.

Chat. XIII] ОХ*+Ъу*+Сгг = 0. 423
arbitrary integers, let a2, /P, y* be the largest squares dividing bc} ac, oh,
respectively, and set aa=pyAf 0b=<*yB, yc=aPC; then the former equa-
equation is solvable if and only if AX*-\-BX2-\-CZ*=0 is solvable, and the latter
falls under the above theorem since AfB,C are relatively prime in pairs and
have ho square factors. For, bc[o?~BC is an integer without square
factor, so that В, С are relatively prime and without square factors»
E. F. A. Minding117 considered x* = Ay2-\-Bz2f where А, В are without
square factors. Let / be the g.c.d. of A=af and B=bf~ The equation is
solvable if and only if At Bt —ah are quadratic residues of B, A, /, re-
respectively.
A. Genocchi118 treated the equation ctzz-\-bxz= (a-\-h)y2t equivalent to A),
by the methods of Lagrange and Paoli*1 of Ch» XII.
G. L. Dirichlet119 treated axz-\-by2-\-cz2—0, where a, b, с are relatively
prime in pairs» If u, vt to are given relatively prime solutions, we can deduce
all solutions. Since auf bvf cw are relatively prime and аи, for example, is
even, we can find integers I (even), m and n such that €ntl-\-bvm-\-cum=l>
Set aP-\-bm2-\-C7il = k> Then u'=2l—huf v'=2m—kv) w' = 2n—hw are solu-
solutions, congruent to w, v, w, respectively, modulo 2» Hence, in
2u"=-vw'—wv\ 2v"=wu'—uw\ 2w" *=u\/—ш\
u/ft v"t wn are integers. If z, y, z are any integers,
F) t=au'x+Wy-\-cw'z, V^aux+bvy+cwZi t" = u"x-\-v''y-\-w"z
are integers and t=lf (mod 2). It is shown that, conversely, if it t', trt
are any integers for which t—t' is even,
", 2y = vi-\-v't'-2cav"t",
2x
{ } 2z=wt+w't'<-2abv>"t")
so that xf yf z are integers. Multiply the latter equations by <tx, by, cz,
add, and use F). We get
си? -\-Ьу2-\-сг2=tt' - abet"*.
Hence if x, yt z are solutions of the initial equation, then t t', V\ defined by
F), are integers for which t=V (mod 2) and tt'=abct' . Conversely, if
t, t'f i" are integers satisfying the last two conditions, the values of x, yt z
given by F0 are integral solutions. Further, by use of the above relations
he proved the following extension of Legendre's114 theorem: If no two of
a, hf с have a common factor and are not zero, ax2-\-byz+cz*=0 is solvable
in relatively prime integers if and only if — be, —са> —ah are quadratic
residues of a, bf c7 respectively, and the latter are not all of the same sign;
further, if —hc=A2 (mod a), —ca=B2 (mod &), ^ab^C2 (mod c), there
exist relatively prime solutions for which
Az=by (mod a), Bx=cz (mod b), Cy^az (mod c),
u* Anfangegitmde der Hoberen Arith-, 1832, M.
"• Aimati di Sc, Mat. e Fia., 6, 1855, 186-194, 348,
иш Zahlentheorie, H 156-7,1S63; ed. 2,1871; «d. 3,1879; ed. 4, 1894.

424 Histoht of the Theory of Ntjmbebs. [Chap, хш
J. Plana120 stated that all integral solutions of xx—7V=Ю1*2 are gjven by
x - p+
a
for a=2 or 1, where 3?, q are arbitrary integers,
G. Cantor121 considered the solution in integers of F=0, where F is
any ternary quadratic form. A formal solution {ф7 ф7 х) is one for which
F{$t Ф, x) =0 identically in 2, у, where ф, • • ■ are binary quadratic forms in
x, y. In particular, let F be
Let the greatest common divisor of the three coefficients of Ф, and those for
ф and x be relatively prime in pairs; then the formal solution (ф, ф, х)
is primitive, and we can find integers tcJs for which
w$^a"xt wx^—а'Ф (m°d a)
ю'х^аф, ад'ф^— a"x (mod a*)
у>"ф**а'$, ю"ф^ -аф (mod a'7),
identically in x, y. By the two congruences in the firet line,
(«*+afe")*x=0 (mod a).
Then w2-haV ^0 if a is odd, or when a is even if ^, x are properly primitive.
The solution (ф, ^, x) is said to pertain to the combination {w, w\ w"} if
v?+a'a"=Q (mod a), v>"+aa"=Q (mod ar), w^'+aa'^O (mod a").
The number of ^ssible sets of roots is 2"+ir, where «is the number of distinct
odd prime factors of the determinant D = —aa'arr of the primary form
[Wa"H> while ??=0, 1 or 2, according as i)/4 is not integral, an odd or even
integer. Then, if —a'a", —а"а, —аа' are quadratic residues of a, a'} a",
respectively, there is a primitive solution (ф, ф, х) of [aa'a"~}=® pertaining
to any chosen one of the 2"+^ combinations {w, w', w"}, and [aa'a"]=-0
has exactly *-2ы+11 systems of primitive solutions, where <r=2 if D=Q
(mod 4), while <r=4 in all other cases.
L. Calzolari122 treated G) «? = Ax2-\-By* by setting (8) u=Yx-\-Xy.
The discriminant of the resulting quadratic in x, у is to be a square, whence
(9) AX*+BY*-AB = U\
Eliminate X between the latter and (8), usingG). We get ZFy*=(uY-Ax)\
A0) Ax = Yu±Uy, By^Xu^Ux.
Thus from a set of solutions of (9O we get one of G), vk., that given by (8)
■» Memorie R, Accad. Torino, B), 20,1863, 107, footnote,
* De Aequat. aecimdi Gradua indcL, Din. Bsiin, 1867.
« Giomale di MaLp 7, 1869, Ш-192.

р. хш] ах*+Ъу7+сг*=О. 425
and
x XY±U
and conversely. Expressed geometrically, G) is a cone with the vertex at
the origin, and (8) is a plane through the vertex. The intersections are two
lines whose projections on the o^-plane are given by A1). If Xo, Fo, Uo
iB a particular solution of (9), and if u^ xq, yn are the values given by (8)
and A0), the general solution is
Х = Хъ — Xatt Y- Yu-\-yatt U= Uq—UqL
Calzolari123 stated a theorem, which not only decides like Legendre's1"
the possibility or impossibility of integral solutions of
A2) u*=Ax*dzBy2 (А, В without square factors),
but determines the general solution without recourse to the process of
Lagrange. We may set A^a\-\ \-<&7 B=b]-\ \-bl (m^4p n^4).
Set х>=а&, yi=b,y. Then
аз) «2=f>!±i;t/;.
bet pi, * • -, p», <fi, * ", qn be arbitrary integers» We may set
Then A3) becomes u—Xxi4='Zyi-\-K=0> where
Ки=(р±д)(^±2у*)-2р>х
In the former give to xit yi their values. Then
A4) Xi=pi-\-Jct yi = qi-\-k} u
Substitute these values A4) into the two expressions for K. Thus
A5) (p±qy-Xp]^2q] = (m±n-l)k\
For k=Q, values pif qi satisfying A5) give^, y^from A4) which satisfy A3).
Set a-2a<, b = 42bi. Then, for k=-0} xa=2j1 = 2pi—p, by=q, u—ax±by*
Substitute this и into A2); we get a quadratic for xjy. Hence A2) is
solvable if and only if c=АЪг±Ва2^АВ = D, and the general solution is
x=ab±c, y=a?—A, и=ЛЬ±ас7 where the signs of а, Ь are ambiguous»
S. H^alis124 stated that, if А, В, С are relatively prime and without
square factors, and if a, ft 7 give one solution of Az*-\-Byz-\- Cz2 =0, the
general solution is
where a, b, с are arbitrary.
вд Giomale di Mat,, 8, 1870, 28-34»
ш Nouv, Соттеяр. Math., 4P1878F 369-71.

426 History of the Theory of Numbers. [Chap, xiii
S. Roberts125 treated the solution of хг-2Ру* = —z* or =Ь2г2, when each
prime factor of P is of the form $m-\-l. If P have one of the forms
the equations
are solvable. If, moreover, P is a prime of one of those forms or an odd
power of it, хг—2Py>=2 is solvable. There are three more such triples of
equations leading to analogous conclusions.
T. Pepin1* proved Legendre's criterion as quoted by Gauss»116
G. Heppell2V treated <Р = 2а2+Ъ1 by setting b = d-2f, whence
Thus д is even, a=2q. Hence express 2q> as a product fh and take d=h+f,
b=h-f,a = 2q.
F. GoIdscheiderl2S expressed in terms of one solution the general solution
of ах*-\-Ъу2+сг?=0 which satisfies the final congruences of Dirichlet11'
He proved that there exists such a solution for which also kx+k'y+k"z
is relatively prime to a given odd integer s, if A, k\ k" are given integers
whose g.c.d. is prime to s*
G. de Longchampsm wrote x2 = y2-hp^2 in tbe form
(x-\-y)!(pz)=zl(x-y)=L
Hence t must divide г. Set z=2\L Thus x = \(ptz+l)t г/=Х(р?-1),
where X and i are arbitrary. For nx* = y*-\-(n— \)z27 see de Longchamps.1"
P» Bachmann130 gave a clear exposition of our subject»
R. P. Paranjpye111 proved that all integral solutions of хг—г*=2у2 are
where X, ^ are relatively prime» Since у is even,
A. S. Werebrusow132 noted that, if a*—D0z=ma2, a second set of solutions
of X2-DY* = mZ1 is given by
A. Cunningham,1" to solve х?+уг=Аг*, used the known solutions
Y=(t2+Au*)d) Z = 2tu/d, x = (t2-Au?)ldt of Y*-AZ*=*x\ where d-1 or 2,
and solutions of t2-Av> = -1. Then (Y2-AZ*)(T*~At?)=: -x2, whence
y = rY^AvZ, z = tZ^vY give the general solutions. A. Holm (p. 70)
!*Froc. London MatlkSoc., 11, 1879-80,83-87.
« Atti Accad. Pont. Ntiovi lined, 32,1878^, 88.
ulr Math. Quest- Educ» Timeay 38,1883y 56.
i» Dae Beziproritat4geeetz der achtea Potcn^reste, Progrrp B«riin, i889y S.
119 El Progreso Mat., 4, 1894, 46; Jour, de math, elem., 18, 1S&4, 5.
»o Arith, der Quad. Formeny 18»8У 19&-224, 231,
™ Matb. Quest. Educ. Timee, 75,1901, 119. Cf. papers 109y Ц6 of Ch. XVI.
ш Mem, Be» Univ, Moscow, 23; rintermediaire dea math,, 9, 1902, 187»
111 Math. Quest. Educ. Timea, B), 9F 1906, 69-70.

сидр. XIII] ах*+Ъу*+сг*=О. 427
noted that А = а2+Ъ% whence (x+bz)!(az+y) = (аг-у)/(х-Ьг)=т/п (вау),
which determine x : у : г.
P. F. Teilhet134 stated that z*+y2-(ms+ftflK2 implies
2=х(Л*+Я2), х = тК(А2-В2)±2пКАВ, y=nK(A2-B*)=F2mKAB.
F. Ferrari135 noted the solution, with if=1, given by Teilhet»1"
A. Gerardin"* stated that the general solution of x2-\-2y2 = О is
x=2P-m2-n2-\-2m(Zn-4l), y
Gerardin137 noted the identities
= {(т-\-п)(шпх2-\-2г2) —
another similar to the last and several for x2+8y2=z2.
A. Thue1*8 discussed the possibility of Ax*+By2 = Cz2, where x, у, г
arc relatively prime in pairs and z^y^x>0* We can determine integers
V> Q.* r without a common factor such that px-\-qy-rzt with p*f q*, r2 all
<3*. Then
(Bp2+Aq2)x*-2Bprxz+(Br*-Cq*)z2=Q,
(Bp2+Aq*)y*-2Aqryz+{Ar*-Cp*)z*=0.
Hence
ax = Cq*-B>r*7 by=Cp2-Ar*, cz = Bp2+Aq2,
аг-\-2Врг—сх, bz+2Aqr=qj,
where a, b, с are integers» Let U be the greatest of [ A \ , | В \ , | С \ .
By the last five equations, | с j < 6£/, \a\< 12t/, \b\ < 12U. But
af Ъ, —с satisfy the initial linear and quadratic equation. Thus the possi-
possibility of the latter can be decided by a finite number of trials.
L. Aubry1M proved that if pA2 = B2-\-rC2, where В and С are prime to Л,
then j>X2=Yz-\-rZ2 for X^2$iZ if r>0, жйХш^Х r<0; if В and С
are prime to p, then also Y^Ba, Z^Ca (mod p)>
Several writers140 solved 13х2+17у*=2Ш2.
С. Alasia141 solved x2—79y2=101s2 [Plana120] by several classic methods.
G. Bonfantmi142 noted that the evident sufficient condition for integral
solutions of x2-\-y2=mzi is that m be a sum of two squares» To prove that
the condition is necessary, consider integers k^ pi, qi such that
By induction, йта=^-И^л+Р1^тJ, where
ш L'interm^diaine dea matk, 12, 19O5S 81-
ш Suppl al Periodico di Mat., 12, 1908-9, 34-5.
|to Левое, frang., 1908, 17» To make10* m=O replace f by l^2my n by n-f 3m.
1X7 Sphmx-Oodipey 19ОГ-8, 109-ЦО.
W8 Sknfter Videnek&pflfif'bik&pet, Kristiftniap 1, 1911P No. 4P p. 18.
x" Sphinx-Oedipey 8, I9!3y 150 (error in his 7y 1912, 8l-2.
"» Wisknndige Opgaven, 11, 1912-4,281-6,
1<J Gioraak di Mat., 53P 1915, 292-302.
lMSuppLai Periodico di МаЦ 18, 1915,81-6. For m=2y ibid., 17, 1914,84-5.

428 Hibtory of the Theory of Numbers. [Chap, xin
M. Weill148 obtained solutions z = a+M, у^Ъ-\-\'6> г^1+Ь, of
by finding б rationally in terms of а, Ъ, \ X'. Again, ax2-\-by2 = (a+b)z2
has solutions of the form s^l+X5, у = 1+\'6> г = 1-\-Ь. То one of these
two is reduced the solution of ах*+Ъу*=г* when а-\-Ь or ab is a square.
E. Cahen1" noted that WeilTs formulas do not give ail solutions and
showed how to find all solutions of х2+у*=5г2.
E. Turriere1" noted that, if а, b} с are the sides of a triangle two of whose
medians are perpendicular, then a^+b^oc2, whose solutions are expressed
rationally in two parameters»
A. Desboves3* gave all solutions of x*-\-y2 = (т?+п*)г\ Cf. papers 133-5,
142-5 above; Catalan63 of Ch. VII; G. К Malfatti19 of Ch. VIII; papers
191, 252, 294p 307, 311 of Ch. XII; and 225 of Ch. XXII.
K- Hoppe» of Ch. V solved р'-З^т2; Eukr109 of Ch. XXII solved
SINGLE QUADRATIC EQUATIONS IN ТЦКЕЕ OH MOKE UNKNOWNS.
Bhdscara116 (born 1114) found four distinct numbers whose sum equals
the sum of their squares. Take as the numbers y> 2y, 3y, fy. Then
10y = 3^,y^l/3.
C. F. Gauss147 considered the solution in integers of
A) f=ax*+alx\+abZ\+2bxiXb+2b1xxi+2b2xxl = ft.
If a = 0, x is determined rationally in terms of xu z2; to obtain integral
solutions, multiply the three xys by the denominator of x> Next, let
We derive the equivalent equation
^1X1=0, L = ax-\-b2xl+hlx2,
If Аъ=>0, ВФ0, we can give arbitrary values to x2 and L and determine
x and xt rationally. If A2*=B=0, either At is not a square and x2^L = 0
or ilj^and L=±kx2, Finally, let а2ф0, Л2+0. Then
where D is the determinant of/, whence Da=B*—AtA^ If i) = 0, we have
linear factors. If D Ф0, criteria for solvability were given by Gauss.115
Given one solution <*, <xlf «2 of/=0, we can transform / into a like form
with a=0 (treated above). In fact, determine integers £, • —, yz so that
Then the desired transformation is
i« Nouv. Aud. Math,, D), 16, Ш6У 351-^5.
Ш/Ш., D), 17, Ifll7p 463-5.
«» L'enseignement math,r 18P 1916, 89-90.
M Vija-ganilft, f 119; Cokbrooke/ p. 200.
»7 Dieq. Aritb., 1801, art. 299; Werke, I, I863y 358; German tr*nal.y Мавег, 344-6.

CnAF-ХШЗ Quadratic Equation in n^3 Unknowns» 429
Aida Ammei,M8 just after 1807, noted that x\-\-2xU -\-ш1=у* has
the solution
xr=2a1ar} У
and that xl+3xl-\-Gxl-\ + %n(n+l)zl^y* has the solution
G. Libri149noted that ax9+&y2-hc^-hd=0is solvable if а'х2
is, where a\ Ь\ c' are any three of o, 6, c, d. For example, if
we get a solution x=np+q, y=rp-\-8>z=mp-\-t, where p is found rationally
m terms of the indetenninates q} &y i. If an2, Ьгг, cm* are relatively prime
integers and if no one of a, b, с is divisible by 4, we can assign the value d=X
to the denominator of the fraction for p and hence get integral solutions
*, V> z*
Every integer can be expressed in the form F=«2+4I«*—U3z* since
F=0 is solvable. Likewise for 23x2+y2—13^ and cu?-\-5z*—2y2, where a
is a prime =3y 13, 27, 37 (mod 40).
A. Cauchy130 treated the homogeneous equation F(x, y> z) -0 of degree
№, with the given set of integral solutions a, b, с Let x, y7 z be another set.
The ratios of u7 vt w are determined by au-\-bv+cw=Q, xu-\-y *-\-zw = 0.
Then
Set у/х=р,Ь{а=Р. Then
Let ^, x, ^ be the partial derivatives of F(x, y, z) with respect to x, y} z>
Then
Thus a«-hfri?-hcw-0 is satisfied by
B) и=ф{а, b, c)-\-hr-cnt v
for m, n, r arbitrary integers» If the latter can be chosen to make Ft — Fz
for a rational p(p+^), we get the new solution x : у :z = w : wp : —u—vp
To apply (pp. 292-301) this general method to
F{x, y, z)=Axi+By*-\-Cz*-\-Dyz+Ezz+Fxy,
note that the condition Ft=F2 now gives p=P or
Replace P by its value bja and use au+bv-\-cw=0, F(a, b, c) =0. Thus
y/x=p=aP!(ba), where 0=Са*-.Еию+Лад*, у^Л^-^1«>-Ь.&гг2. Then all
™ Y, Mik&miT Abh. Geach. Matb, Wwa., 30, 19l2, 24$. See рарегз 59, 66 of Ctu
141 Memoria ftopra la teoria dei numeri, Игап^е, 1820„ 10-14.
w Erancioee de matb^matiquefl, Paris, l826i Oeuvree, B), VI, 286.

430 History of the Theory op Numbkbs. £Cha*. xiii
solutions of F(x, yy z)=0 are given by ax/<x=bylP—czly, where a, 0, у
have been defined and u} v, w are given by B). In particular, x—ajaf
y = fi!b, z=y{c are solutions.
To apply this method for 2V^3, we remove the factor p—P from Fx—Fi
and have a quadratic in p, whose discriminant ia to be made a perfect
square if new rational solutions exist. To avoid treating this quadratic,
Cauchy287 of Ch. XXI gave a method independent of the above»
G. Poletti1" treated the genejral equation of degree two in three un-
unknowns» Firet, for solution in rational numbers, solve for one unknown и
in terms of the other two v, w. Since the radical Z is to be rational, a
quadratic function of v, w is to be a square Z1. Solve the latter for v;
a new radical Y is to be rational, whence
Solving this for wt we see that a radical X is to be rational:
(F) X^AYt+BZt+C.
Hence the rational solution of the initial equation is equivabnt to that of
(F), where Л, В, С are given integers. This in turn is evidently equivalent
to the solution in relatively prime integers of
(G) x*=Ay>+Bz*+Ct\
8еЬ1т=х*-Ау2 and call Ф the g.c.d. of x, у; ф that of z7 t. The quotient
n"! of 7г by фЦ2 is an integer» Thus the problem reduces to
(H) ti-Ajfi-T^ Bz\+CH^t^\
where Si^x/ф and у^ = у(Ф are relatively prime, and likewise also zh tu
From Legendre's theory of the quadratic forms of divisors of x]—Ay\\
we get iTi as a quadratic function of two parameters y't z\ and ф as one of
y\7 z\; then by the composition of quadratic forms, we get zb y± as functions
of the lour parameters y\ z\ y\ z[. To get the linear forms 4Л£+Ь< of
the divisors iru use Legendre's text. By (Ha) these must divide р*-\-ВСа*>
whose divisors are of certain linear forms 4ЯС&+0.'' Equate each of the
latter to £A£-\-bi and solve for integers £, £b For each such set of solutions,
we can tell by a theorem of Legendre whether or not (H2) is solvable in
integers* There is a similar discussion of the solution of (F) in integers,
A. Cayley152 treated the generalization of Euler's*2 equation E), viz,,
C) *(s, У)=<*х2+рх+'у-{у>-~г}у-в=ф(а, Ъ).
This is a special case of
D) {abcfgh)(x'y'zy = (abcfgh) {xyz)\
where the second member denotes az*+by*-\-cz1-\-2fyz+2gxz+2hxy> It is
assumed that the latter has a linear automorph (transformation into itself),
which may be taken to be such that zr-z. For*'=* = l, fc=0, D) becomes
C). We can find a solution of D) by Hermite's method: set x'=2J-x>
«iMemorie Accad. Sc, Torino, 31, 1827, 409-49» <X Atti della Societi Ital. delle Sclent
f«eidente in Modena, VoL 19.
™ Nouv, Ann. Math,, 16, 1867, Ш-5; ColL Math. Papere, Шг 205-8.

Chap. ХПГ] QUADRATIC EQUATION IN П^З UNKNOWNS. 431
F=gh-af,
G=fh-bg,
where q is arbitrary. Multiply these by £, tj, f and add. Thus
so that we get D). Using the multipliers C7 F7 G, we get z = £. Then the
first two equations readily give
which satisfy D) identically with z'=z = l. Taking h=Q, we get values
making ax*-\-2gz+c-\-by*-\-2fy identically equal to the same function of
x'} y'- To pass to formulas exactly equivalent to Euler's, set
H. J. S. Smith163 stated criteria for the solvability of A) in integers,
whether the coefficients are real integers or complex integers p-\-qi. It
suffices to consider the case in which the coefficients a, • • >7Ъг of / have no
common divisor, while / is an indefinite form of determinant Ф 0. Let Й be
the g.c.d. of the nine two-rowed minors of the determinant SPA of/» Let
&F be the contravariant (&*—aia^H of /. Let Ц Д, IS be the quo-
quotients of ft, Д, UA by the greatest squares contained in them respectively»
Let o> be any odd prime dividing Д but not £; 5 one dividing A but not
И; в one dividing both Ut A. Then /=0 is solvable in integers =#0if and
i
where the symbols in the left members are those of Legendre, and those
in the right members are generic characters of / (Eisenstein, Jour, fur
Math», 35, p. 125). This theorem is a generalization of the criteria for the
solvability of ах*+агх*+а&1=0*
A. Meyer1*4 proved the preceding theorem for forms of odd determinant.
P. Bachmann165 proved that, if F is a ternary quadratic form, all solu-
solutions of p*—F(q, q', q")=2% in which p is divisible by 5, are obtained
by a definite rule from any one solution and all solutions of
*-F(tt, *',«") = 1-
The left member repeats under multiplication.
S. Kealislfii stated that all integral solutions of z2-\-ny2=it*+nv* are
given by (cf. Gerardin187)
c. Boy. Soc. London, 13, Ш4, ЦО-1; СоП. Math, Pspcra, 1,1894,110-1.
"» Jour, fur Matt., 98, 1885, 177^9.
» Jour, for Math,, 71, 1870, 299-303.
"■ Nouv. Ann. Math., B), 18,1879, 508.

432 History op the Theory of Numbers. [Cejap.xiii
E. Cesaro157 found various sets of solutions of
v*-v(x+y+z)-\-xy+yz+zx = 2w*.
Realis stated and Rochetti1*8 proved that
has an infinitude of solutions. Solving for z, we are to make ту—п* = П,
whence n*+& is to be expressed as a product of two factors. Or, choose
p, g, r, 9 so that n=pr±qs] then four solutions are
p4g, v+, (p)+C^)
A. Desboves1" gave the complete solution in integers of the general
homogeneous quadratic equation in n variables, when one solution s, yt
• • ■ is given. Regard mx, my, - <- as the same solution as x, y, —. First,
l3
E) ++++f
Let X = px, Y=py+p, Z=pz-\-qm Then E) gives p as a rational function
of p, g, x, y, z, so that
F) Y
Z = -bzjj?+(ez-\-fy+cz)q2-\-(dx+2by)pq>
This is the genejral solution of E), since we can find p, q such that F)
becomes any assigned solution» A convenient modification (pp. 233-5)
of the method of Gauss4' leads to F). Special cases of F) have been
noted above (Desboves38).
For any n, set Х=ря+г, Y^py+p, ••• in the proposed equation
F(X, Ff..-)e<). We get p and then
X=Mr-Nx, Y=Mp-Ny,
The results are no more general than those for the case r=0.
A. Meyer180 gave criteria for the solvability in integers of
G) щ?+&^+сг?+йг*^0,
where a, 6, c, d are integers not zero without square factors and such that
no three have a common factor» Write (a, b) for the positive g.c.d. of
a, b, and set
в-fo b)(a, c)(a, d)a, b- (Ь, а)(Ь, с)(Ь, d)ft
Then necessary conditions for the solvability of G) in integers not all zero
are (I) a, * ■■, d are not all of the same sign, and (II) — (a,c)(a,d)(fr,c)(b, d)yb
is a quadratic residue of (a, b), with five similar conditions derived by per-
permuting the letters. Again, G) is solvable if and only if conditions (I)
v. Correq>. МаЛь, 6,1880, 273.
Ьева. (I)p 1, 1881, 165.
-•Noov. Ann. Malb., <3), 3F1884. 225-39,
MvV^rf-JUbrvrbHft NAturforech. GoeO. Zorich, 29,1884, 209-222.

Chap. XIII] QUADRATIC EQUATION IN Л^З UNKNOWNS. 433
and (II) hold and either abcd^2, 3, 5, 6, 7 (mod 8); or abcd^l and
a-\-b-\-c-\-d=0 (mod 8); or а6сс?=4 (mod 8) and, if a and b are even and
с and d odd, either $abcd^3, 5, 7 (mod 8), or Jo&cd=l (mod 8) and
llj^ (mod8).
He gave necessary and sufficient conditions for integral solutions of f=Q,
where / is any quaternary quadratic form. Finally,
a*?+by2+cz*+du2+evi=Q
is solvable in integers not all zero if the coefficients are odd and not all of
the same sign.
H. Minkowski1*1 defined an invariant D in terms of the prime factors
of the determinant of the quadratic form and proved that zero can be
represented rationally by every indefinite quadratic form in 5 or more
variables, by one in 4 variables if D is not divisible by the square of a prime,
by one in 3 variables if D= 1, and by one in two variables if D = — 1.
G. de Longchamps162 would solve x22Xi = SX1-^; by choosing integers
x9 au • • •, a» for which SX^af = 2х2Х,чх^ (for example, by taking <*i, - • •,
an-2 arbitrary even integers and choosing a*_i so that SS^'Xi-of*—2 is divis-
divisible by \nt and taking ая to be the quotient) and then finding yh - - -, yn
from x—Vt=<xi. Application is made to nx*— y2+(n—l)zz and to
x2—xy+y2=z2.
The discriminant of the latter equation in x is у1—Цуг—z2), which must
be a square k2; whence a solution is z=-7, у=5, &=11, x—8»
P. Bachmann181 proved Meyer's1" theorems.
A. Meyer164 discussed the solution of pt—UF(qJ q\ q")=e [cf. Bach-
mannlw]. For this and the next paper, see the chapter on quadratic forms»
G. Humbert1** treated the integral solutions of x1—4yz—±tu = A.
Anonymous writers16* stated that x1-\-y2—z*=2u2 has the solutions
2= {с2-2а(а+Ъ)}*-2а*(а*-2Ъ2),
Or we may compare the known solutions of у*—г*=№, x2-\-h*=2u2 and
tske u— az—b2=2rri*— P\ hence an infinitude of solutions can be found
from one.
A. Gerardin167 stated that x2-\-hy2=z7+kP has the solutions (Realis1*)
х=т2-\-п*+кр*-2т{п-\-кр), y=
2=m*+n2—hpl+2n(kp—m)f 1 =
Нт2, y^ni-\-hpi-\-hnt2~2m(n-\-hp))
И| Jour» fur Math., 106p 1890, 14. GeBamm. Abhandh, Iy 227,
ш Ei Progreeo MatM 4, 18WP 40-7^ Jour» de math. Яёга., 18,1894, 5»
вд Arith. der Quad. Formen, 1898, 26^266, 653»
ш Joux» fQr Math., 116P 1896, 32b
m Jour» de Math,r E), 9, 1903, 43.
ш Sphinx-Oedipe, 1907-8, 30, 95-6.
|И 1Ш., 107-$.

434 History of tee Theory of Numbers»
F, Ferrari1" noted that the solution of xl+A&U +Anxl=xl+1
reduces to the solution of Zx\=xl+\.
0. Degellw noted that, if x, y, z1 8 are homogeneous coordinates, the
surface
can be represented on a plane (Clebsch, Jour, fur Math., 65, 1866, 380) by
Take &=0 as the plane and set ps = a in the initial equation. We get a
rationally. Hence px, - -, ps are expressed as homogeneous quadratic
functions of fiT Ь, Ь- By the same method he1™ treated
and found
Several writers (pp. 164-6) gave solutions.
A. G6rardinm found m from (l+waJ-h(l-|-w^J-{wc)!=2, whence
He noted (p. 22) the identity
AlsoP-fgg /ppg$
Gerardin372 gave solutions of cases of
x2+2(^-hce)x+m2/42a^+n^= EL
0. Degel stated that all solutions of
are given by px^A+2aB, ру=А+2ЪВ, рг=А+2еВ, pw=A+2dB,
pu=A-\-2eB, p&^A-\-%fB, pt^A, where a, --,/are distinct and Ф0, and
Л = 11а2-Ь2+Зс»-|-^-2е2-?Г, B= -lla-|-b-3c-<l+2e-|-^
" V. G. Tariste "IU noted that, if alf •.., art is one set of integral solu-
solutions of wiixjH Нпця£=0» all solutions, are given by
{* * • 1
where the a' are any rational numbers and M is such that the x's are
integers. Gerardin (pp. 136-7) remarked that this result follows by taking
L. Aubry discussed the integral solutions of xxyx-\ !-*«#«=0.
W. Mantel treated xy+xz+yz=N.
For x2H-i/9=2a*-|-2b1, see papers 83-87 of Ch. V.
of Ch, XIL On яг'+Зу'^ц'+Зи», ^ papers 201 and 211 of Ch. XXIL
u'SuppLalPeriodioodtMat., 11, 1908, 129-131.
"»L'i"oteno6diain? dee math., 15,1908,151-2.
"•/W*ieie09 1в7
w Sphmx-Oedi^ 6, 1911, 74^-5,
1W L'mtermediaire dee math., 18,1911, 202-3.
bul., 20, 1913,226.
Ш,21, 1914, 49.
Wd., 23, 191^1334.
kdige Opgaven, 11, 1914, 44^90.

CHAPTER XIV.
SQUARES IN ARITHMETICAL OR GEOMETRICAL PROGRESSION.
Thkee squares in arithmetical progression, x2-\-z2=2y*.
This topic is closely connected with congruent numbers, Ch. XVI,
especially papers 41, 67, 68, 120, 141. It may be stated in terms of tri-
triangular numbers (Ch. I1W).
Diophantus, III, 9, used three special squares in A. P. (see Ch. XV).
Jordanus Nemorarius1 in the thirteenth century found that
make f2—г* —<?—rJ=2b3c+3b2c*+bc3. Here h is any integer, с any even
integer» In his notations, set a = b-\-cf d=a-\-bf h=ac7 k = bc, e = adlf=bd>
Then €—h+k+fj and a solution is v=№+/)/2, r=f—v, q=e—v.
Regiomontanus,2 or Johann Miiller A436-1476), proposed in letters the
problems: Find 3 squares in A. P., the sum of whose integral roots is 214;
find 3 squares in A. P., the least being >20000; find 3 squares in harnionical
progression.
F. Vieta8 took A*, (A-\-BJ and (D—A)* as the squares- Hence
7J ОШ
A
Hence we may take DZ-2B\ D*+2B2+2BD and D*+2B*+4BD as the
sides of the three squares.
Fermat4 proposed to St. Croix, Sept., 1636, that he find three squares in
A. P. the common difference being a square.
Fermat* knew a rule for finding three numbers whose squares are in
A» P. Apparently the numbers were r2—2s2, r2Jr2rs-\-2&2i r*+£r&-\-2s2*
Replacing r by p—q and s by qt we obtain Frenicle's set p2—2pq—q2,
jp+<ft p2-\-2pq—з2. To derive the latter, Frenicle* noted that the squares
of a— bt ct a-\-b are in A. P. if а*-\-№=<?t and took a-p^—g2, h — 2pqy
c=p2-\-q2.
L. Euler7 deduced from the solution у = 1 of 1 +з3 = 2у4 the solution
у = 13, etc. Cf. Cunningham79 of Ch. XX.
To find three integers the sum of any two of which is a square and whose
squares are in A. P., "Amicus'Te took 2а2Ь2±(а4—Ь4), аА+Ъ4 as the
1 Elements Arithmetics, deeem libria, dcmostrationibim Jacob! Fabri 3tapulensi9k Porw, 149G,
1514, Book 6, Theorem 12.
5C T. de Murr, Memorabilia BibL publ. NorimbergensiQm, Pars I, 1786, 145, Ш, 201.
Cf. M. CaDtor, Geachichte der Math,, II, 1802, 241, 263.
1 Zcteticar 1591, Vp 2; Орстл Math., 1646, 76. Same by J. Prestet, ElemeaH des Math., ou
Prmcipee . . ., Parie, 1675, 320.
4 Oeuvres, U, 05; III, 287.
'Oeuvrte, II, 234; letter from Fremcle to Fermat, Sept. 6, 1541 {tables by Fremcle, p. 237).
1 IVianglee rectangles en nombres, prop. XI. Full referepce in Ch. IV."
7 Algebra, 2, 1770, Ch. 9, Art. 140; French trand,, 2, 1774, pr 167; Opera Omaia, A), I, 402.
■LadieH' Diary» 1795, 38, Quest» 974; ЬеуЬоигп'я Math. Quest, from L. D., 3, 1817, 297»
435

436 History of the Theory of Numbeks. [Chap, xiv
numbers. Their squares are in A. P. and their sums by twos will be squares
tf2az+2№=0> which is known to hold if a, b=2mnd=(?n?—ri*). The same
problem was treated by A. Cunningham and F. Phillips,9 A. E. Jones10
started with any three numbers
whose squares are in A. P., and called P, Q, R the values obtained from them
by replacing m by x*9 n by г2. Then P, Q, R are the desired numbers since
C. Campbell11 treated the similar problem to find three numbers x, y, z
the difference of any two of which is a square and whose squares are in A. P.
Let x—y=rri*t x—z=niJ y—z=p\ Then n2—p?=m\ Take n+p=mst
n—p-mfo. Since х*+г2=2у* &ve& zt we get у and z in terms of m, n.
J. Cunliffe12 treated the problem to find 3 squares in A. P. such that the
sum of each and its root shall be a square.
J. Wright13* found three squares x2, y2, z2 in harmonical progression
such that each exceeds its root by a square» If a)b = 2rs±(ri—$*),c=r*+$2,
a*+b*=2c2 and the sqiiares of x=*n*jd} y=lm?l(cd)} z^bn2/(ad) are in har-
harmonical progression. For d=mBn—m), x1—x=O. Also, у1—у = П if
}№-Ш=П = (Ьп-ртJ, which holds if m^2bn(c-p)/(bc-pz). Then
г2—2=П if {be—p1J—iap(b— p)(c—р) = \3 = (Ъс+2ар-р2)*, which gives
(
р)
J. Ivory121 found two sets a2, ¥, & and a*, b\} c\ of three squares in A. P.
having the same sum. The conditions are ai-\-c2^2b7=2b\ = a*l-\-c2i) or
4b2 = S(a±cK=S(a1dtCiJ. Hence we require a square which is a sum of two
squares in two ways» The least numbers are obtained from
To find three numbers whose sum is 117 and whose squares are in
A. P.,3. Jones1' took xt 5x, 7xas the numbers, whence z=9. S.Ryley took
2mn±{™2-n*), rtf+n2 as the numbers. Then («+2mJ = 117+3m2= □
for m=3; the resulting numbers 9, 45, 63 are said to give the only solution
in positive integers.
R. AdrainH used the squares u2—y = (u—pJ, u2} u2-\-y = (u+qJ, whence
2pu—y-pi> y—2qu=g\ Solving, we get u=(p2-\-tf)l{2(p-q)\. There
results Frenicle's5 solution.
J. Surtees1* noted that (a-n)\ a2+n2, <a+nK are in A. P. and
D if а~г2-\гп=2т.
J. R. Young16 found three squares in A. P. such that the roots increased
»M&th. Quest. Educ. Times, 24,1913,107.
"Math. Queet. and Solution», 5, 1918,62-3.
* The Gentleman'e Diory, or Math. Repository, London, No. 65, 1805, 40-1, Quest. 873,
« Math. Repertory (ed., Leyboum), London, 3, 1804, 97^106, Prob. 7.
^ New Seriee of Math. Repository (ed., T. Leyboum), 1,1806,1, 99,
»»Ibid., 121-3.
я ТЪе Gentleman's Math. Compaction, London, 2, No. 9, 1806,15-17.
14 The Math. Correspondent, New York, 2,1807,14.
a Ladies' Diaiy, 1811, 39, Quest. 1217; Leybouro'a Math, Quest, L, D.s 4,1817, t39r
" Algebra, 1816; Amer. ed.v 1832, 2334 C29-31).

Chap. XIV] ТВШ1Е SQUARES IN ARITHMETICAL PROGRESSION» 437
by 2 give squares, the sum of the first and third of which is also a square.
Take q=l in Frenicle's set; we get p*4-2p—1, p*+l, pz—2p—1- Hence
thecoDditionsarep*H-3 = D, 2р*+2^П. Set p = m+l) and let the second
equal (mra+2J, whence m=4(l—n)f(nz—2). Then
if n = 5/4, whence p = 23/7.
To find three squares in A. P. such that any root plus unity is a square,
H. Clay17 took s2, a***, &x2 as the squares. Set z+l = (r+l)*. Then
ax+l = (sr+iJ determines r. Then bz+l^O if a certain quartie in s
is a square, which is the case if s = Bpg-4&)/(g2-l). Finally, choose a
and Ь so that I, a?, b2 are in A. P. A. B. Evans18 took a=5, b = 7 and
proceeded similarly. S. Bills19 employed the numbers а, Ъ=2рд±(р2—д2);
c = pi+(ft whose squares are in A. P., and took ах/у*, Ьх/у2, cx/y2 as the
roots of the required three squares» Then. ax-\-y2— (г-\-у)*, Ьх-\-у*** (&-\-уУ
determine x, y* Then cx+y*= D if a quartie in r is a square, which is the
case if r—s(a+b—c)/Bb). W. J. МШег1* called the numbers xt yy z and
set x-\-l = m*t у+1 = п*> 2-|-l = p2, m+n = r(n-\-p), m—n = s(n—p), whence
m
Then х*+* = 2у* reduces to k*=j(rt s), which is solved. D. T. Griffiths»
took x2— 1, 2/2—1, a?*—1 as the numbers. Their squares are in A. P. if
x2+2/*—2=а(у2+г*—2), y*-z*=a(xz-yT)> Taking a = 1/2 (the value when
the squares are 1, 5?, 72), and eliminating г7 we get 5x*-y2=4. This holds
if z=5, у =11, whence z = 13.
To find three squares in A. P. such that each less its root is a square,
Smyth21 took aV, bV, с*х27 p = l/a, etc. Then z*—px7 »— are made
squares in the usual way» " Epsilon " used the numbers I/A", a/X, b/X,
where X=2x-x* and where 1, a\ b2 are in A. P. Now 1/X2-1ДХ> □.
Again, F—1= П if t= (fc+Q7DM). It is shown that a/X and b/X are of
the latter form if
^ \4ab-(ab-a-b)*}*
X~$ab(db+b-a)(db+a-b)(a+b-aby
To find three squares in harmonical progression the sum of whose roots
is a given biquadrate dA, " Epsilon"n took a, c^2mn±(m*~n*) and
Ь=тп?+п*. Then the squares of A/a, h/Ъ, Л/саге in harmonical progression;
equating their sum to d*y we get h.
A. Guibert28 notsd that the common difference of 3 squares in A» P.
is a multiple of 24, and similar theorems. The general solution in positive
relatively prime integers of a*H-c2=2&2 is stated to be
17 The Gentleman'* M&th. Companion, London, bt No* 25, 1S22,
l» Main. Quest. Edue. Times, 16, 1872, 27-28.
"/buf., 11, 1869,88-91.
"IbuL, 63,1895, 46-7.
я The Gentleman's Math. Companion, London, 5, No. 26,1823,
* The G«ntlemanJe Math. Companion, London, 5, No. 28, 1825* 365-6.
» Nouv. Ann. Math., B), 1, 1862r 213-9.

438 History of the Theory or Numbers. [Chap, xiv
where p, q are relatively prime and one even- Extending the A. P., he
proved that the nth term is a square if </=l, 2p = n—2, or g = 2, p = n^2.
" Civis " 2I proved that the commcu difference of three rational squares
in A. P. is never 17. For, if so, 4ab(a*—b*) = 17g2. Put a=r*, b = s2,
r2_s2=8l^t whence r = 2^-hi, s=2t?-l. Put u=q/(&rsv). Then 17u2-l
= 4t>4, which is impossible in view of the formula for z in the known solu-
solution of 17u2—1 = г\ A. Martin25 noted that the theorem is evident for
integers since a multiple of 4 cannot equal 17.
To find2* three squares in A. P. such that each exceeds its root by a
square, employ Frenicle's numbers (say lt то, n), and take fcr, mxt nx as
the roots of the required squares. Then ftp1—lx=\3t etc., are solved as
in Ch. XVIII.
D. Andre"** noted that, if three squares are in A.P.j
(
-~) +{-^ )=<?+<?, x=a+c, z=a-c
G. R. Perkins2* treated the problems 1 p]: Find three squares in A. P.
such that each less Q)lus] its roots is a square. Take the numbers to be
the squares of $±i, ?jd=J, f±J, where 4£ = xH-x, ^=y-\-y~xt 4f=z+z^x,
and the signs are + or — according as the problem is 1 or 2. Then each
square it its root is a square. The squares are in A» P. if
t r,
These give t, ^, f and n=2m(p-\-l)INt where N^2pBp+1). The desired
numbers are the squares of i±\ = mfal mjbf mfct where
=iV.
It remains to make x, ^, a; rational, using 4£—x+arl, etc. This requires
that ?n2^£m be a square for t=af bf c. Now
if w= -•• • t ;■
ztBfc—a)
Then m2^^n=D if А;2—ЬBк—а) = П, say (ft—JJ, whence A is a rational
function of I Then A2—cBk—a) = П if /-(e+b—c)/2. For Prob. 1T
p>2; ifp = 3,?n/aJ ••-, m/c are quotients of numbers of 14 digits fcf. Hart5
of Ch. XVIII^. Three times as many digits are involved in the answer by
D. Kirkwood,2* who started with x2, 25x*, 49z*. For Prob. 2, the use of
p = 1 gives the answers due to Williams6 of Ch. XVIII.
tl The Lady'H and Gentleman's Diary, London, 1866, 56-7» Quest. 204b
» Math. Queet. Educ. Timee, 52, 1890, 87.
** Ibid*t 14, 1871, 54. A Collection of Diophantine РгоЫеше by J, Matieeon, pub. by A-
Martin, Washington, D. C, 188S, £ 10, pp. 14-16.
17 Nouy. Ann. Math., B), 10, 1871, 295-7.
"The Analyst, 1, 1874, 101-5.
M Stoddaid and Henkle, University Algebra, N. Y.> 1861, p, 404.

Сна*, XIV] ТнЯЕЕ SQUARES IN ARITHMETICAL PROGRESSION. 439
A. Cunmngham" investigated the sets of three numbers < 10000 whose
squares are in A. P. the ratio of the greatest to the least being as great (or as
small) as possible.
W, A. Whitworth" noted that if three squares without a common factor
are in A. P., the middle one is ^ 1, 25 or 49 (mod 120) and each of the others
is ^1 or 49 (mod 240).
J, Keuberg32 and J. Deprez83 investigated " automedian " triangles,
viz., those whose medians are proportional to the sides a, b, c. If a>b>c,
the condition is а?+<?=2Ъг.
G. Bisconcinr54 noted that, if A is the common difference of three
squares x\ in A, P., then x\—x\=A> x\-x\ = 2A. By the latter, xlr
Xs = BAT\*)!(?\). Thus \ = 2аь А=2а1а^ xv=ax~az7 x^a^a*. By
the first condition, x\ = a\+a\> It is stated [Incorrectly35] that av=r>—$2,
az=2rs, whence4=4r5(r*-s2), which he called a number of Fibonacci»
C. Botto*5 noted the incompleteness of the solution by Bisconcini. To
obtain all relatively prime solutions of хг+]р — 2z2t note that x and у are odd,
and set p==(x+y)/2, q=(x—y)/2. Then p*+q*=z\ Since p and q are
relatively prime, p, q^u*-i?, 2uv, and z=v?+ifi. The same substitution
reduces г*—у7 = 2г1 to 2pq=z2, whence p, q=a?, 2b2 and z = 2ab.
G. Me4rodM noted that v2—2t^= ~z2 has the solutions
E. Turriere" noted that the sides of an automedian triangle are
A» G^rardin88 noted that for the automedian triangle with the sides
31, 41, 49, the sum of the sides is a square 121. J. Rose3* noted that by
Turriere's formula, а+&+с=ХC—Р) becomes a square by choice of X.
R. Goormaghtigh40 restricted the last problem to relatively prime integral
sides, whence these axe the absolute values of о*-^±2а^, л*+02. The
perimeter is a square if а*+02+4л0=:и*, whence a-\-2p=v, i^
Th
)/
See papers 15, 35, 62 of Ch. XV; 20 of Ch. XVII; 5, 8, 16 of Ch.
XVIII; 7, 8, 48, 49, 57, 114, 143 of Ch. XIX; 11 of Ch. XXII. On
x2+l=2y\ see papers 112-129 of Ch. IV; 154, 188, 215, 234, 298 of Ch.
XII; 92 of Ch. XXIII.
Papers without novelty on
A. Boutin, Jour, de math. elem>p D), 4 [19], 1», 12 [Victa'J,
Plakhowo, ibid., E), 21, 1897, 95 [Frenicle»].
*° Math. Queet Educ. Timea, 71, 1899, 56»
МШ„ 72,1900, 98.
я Matheeifi, 9, 1889, 261^; C), 1,1901, 280,
u Matheeis, C), 3, 1903, 196-200, 226-30, 245-S.
u Pcriodko d[ Mat., 24, 1909, 157-70.
*!bid2324
"Sphinx-Oedipe, 8,1913t 130-1.
17 L'eneelgnemeDt math., 18, 1916r 87-8.
M L?[ntcrm6diaire dee matii., 23, 1916, 173.
»lbid.t 24, 1917,20-22.
4С1Ш„ 88-90.

440 History of the Theoby of Numbers. есяар* xiv
H. S. Vandiver, Amer. Math, Monthly, 9, 1902, 79-80; othere, 7, 1900r 82-3p 112-3»
A» Gerardin, SphinxOedipe, 1906-7, 95, 161-2 [Victa,' bibliography}
F. Ferrari, Suppl. al Periodioo di Mat,, П, 1908, 77-* tFrenick*]'
A. Gdrardin, Assoc. franc,, 190&» 15-17 [bibliography].
A. Tafelmacher» nnterroediaire dee math*, 15, 1908, 102, 259.
Webch, ibid., 16, 1909, 19, 156 [no novelty in authors cited].
A. Martin, Amer. Math. Monthly, 25, 1$18, 124.
E. Bahier, Recherche „ „ . Trmngtee Rectangles en Nombrce Entiera, 1916, 212-7.
FODE SQUARES IN ARITHMETICAL. PROGRESSION*
Fcrmat*1 proposed the problem to Frenicle May (?), 1640 and stated
(Fermat11 of Ch. XV) that it is impossible* Euler109 of Ch* XXII, P.
Barlow,41 and M* Collins4* proved the problem is impossible.
B. Bronwin and J» Furnass43* took relatively prime squares x2, y2, z2, w\
By yz—x2=^—y2=wi—z*> we must have y+x-2abt y—x=2cdf z-\-y^2ac,
z—y=2bd, w+2 = 2bc, w—z=2ad. By the two values of у and those of 2,
(a+d)& = (a—d)c, {c+d)a=b{c—d). But the g*crd. of the four numbers
aid, c±d is 1 or 2. Hence a+d^5c, a—d= &b, c+d=tbt c—d=ea, 5—1
or 2, € — 1 or 2. These are inconsistent since о is prime to d>
A. Genocchi44 proved the impossibility of 4 squares in A. P. and the
following generalization (of the case p=2). The four expressions^^ (p+1)^
and x41(p—l)y are not all squares if p is a prime 8m±3 such that p+1 and
p—1 admit no prime divisor 4тп-|-1, and xt у are relatively prime.
Several writers46 failed to find a solution»
L. Aubry48 proved by descent the impossibility of 4 squares in A» P*
E* Turriere^ gave a proof.
Numbers in artthmbtical progression all but one being squares*
A. Gnibert'8 noted that if А2, В2, С, LP (all but С being squares) are
in A* P., they are the products by a square of a similar progression of odd
integers relatively prime by twos. From the conditions Л2+С=2В2,
B2+D1 = 2C> eliminate C. Then Dz^3B2—2Ai. The known method of
solution gives
A ^2p*-2pq-~q\ Ъ=*2
A* Cunningham49 found five integers in A» P., four being squares* If
t?t u?2, X, у*, г2 are in A* P*, 1^+32^ By)*, З^+г*- BwJ, which require that
the five numbers be equal (Collins,43 pp. 17-23). Next, let all but the fourth
«Oeuvnsi, II, 195*
"Theory of Numbers, 1811, 257.
" A Tract on the pousible and impossible rases of quadratic duplicate equalities » . », Dublin,
185$, 16* Abetiact in British Asaoc. Reports for 1855, 1856, Trans, of Sections, 4.
The Ladies' and Gentleman's Шагу, London, 1857, 92-6.
<» The Gentleman's Diary, or Math. Repository» London, No. 73, 1813, 42-43.
« Comptee Rendug Pan», 78, 1874, 433-6.
• Amer. Math. Monthly, 5, 1898, 180.
" Sphinx-Oedipc. 6,1911,1-2.
« Ueneeignement math., lfl, 1917,240-1.
« Nouv* Ann* Math., B), 1,1862, 249-252. Cf. Pocklmgton" of Cb.
11 Mfttb Quest. Bduc. timee, B)r 9» 190&, 107-8.

Chap. XIV] SQUARES IN GEOMETRICAL PROGRESSION. 441
be squares, the first three being tfi, w2, x2. As known, t>, x = P—u2T-2tul
y)=F-\-u*. Since the common difference of these squares is b=Abu{P^u*),
the fifth number is w*+Zb=z7'* This has an infinitude of solutions t, u, z
derivable in succession from the minimum solution. From the solution
Vy 132,17s, 409, 23s, there are deduced two solutions in much larger integers.
Squares in geometrical progression.
Beha-Eddinw A547-1622) included (as Prob. 6) among the 7 problems
remaining unsolved from former times: Find 3 squares in G. P. whose sum
is a square. Kesselmarm noted that the problem is impossible since
х2-\-хгу2-\-х2уА = П has no rational solution [Adrain,11* Anderson,114 Genoc-
chi/1* Pocklington1" of Ch. XXII].
To find three squares in G. P. and three numbers in A. P. such that the
three sums of corresponding terms are squares, W. Saint51 took a*, a?z2,
a2z* as the squares in G. P. and 2a-\-lt az*-\-a-\-\, 2ах*-\-1 as the numbers
in A. P. It suffices to make aV+ar^-l-aH-I = □ = (ax-\-x/2J, say, whence
а = \хг— L Others took x\ 4xs, 16x* and either 1, 4x+l, 8x+l or2os+as,
W. Wright52 found three squares zi, а?х1, а*х2 in G. P. each plus its root
being a square. Thus z2+x — Пт хй+х/а=П, х2+х/аг=П, which are
satisfied in the usual way (Ch. XVIII).
To find three squares in G. P. each less its root being a square, X Ander-
Anderson63 took x2, xy} y2 as the roots and хг— 1 = (р—яJ, у2—1 — (q—уJ, which
givexT#. Thenz2^—:вд= П leads to a quartic in p which is solved as usual.
Isaac Kewton {I c.) took {r*/Br-l)\2, r\ Br-lJ as the numbers. The
first of the three conditions is satisfied identically. Take r2—r=n2r2J
whence r = l/(l-n*). Then Br-lJ-Br-l) = П if 2«2-|-2 = П. Set
n = m-\-l. Then 2rc*+2 = (stn+2J determines m.
S. Ward54 found three squares s2, ix2t 16\r2 in G. P., such that if any one
of them is increased by its root, the sum is a square. Take x2-\-x=p2z*.
The remaining two conditions become 2p2-\-2= П, р^З— П, which hold18
if p = 23/7.
der RechenkmiHt yon Mohammed Beha-eddin ben Alboeeain aus АгоЫ, &rabiech
u. delitech voq G. H. F. Neaaelmano, Berlin, 1843, p. 56. French trans], by A. Marre,
Nouv» Ann. Math., 5P1846, 313.
и The Dkry Companion, Suppl. to Ladies* Diary, London, 1808, 36-37.
и The Gentleman's Math. Companion, 5, No. 34, 1S21, 4144.
" Ibid., 5, No. 27, 1824, 274-7r
rt J. R. Yoang'e Algebra, Amer. ed., 1832, 341.

CHAPTER XV.
TWO OR MORE LINEAR FUNCTIONS MADE SQUARES.
Diophantus, II, 12, solved x+2=\3, x+3 = D (the first instance of
a " double equality ") by resolving the difference of the two linear functions
into two factors in a suitable manner; here he took 4 and 1/4. Take the
square of half the difference of the two factors and equate it to the smaller
expression, whence 225/64=^-1-2. Or equate the square of half the sum
of the actors to the greater expression» To solve without using a double
equation, take x^y2—2 and make x+3=^+l a square, say by equating
it to (y— 4J, whence у *=15/8.
Diophantus II, 13 relates to 0—s=D, 21-x = D; while II, 14 relates
to x—n=m,x—m=EL
Diophantus, III, 5, 6, required three numbers such that their sum is
a square and the sum of any pair exceeds the third by a square. Hence
the sum of the three squares is a square, as for 4, 9, 36.
Diophantus, III, 7, 8, required three numbers whose sum and sums by
pairs are squares. Let the sum of all three be (z+1J, the sum of the first
and second be x2, the sum of the second and third be (z—1)*. Then the
sum of the first and third is бз+l and equals 121 if s = 20.
Diophantus III, 9 relates to three numbers in arithmetical progression
whose sums by pairs are squares. Since x2, (x-\-l)\ (x—8)* are in A.P.
if x = 31/10, we seek three numbers whose sums by twos are the numbers
961, 1681, 2401 just found.
Diophantus III, 10 relates to three numbers such that the sum of any
pair of them added to a given number a gives a square, and such that the
sum of the three added to a gives a square. For a = 3, take the sum of the
first two to be x*+4z+l} the sum of the last two to be гг'+бх+б, and the
sum of all three to be я*+8я+13. Then the numbers are 2я+7, х*-\-2х—6,
4z+12. The sum 6x+22 of the first, third, and я, is the square 100 if
г = 13. In III, 11, a is negative,
Diophantus, III, 18 and IV, 35, noted that his method does not make
ах-\-Ъ and cx-\-d squares if a : с is not the ratio of two squares»1
Diophantus, IV, 14, made x+lt y+1, x+y+1, x-ij+1 squares.
Diophantus, IV, 22, found three numbers in G. P., the difference of
any two being a square. In V, 1 [2], he found three numbers in G. P.
such that each less [plus] the same given number is a square.
Diophantus, IV, 45, made 8a+4 and 6z+4 squares by subtraction.
Diophantus, V, 12, 14, treated the problems to divide unity into 2 or
3 parts such that, if the same given number is added to each part, the sums
will be squares [see Chs. VI, VII].
i Cf. G. H. F. Neraelm&nn, Algebra der Griechen, 1842, 335-40. Cf. 86 of Ch. XIX.
443

444 History of the Theory of Numbers. [Chap. XV
Brahmegupta2 (born 598 АЛХ) made ax-\-l and Ъх-\-1 both squares,
viz., of {Za-\-b)!(a-b) and (а+ЗЬ)/(а-Ь), by taking х = 8(а+Ь)/(а-ЬJ.
He made (§§ 80-81, p. 360) x+yy x—y, xy-\-l all squares by taking
whence
He made (§§ 82-85, pp. 370-1) x-\-a and x-\-b squares by taking
x-[l\
whence
To make ax-\-b a square (§§ 86-87, pp. 371-2), put it equal to an arbi-
arbitrarily assumed square and solve the equation for x.
Bhascara* (born 1114 A.D ) made Zy-\-l ^nd Sy-\-l squares by equating
the first to Cn-hl)S whence %+1 = 15теН10л+1 = а forn = 2or 18.
Alkarkhi* (beginning of eleventh century) solved x-\-\0=y*7 x-\-l5=z*
by settings=^+2/3 in y*-\-5=z*] also, х+3 = у*ух-\-5=г*Ъугэкш%г-\-у*~±7
-2/= 1/2.
G. Gosselin5 found three numbers A3/9, 133/9, 253/9) in A. P. which
become squares when increased by 4; three numbers A/9, 15/9, 48/9)
whose sum is a square, the first a square, and the sum of the first and either
of the other two is a square; four numbers B5, 16, 12, 11) whose sum is a
square, while the excess of the first over the second, second over third,
third over fourth are squares,
Rafael Bombelli8 required three numbers, the sum of any two of which
increased by 0 and the sum of all three increased by 6 are squares. He
gave 384/*, SS1^, 14^/юо. Не found (p. 458) a number which added to
4 and to 6 makes two squares.
F, Vieta7 generalized the method of Diopbantus III, 10 [11]. If the
numbers are xt y> z, let
х+у={А+ВУ-а, y-\-z
Then
say F2, by choice of a rational A.
* Brahme-ephut'a-eidd'hAnta, Gh- IS (Algebra), $§7!v-79. Algebra, with arith. and mensura-
mensuration, from the SaDecrit of Brahm^gupta and Bhdacara, travel, by Colebrooke, 1817,
pp. 368^9,
» Vfj&rgafl'ita, 5 197; Colebrooke,5 p. 259.
* Extrait du Fakhri, Ftench traiul. by F. Woepcke, Paris, 1853, 86r 101»
* De Arte m&gna, sen de oocuHa part© Oumeronun, PariB, 1677, 7^-5.
« L'algebra opera, Eologaa, 1679, 496.
1 Zetetica, 1591, V, 4£5], Francisci Vi^tae opera mathematics, fed. Franciflci д Schooten,
Ld. Bat., 1646, p. 77.

Chap. XV} IjKEAK FUNCTIONS MADE SQUARES. 445
C. G. Bachet* treated ах-\-Ъ=С\, <кг+с = П, by folding two rational
squares whose difference equals Ъ—с. То solve &г+4 = П, dr+4=EI,
take the double 4 of the side 2 of the common square 4, and the difference
2x of the left merabera, and one fourth of 2s. Then the square of i(§x+4)
equals &г+4 and the square of \{\x—4) equals dr+4. By either condit!on,
x=H2. Next, let the constant terms be distinct squares, as in 10я-|-9= СИ,
5z+4= EL Seek two numbers E and 1) whose sum is double the root 3
of the larger square and whose difference is double the root 2 of the smaller
square. Take one of these numbers 1 and 5 as one of two factors whose
product gives the difference 5x-\-5 of the given functions. From x-\-1 and 5,
we get
(^)! (^У = 28.
But the factors 5x-f5, 1 give {§Ex4-6) !2>10я+9. Next, for 65-6* = □,
65—24x = □ 9 multiply the first by 4 and we have a problem of the first type.
For 16—х = П, 16—5х=П\, seek two squares whose difference is the
quadruple of x. Take 4—N as the side of the larger square» Then
D-ЛО*-4{16-D-Л02}=16-40ЛГ+5ЛГ* is the smaller square, say
D-7JVJ, whence ЛГ=4/11. Thus the squares are D0/11J and A6/11J,
one fourth of whose difference gives x=336/12L [Bachet here used the
same letter for x and N and put 4—UN erroneously for 4—7N.2
Fermat,* commenting on Diophantus III, 10 and V, 30, desired four
numbers such that the sum of any pair increased by a given number a
gives a square. Let a=15. The three squares 9, 1/100, 529/225 are such
that the sum of any pair increased by 15 gives a square (as found by
Diophantus, V, 30, who took 9 aa one square and solved х2+24 = П,
=П, х2+у*-\-15 = П). Take as the four numbers
(the last three being of the form 2nx+v?)\ Then three of the conditions
are satisfied identically. The remaining three conditions are
a "triple equation'* in which each constant term is a square. To treat10
such a problem, x+4= П,2х-|-4= □,5х+4— □, replace x by an expression,
like x*-|-4x> which if increased by 4 gives a square» Then it remains only
to solve the "double equation1' 2x2+&r+4= □,5x2+2Oc-|-4= □, from one
solution x = c of which we can deduce a second, by replacing x by x-\-c.
Fermat11 later explained this method in detail. It is stated (§§9-11)
that the method fails for
A)
8 Diophanti Alexandra Aritb., 1521, 435-9. Comment on Etiop., VI, 24 (p. 177 above).
• Oeuvreer I, 292, 326-7; French tranaL, III, 242, 263-1.
10 Оешггея, 1,334-5; Ш, 209-270, Cgmment on Dbphantua VI, 24. For further examples,
11J. de ВШу, Inventum Novum, Toulouse, 1670, Part II, SI 1-28; German trand. by P.
-von Schaewen, Berlin, 1910; Oeuvns de Fennat, III, 360-374 (p. 329 for2i+12 = Q,
2+5D)

446 History of the Theory of Numbers. [Chap, xv
Thus, if a=2,6=3, we substitute 2z*-\- 2x for x to satisfy the first identically;
then the other two become ftcHftr+l-G, 10x2+10x+l = □, one solution
beings — —1; but this makes the unknown 2x?-\-2x zero [von Schaewen*1].
Although the method fails for a=5, b = 16, s = 3 is a solution. For
a=l, 6»= 2, there is no solution, whence foursquares (the firstJbeing taken
as unity) cannot be in A. P.
M* Petrus14 found three squares A*, B2, C* such that the difference of
any two is a square and the difference of the sides of any two is a square.
He first gave a process to find four numbers pt 8, t, q such that £*+«*,
P+q2 and psiq are squares, while p/a>*/#, solutions being 112, 15, 35, 12
and 364, 27, 84, 13. From the former he derived the answer to the first
problem:
A =26633678, £ = 29316722, 0=40606322.
In general, we have the answer18
since
С~А=Црд+81)\ В+А
C-B-lQpt&q, C+B
Renaldini14 A615^-1698) treated Petrus'1* initial problem (in Fart II)
and (in Sec. 1 of Part III) duplicate and triplicate equalities.
J. Prestet16 treated the problem of Diophantus III, 7. Let the sum of
the first and third be s2, that of the first and second y2, that of all three г2.
Then the numbers are x2+y2—z*t гг—хг7 z*—y*> The sum of the last two is
not easily made a square» Since 2 = 1/25+49/25, set x = zj&. Then the
sum of the last two is 49г'/25-у2-(а-7*/5)г if г = 5(вЧЛ/И. But the
numbers obtained this way are larger than those of Diophantus and Vieta.
For Diophantus III, 9, he used (p. 326) z, г-yd, z+2d} with 2z+d=y\
$d*f which give s and d. To avoid fractions, multiply the numbers
by 4. Hence the numbers are 3yz—x*7 y2+z2, — y*-\-Zx2. It remains to
make the sum 2#2+2z2 of the first and third a square» Express 2 as a sum
of two squares, the smaller between 1/2 and 1. By Diophantus II, 10,
the root of the smaller is (ci-2c~l)/(c2+l). By trial, 9 is the first integer
с giving a fraction C1/41) >3/4_ Thus 2C12+492) = 82*. Hence x2 = 2401,
i/* = 96I. He gave also a less special solution. He treated (p. 329)
analogously Diophantus III, 10»
J. Ozanam" found two numbers, such that each when increased by a
square (say unity) gives a square, and such that their sum and their dif-
difference increased by another square (say I2 = z?+2x+l) shall give squares.
The required numbers are taken to be 168^ and 12W. Then the final
a Arithmeticae Rational!* Mengoll Petri, Boncmiae, 1674P 1st Pref, Cf. Eider."
18 Reconstructed from the author's inadequate notes on Petrus.
"Caroli Renaldimi Matbematum Analytics^ Artie Para Terti^ 16Я4; reviewed in Act»
Eniditorum, 16S5, p. 178.
11 E3^mens dee Math, ou Principea Generam . . ., Paris, 1675, 325.
» Utter, Oct., 13,1676, to de Bflly, Bull, Bibl. Storia Be. Mat. e Fie,, 12,1879, 517.

Chap. XV] LlNEAB FUNCTIONS MADE SQUARES. 447
conditions are satisfied since 168+120+1 = 17*, 168-120+1 =7*. To
make 168^+1 and 120/2+l squares, we have a double equality, satisfied
by x - -1648825564/1242622079.
G. W. Leibniz17 discussed the problem to find three numbers the sum
and difference of every pair of which are squares.
M. Rolle" found four numbers the difference of any two of which is
a square, and the sum of any two of the first three is a square:
D = A+B+C.
For y«l, z=2, A =2399057, B-2288168, 0=1873432, £=6560657.
T. F- de Lagny19 solved 4x-\-G = y*, 9x+tt=z2 by a " new method.'1
Eliminating x, we have 9y7/^—l/2=zK Hence 9y*—2= □ , say the square
of Zy - a. Thus у is found in terms of a.
P. Halcke2* divided б into two parts such that each increased by 6
gives a square, and made 6+z, 12—x both squares.
Malezieux*1 proposed the first problem of Fermat.* It is a question of
finding three equal sums of two squares.
The problem to find three numbers the sum and difference of any two
of which are squares received at the time of its proposal no comment
except the mere statement by С Bumpkin42 that 1873432, 2399057, 2288168
furnish an answer.
J. Landen" took as the numbers
Then z±y, z±z, y—z are squares. It remains to make
E=fV-g*-P±l
a square. Set g=f+r. Then E= □ if
which gives r and hence 0=/(/б+6/*-3)/A+6./ч~3/6). The case /=2
gives Bumpkin*&* answer. Or we may take /V+l, P+g*, 2fg as the
numbers, whence x±zf y±z are squares. For the preceding value of g it
is verified that /V±(<^+/*) + ! are squares, whence their product E is a
square. Or we may make E = {fA—l){gA — \) a square by equating it to
(/4V whence j=f/V2^*. Set/=l-d; then 2-/4 becomes a
17 MS. dated Apr. 1, 1676, m Bibliothek Hannover. D. Mahnke, BibHotheea Math., C),
13r 1912-3, 39. a, Euier."
11 Journal dee Savans, Aug. 31, 1682; Sphinx-Oedipe, 1906-7,61-2, Cf- Coccoz," Rignaux.»
19 Nouv. Elcmcns d'Aritk. et d'Algebrer Paris, 1697t 451^5.
» Delieiae Math., oder Math. Sinnen-Confect, Hombiu^, 1719, 235.
" Elements de Geom^trie de M. \e Due de Bourgogne, par de Malezieux, 1722; Spbiox-Oedlpe,
1906-7, 4-5, 45,
« Ladiea* Diary, 1750, p-21, Quest. 311. Cf. Euler »
21C- Hutton'e Diarian Miscellany, extracted from Ladies' Diarjr, 3, 1775, 39в-401, Appendix.
Leyboum's Math. Quest, propoaed in Ladies' Diary, 2, 1817, 19-22. a. Euler »

448 History of the Theory of Numbers. [Chap. XV
quartic in d which is the square of l-|-2<2—5d2 for d—12/13. 0. Wildbore's
solution is the same as Landen's second with /*=a/&, g=xjy. C. Hutton
took 4z, 4-\-x*, 1+4я2 as the numbers. Then 5V+5 and Zx2—3 are to be
squares» The product 15x4—15 is a square for x = 2, and for x—z—2
becomes a quartic in z which is made a square by the usual method» He
obtained Bumpkin's2* answer» T. Leybourn24 took x-\-y = u2y x-\-z^x?}
y-\-z = v>*\ it remains to make u*—tf2, v*—w2)v?—'to2 squares, which is known26
(Lowiy** of Ch. XIX) to be the case if
P. Cheluccii* treated Diophantus III, 7. From x+y+z=*r*f х+у-8г}
x+z=&, y+z = v* follow x = ?—r2+$\ y=r2-P> z = r*-s*} 2r*—Р-з*=1?.
8ett=r-m7 8 = т-п. Then т^^+пё+п2) I {2rn+2n)>
L» Euler27 treated the problem to make х-Ьа, x±b, z+c all squares.
Set x=z2—a, z=p/q, b—a=m, c—a=n* Then p*-\-mq2 and р*-\-щ* are
to be squares.* This is impossible if m——n=P or 2Д and if m = I, n = 2»
Several solutions are found when m=2, л = 6» In §§ 213—8, pp. 26*4-271
(Opera, 446-9), he made x+a7 х+Ъ squares, also a-\-x, a—x*
Euler5** treated the problem to make x±y, х±г, y±z all squares. Let
y=x—p*> z^x—ф, and vi-\-r2=q27 whence y—z = r*y у-\-г=2х—^2—д^^
Equate the last sum to P, whence 2x = t2-\-p2+f. It remains to make
x+y=&+q* and z-\-z = F+p2 both squares. To satisfy p2+72 = q*J take
p = a?-b\ r=2db, q = a?+b?. To make P+g2 and t2+p2 squares, viz.,
^+a4+b4db2o8b»=Q, it suffices to make *s+a<+b4 = c2-f-#, 2a2b2=2cdt
which are satisfied if a^/fc, Ъ = дк, с—Рд*7 d-h2k\ and
B) 12=(Г-к*)(д*-Н<).
By means of a table of values of m*—nA for пг^15, n^9, n<m, he found
the solutions 5202 = C^24)(9l-74) and 9752 = C<-2<)(U4-24) of B)
and hence
x=434657, у=420963, г = 150563,
x=2843458, у = 2040642, г = 1761S53.
J. L. Lagrange* treated a-\-bx = P9 c-\-dx=u* by eliminating x; thus
the second member being made a square in the usual way. To make
u*, hx-\-Jcy=s2>
u Math* Quest, proposed in Ladies' Diary, 2, 1817, 19-22.
* New Series of Math- Repository (cd.r T. Leybourn), 3, 1814,1, 163, Quest. 310.
* Icatitutiones analyticae, Vienna^, 1761, 135.
"Algebra, St. Petersburg, 2, 1770, $223; French tmnfll., Lyou, 2, 1774, pp. 281-5. Opera
omnia, A), 1, 454-6. Cf, Haeiita&cheP» of Ch. XXII and paper 82 below.
* EiiWe further diecuteioQ will be given under concordant forma, Ch* XVI.
* Algebra, 2, 1770, 5 235; 2, 1774, pp. 314-fl. Opera Omnia, (l)r 1, 470-3. Same problem
in papers I2r 14, 17,18, 22, 23, 24, 30, 33, 34, 57, 74, 85, 89» See papers 40^45 of Ch.
XIX
«Addition VI, arts. 62-63, to Eider's Algebra, 2, 1774, 657-561. Eulerb Opera
A), I, 595-7. Oeuvres de Lagrange, VII, 115-7-

Chap, XV] LlNEAR FUNCTIONS MADE SQUARES. 449
eliminate x and yt and choose z^ujl so that
ak^bh л ck—dh ,_,
ad—со ad—со
In the " Kepository solution of the problem to find three numbers the sum
and difference of any two of which are squares,0 Idb5x—2хй^2х?-\-5хА±х*
are taken as the square roots of the sum and difference of the first and second
numbers, while li&r-f-fo^far'—Зх4^5 are taken as the square roots
of the sum and difference of the first and third numbers» Hence the three
numbers are determined. Неге х is any square» Taking x=9r we get
numbers 4387539232, etc», of ten digits each.
C. Hutton*1 noted that y±l = □ if y = 4x2-4x> Then $y+l~Bax-l)*
gives x.
Euler32 solved the problem to make z—a2v, • - >, z—d?v squares, where
a2, * •', d* are four given squares, by investigating a quadrilateral the sines
of whose angles p, q, r, s arc ax, - • -, dx, where ar - - - f d are given numbers»
Let Л, — », D be their cosines» Since sin (p+g)-|-sin {r-\-s) =0, etc», we
get aB-\-hA-\-cD-\-dC—0 and two similar relations obtained by inter-
interchanging by cf and By C] or bt d and B, D> Hence we get the ratios of
At -•»,£) as cubic functions a, • ■ -, 5 of o, * —, d» Thus Л = c^, • - -,
D-%. TlieiiaV+aV = l,b2x2+/9VeI,ftndwe find that x2 = v!z,y2 = llz,
where
г=4 Fc—ad) (ac—M) (ah—cd)»
Hence
ОС
COS ©=—F,
Euler33 required three numbers xt y, z such that the sum and difference
of any two are squares. Let x>y>z and set
C) x=p*+q*~r*+s\ y = 2pq, z = 2rs.
Then x±y = (р±аУ, x±z = (rd=5J» Also p2-\-q2=r2-f-s* if
D) p=ac+bd, q=ad—hc, r
Thus x = (a*+b*)(c2+<F). It remains to make
both squares. Their product is a square if
Take d=a. Then а2=(п2№-с?)/(п2Ь-с). Take a-6=fcc, and take b equal
to the numerator of the resulting fraction for 6/c» Thus
" The Diarian Repository, or Math. Register . „ . by a Society of Mathematicians,
1774, 522-3- Cf- Eiiler,"
11 Miecell&Dea Math., London, 1776, HO»
« Мёад. Acad. Sc. St, Peterab., 5, anno 1812, 1815 A780), 73; Comra. Aritlu, II, 38Q-5.
" Ibid., 6, 181ЭЧ A780), 54; Comm- Aritb,, II, 392-5. Cf» Eul^r.»

450 History of the Theory of Numbeks. [Chap, xv
It remains to make у—г a square. Since d=-a,
ab№-c2)=3n4=bl)B=Fn5J.
Choose the lower signs» Then 3(n2— 1) is the square of (n-\-l)f/g if
Multiply the resulting values of а, Ьу с by Cgz—/*J; we get
For f—g = l, we get p—</=5, r=7, 8 = 1, whence x = y=50, z= 14. From
one solution x, #, 2, we get (§ 15) a second solution
E)
X ^ ■ У 2' Z=2"
In the " additamentum u (§16), Euler treated the problem to find
three squares x*7 y2, zz whose differences are squares» Using C) and D),
we have
the last being a square if obcd{a*-ht){<P-ct) = □. This is satisfied if
From one solution we get a second by E).
E. Waring" noted that, in the problem to find three numbers the sum
and difference of any two of which are squares, four of the conditions are
satisfied if we employ either of Landen's*3 notations for the numbers or
the notation a^+fcV, 2atey, а*уг+Ъ*х2, but gave no discussion» He
recalled Rolle's18 values A, B} С
Euler35 treated the problem to find four positive numbers m arith-
arithmetical progression such that the sum of any two is a square:
Hence all are expressible in terms of p, qy r, subject to two conditions
Thus r= Ш ~x2+y\ We get 2r*= 12 and satisfy 2r* = p*+** by taking
the first term being positive, whence p<L Similarly, we satisfy 2r*
by taking r=xt+y\ and
Then x2+y%=xl+}fi is satisfied by taking
x=fz+l, ^=/z-l, y=z-f, yi = *+f,
as may be done without loss of generality by removing a common square
« Merfitationea AJgebraicae, ed. 3t 1782, 328.
"Poethumciue paper, 1781, Comm. Arith,, II, 617-25; Opera postuma, 1, 1862, 119-127.
Reprinted, Sphmx-Oedipe, 4f 1909t 33-42.

Chap. XV] LINEAR FUNCTIONS MADE SQUARES» 451
factor from our numbers» Then At B} C, D are all positive if p2-\-<p>raf
a condition expressed in terms of/ and z and treated at length by Euler.
For z = 4,/=7/2, we drop the factor 1 /2 and get x = 30, х1 = 2в, у = 1,2/1 = 15,
p-839, $ = 329, r=901; multiplying the resulting A7 ■ - ■ , D by 4p we get
the integral solutions
722, 432 242, 2 814 9G2, 3 246 482.
J» Leslie36 made 2+1-П, »+1 = П, г-И+l =given □ by setting
z=2--l, v=y2— I.
P. Cossali37 made F=/u-|-n2 and F+/x squares by taking F=(y-{-n)qy
thus finding #. Next, if (ad—bc)f(a<-c) is a square r2, ox-j-b and cx+d are
made squares» Set cx-$-d = (y+rJ; for the resulting a:,
\ (
с
If (bc—ad)/c is a square $г, set cx-bd = ^2; then <ix-{-b = y2a[c-{-q2 can be
made the square of q—ky. To make (pp. 14545) H+x — □, Я—x = D,
according to L. Pisano, we have only to express 2H as a sum of two squares.
To find three numbcre in geometrical progression the difference of
any two of which is a square, R. Nicholson35 took nxt n2x, n3x as the
numbers» Since the ratios of their differences are 1 : n+1 : n} take n=tf,
t2-f-l — П = (u+$J. For the resulting v, n— 1 is a square» Takings to be a
square, we get an answer» J. Cunliffe took naAt «a2^2, nbA as the numbers,
where n = dr—Ь2\ the single condition а?-\-Ъ2=П is satisfied if b = r2-$2,
a=2rs.
To find three numbers in geometrical progression whose sum is a square,
several20 took x\ nx2, n*x\ 1+я+п*-= □ = (ne-l)\
To find40 three numbers the difference of any two being a square,
take x—y = l$v*y x—z = 2bv*} y—z=9v*, where v and z are arbitrary; or take
5x\ x2, ¥+x% where 4x*-b*= Bx-ny gives x; or take (r+1J, 2xH-l, 4x,
where 2j;—1 = D.
J, Cunliffe41 made x — y, etc., and x+y—z, etc», squares» Take
x-\-y-~z=d?7 x-\-z—y—b2y у+г—х = с*.
Equate x-y = i(b*-c?) to e\ x-z = l{a*-c2) to d\ Then
y-z = l{a*-b2) = d2-e2
must be a square, whence d = 2rs(m2-\-ni)t e = 2r&Bmn)> Set
(a+b)r = 2s{d-\-e)y (a-b)s=r(d-e)y
which give
* Trans. Roy. Soc. Edinb., 2, ПЩ 1»3, Prob. IV.
" Origins, Trasporto in Italia . , * Algebra, t, 1797, 106-7.
11 The Gentleman's Diary, or Math» Repository, 1798, No, 68; Davia* ed, 3, 1SU, 290»
J> The Gentleman's Math. Companion, London, 1, No. 2, t799p IS,
"Ibid., 2L
41 The Gentleman's Diary, or Math. Repository, London, No, 61, I8OI, 43, Quest, 80Й»

452 History of the Theory of Numbers» [Chap, xv
T&ke с=2тп(г2+2$*)-(т>~п2)(г*-2а*). Then c? = b2-2e* gives
CunlifiV2 treated the last problem and Prob. 8: Divide n into four parts
the difference of any two parts being a square. Also Prob. 9: Find four
numbers whose sum and sums by twos are squares»
R. Adrain41 made two or three linear functions rational squares as had
Lagrange.*
Several44 found two numbers such that if unity be added to each and to
their sum and difference, the sums are squares. The numbers x^2x
answer the first two conditions. Then 4х+1~П=р*, 2s2+l = El3. Take
p=r+l. Then 16{2^-hl)-(r2H-4Jif r--8, whence the numbers are 120,
168.
S. Johnson45 found integers x, у, г, v such that their sum and the sum of
any two are squares and 2(v+z+y) = □- Set x-\-y+z+v=a*y х-\-г = Ъг7
y+z=c?, z+y^<P. Thus 2z = b2+c?~<P. Then v+x=az-y—z^a2—c2,
v+y = a2—Ъ*у v+z=a2—d2 must be squares» Set a%—<?=e2y a*—<P=P>
c=rp-ft e=sp-\-d. Then сЯ+е2-^/2 gives p = B7/-2sd)/Cr2+s2)> To
obtam integers, take/=(n2-m1)(r2+82), d = 2mn(r2+s2). Then
By tf-dH/2, a = {?i2+w2)(r432)- Thus а*-Ы=П if Ъ= (п*+т)>2г$.
Finally,
if n :m=2ra{r2+si¥ le^
Johnson46 used the same methods to find x} y, z, v whose sum is a
square and difference of аду two is a square. J. Cunliffe took v—s^c2,
v—y = b*, v~z*=a2, v+x-\-y-\-z=n; it remains to make z—y=b*—c2,
x—z=a2—c2, y—z=az—b2 squares. Hence we desire three squares a2, b2, c2
the difference of any two of which is a square. This is stated to be true if
a2=485809, ^ = 451584, ^=462400.
The problem4*" to find three numbers in A. P., the sum of any two of
which exceeds the remaining one by a square, reduces to x2-\-z2 = 2yz (Ch.
XIV).
J. Cunliffe466 found two rational numbers (x*+n and y*+n) such that
each and their sum and their difference exceed a given number n by squares.
The condition x2+y2+n=O = (x+vJ gives x in terms of y, if, n. Then
s«-lf-*«ae(n-»*-y*)VDt*) if n'-2m^a = (rt>-n)S which deter-
mines v.
«The M^th. Repoeitory (ed,t Leyboum), London, 3, 1804, 97-106,
^ТЫ Math. Correepondent, N«w Y«kf 1,1804, 257-241; 2, 1807, 7-11.
« Lftdiee1 Diary, 1804, pp. 38^9, Quart. 1111; Leyboum'a Math. Queet. L. D., A, 1817, 23.
« The Geatlemao'fl Math. Companion, London, 2, No. я 1805, 46^-8.
*1Ш., 2, No. 9,1806, 35^6\
♦*• New Seriea of Math. Repository (ed., Leybourn), I, 1806r I, 7-10.
«•/Wrf.,2, 1809,1,9-11.

Сяар. XV] Linear Functions Made Squares. 453
C. F. Kausler4* treated the problem to divide a given number a into n
parts such that the sum of any n—X parts shall be a square. [The treat-
treatment by Diophantus, V, 17, of the case n=4 was given in Ch. VIII.]
The treatment for n is similar to that for his first case n = 5. Then, by
addition, the sum of the 5 squares s?, - - -, $1 is 4a. First, find a square P*
approximately equal to 4a/5, say
Since every number is a sum of 5 squares, set
Thus a* = M-giN. Since 2s* = 4a, we get 2=-~2Zi,tt</2"<- Thus, if
a=21, the nearest square root of 20a is m—21, whence z = 2, Af=41, №=10.
Since 4*1 = 1+9+25+49, 1 = {9+16)/25, the $'в are 3/5, 4/5, 3, 5, 7, the
«'дагеЗб, 33, 11, -9, -29, and 3 = 1676/16785.
To find48 three numbers x, vxf v*x in geometrical progression such
that each increased by a given number n is a sqnare. From z+n=c?f
w:+n = (d+c)8, we get x, v. In the resulting value of *Рх-\-щ put c?—n=r3;
then
d. The desired numbers are r2, Jr2—ti, (£r2—/iJ/^, where
r= (rt-$*)/{2$) makes r2+n= □ ^c2.
Several451 found four integers whose sum is a1 and excess of the sum of any
three over the fourth is a square Ь\ с2, d2 or e2. Hence ^-h^+^+c3—2a2,
which determines a rationally if we take c=p—a, d=q—a.
To find*9" two numbers (i?—n and w2—n) whose difference is a square
and such that if each and their sum be increased by the same number n
there result squares, we have to make tP—w* and tP+w*—n squares and
hence a certain quartic function a square.
J. Wlnward50 found N integers whose sum is a square m2 and sum of any
tf-1 of them is a square. Take Bm-n)nf Bm-2n)Bn), Bm-3n)Cn),
■ • *, [2m—(N—l)n](N—l)n as the first N—1 numbers, and m2 less their
sum as the Nth. Then m2 exceeds the jth number (j<N) by (m— jnJ.
Equating the excess of тг over the Nth number to (nr)\ we get m in terms
of n, r.
Several51 solved х-\-а? = П, г/п-$-а* = П by known methods»
To find*2 four integers whose differences are squares, let x = 2lmn,
у=1(т*—п*). Then five of the differences of u, t/-bx2, и+х*-{-уг9
47 Мёю. Acad. Sc St Feteiaboai& 1, 1809,271-282.
u The Gentleman's Math. Companion, London, 2, No. 13,1810, 264-5.
" The Gentiemaii'e Diary, or Math, Repoetany, London, No. 71,1811, 35, Quest. 963. For
3 numbetB, Gentleman's Math. Companion, 5, No. 29, 1826, 362-4.
«» New fence of Math. Repository (ed^ Leyboum), 3,1814,1, 105-8.
и Tbe Gentleman's Math. Companion, London, 6, No. 25, 1823, 141-2.
» Ladies Diary, 1823, 35-36, Quest. 1390.
"ibeGentlemaii'fl Math. Companion, London, 5, No. 25, 1823, 202-4.

454 History of the Theory of Numbers* [Сна*, xv
rt*)* are squares. It remains to make (JW+n2J—Р(тп*-И12)г= D.
Take 1=2. Then 3Dw4—n*) = LL From the case ?й=л—1, we get the
new solution m=37, те=23 by Killer's67 method of Ch. XXII-
W. Wright" found three numbers у3—1, x2—1, y*— 1 whose sum is a
square, each plus unity is a square, and the sum of the roots of the latter
squares is a square. Take tP+xt+y*1—3— (v+pK, v-Vx+y—q*.
To find three numbers such that the sum of the first and second and
difference of first and third are squares, the sum of whose roots shall be
a square and equal to the sum of the required three numbers, F. N. Bene-
Benedict5* took the latter to be а?х*—х\ x\ (b*+a2—l)s*. Then ах+Ъг^сх*
determines xt where c^2a*-bb2—1. Finally, c= □ = (&—mJ ipves b.
Several56 found three numbers x2—1, y*—1, 2*— 1 in arithmetical pro-
progression, whose sum is a square and each plus unity is a square. Use the
known solution x, z = zt{rn2—n3)-\-2mn; у = т'1-\-п* of x*-\-z2 = 2y!i. To
make з?+У*+23-3=3(™2+^)*-3 = П, take n-1 and solve Зтп2+6*=П
аз usual.
W. Wright6* found three integers x, у, z, double the difference of any two
being a square, also double the difference of the sum of any two and the
third. First, solve n(a2-b*) = p2, «(c2-^)^2. Since р*-д2=п(аг—c2),
take a+c— 0>+9)^/(vn), a—c — (p—q)vlt, which |pve at c. Then pz—q^= □
if р=2*мг(еР+£2), q=2tvn-2de. For brevity, set г^Ч-л»2, $^*2-m>*.
Then a=r(d2+e2)+2des, с=5(еР+е^+2йег. Then n(c"-bl)«g2 or
c?—if In— П, becomes a quartic in d, which is satisfied if d = 2rselDtiv2n—s2).
The case rt=l/2 leads to a solution of the initial problem» Set
2{х+у-г)=>а2, 2(у+г—х) = Ъ\ 2{x+z-y)^<?, which ^ve x, y, z. Then
tbe init al three conditions require that Цс*—Ь2), > • • be squares.
J. R. Young67 treated Diophantus III, 7, 9 somewhat as had Prestet.1*
To make (pp. 347-51) x=fcy, xzkz. y±z all squares, take x+y = u2,
Then x^y^if—w1, x—z=u*—wz are squares if
d^49, we get Euler's2* first answer, believed to ipve the smallest possible
numbers. Or we may make а^Е^ — П by taking o^m2+n2, Ъ = 2тп and
similarly for c2—-d2. Other methods are based on the choice
He {p. 345) treated Diophantus IV, 14.
F. T- Poselger58 treated Л =* П, В = □ for the case in which A - В is
factorable into pq (cf. Diophantus II, 12)- We may set
since
»ТЪе GentlemaD1* Math. Companiob, LondoDt 5, No. 28, 1825, 369-71.
« The Math. Diaiy, New York, 1,1825, 27.
« The Gentleman's Math. Companion, LoDdoii, 5, No. 29, 1820, 361-2.
« /bi^t 5, No. 3(^ 1827, 574--5.
^Algehm, 1816. American edition Ьу S. Waid, 1832,324-6,355-6.
» Abh- AJumL W». BerEn (MAth.), 1832,1.

Сна*. XV] LlNEAH FUNCTIONS MADE SQUARES. 455
S. ByleyM found three numbers whose sum, sum of any two, and dif-
difference of aoy two plus unity are squares. Take x+y^a2, x+z = b*f
1. The remaining conditions reduce to
Then 4b* = nr—2r*=(n—rmJ if n = r{m2+2)/Bm). Take r = 2m. Then
4а*=л'+2г2—4 = П if т*+12=П = (в—m)\ say. Several used the num-
numbers 2x*+2y*-%, 2s*-2t/2+i, 20*-2s*+i, which satisfy five of the condi-
conditions. To satisfy itf—ty+l^v*, take x+y^v+1, 4(x-y) = v—l. For
the resulting x, y7 16B^+2y*+§) = lW+30t>+25 = (av-5)\ by choice of v.
Ft» Buchner* solved s+1 =* p2, x —l = g* by setting p+q = m, p—q = 2(mt
and simlarly for z-\-a=p*t x—b^q*.
T. Baker61 found four numbers p2—s, g*—s, r2-*, 5 such that the sum
of any two is a square, the difference of any two increased by a square r*
(which is to be found) is a square, and the sum of all four diminished by r2
is a square. Set 2* = ^—U We need only m&ke p2+t, q*+tt r*-K
A — ^-q2+r\ B^p*+q2—r2+t squares. Equate the first three to the
squares of y+t/xt q+t/yt r+tjz respectively, thus finding p> q7 r. Then
Л= \p—v(q+r)\2 determines t, and В^П holds if
S. Jones*2 found four positive mtegers xt у, г, у+г—х half of whose sum
is a square, the sum of any two is a square, the difference of any two in-
increased by a given square e2 is a square, and the sum of the four diminished
by e2 is a square. Take x+y=a2, х+г=№, y+z^c?f 2
y+2z—x=e2, whence а?+е*=*Ъ2+&=*2с1, which are satisfied if
{2pv^(p-v))c
Then all further conditions are satisfied if b2—с*-\-е*= □, i. e.,
f ^m2p*+4nYv+2mYv*-*n7pi?+m2=nt m=p2+2p—l, л=рг+1.
Now /is the square of mp*—ЯпРрь/т+ти2 if v—2m1lp(n2.
T. Baker88 found five integers p*—tt <f-t> rs—tJ 32—t, t the sum of any
two of which is a square. Set 21=р*+(?—т\ We need only make
squares. Equate the first four to the squares of
r+xim-q), r+zim-p), *+y(m—q), 8+w(m-p),
respectively. The resulting relations serve to express г/тл, s/mf pfm, q/m
rationally in terms of xr y, zt w. The condition A = D is satisfied by making
special assumptions.
« Ladiee* Diaryr 1S36, 34-5, C^iest. 158&. ^^ "~~
M Bcitrag жиг АоАб& tfnbest. Aufg. 2 Gr., Progr. Elbing, 1838,
я The Gentlem^n'a Diary, or M»lh. Repoeltoiy, London, 1838, 88-9, Quest. 1360.
»/KdL8ft8
, 1838, 33-5, Q^et-1385.

456 Histohy of the Theory of Numbers.
С Gill" found five numbers the sum of every three being a square. He
used trigonometry.
To find three integers in geometrical progression, such that each plus
unity is a square, Judge Scotttt took ж2—1,2х(^—1),4ж2(^—1). It remains
only to satisfy 2z(^~l)+l-D = p*; take 2z-b2^p±l, tf-x^p^h
A. Martin used xf zy, xy* and took y=a2x+2a. Then ху-Н = П,
ху*+1=*{1+2а2хУ if a=Da-4)/a, and x+l^V* gives a. D. S. Hart
used x, xy, xy* with x=?n8H-2m.
A. Emmerich," to solve 4x-\~5—«*, 5a+4=tf2r eliminated x to show
that w^3ar t?=30, 5a2—402=1> every solution of which is given by
To find" three integers in arithmetical progression such that the sum
of every two is a square. To find6* two numbers such that if unity be
added to each of them or to their sum or to their difference, the resulting
sums are all squares.
A* Martin** found three numbers the sum of any two of which is a
square and the sum of the resulting three squares is a square.
etc. The condition f^+gf+H-w8 is satisfied if
Several7* solved a2+x=y*t vP+xlp—z* by use of
y*—-pz*=a2—pa*= (am±panJ--p{am±anJJ iri>—pn1=\*
H. Brocard71 discussed three numbers in geometrical progression, each
plus unity a square.
P. W. Flood72 found three numbers, the first two being squares, the
sum of all and the sum of any two being squares. Take lfa2, Эх2, у*—
It remains to satisfy 9x2—10xy+y*=U, lGx2—lOzy-bt/*^ □ ; eliminate
RWCrii^l^^
Take е=д*-Н\ f=2gky a=g2+k2* Then
if <7=^(Л±г), where г2=5Л2Н-4 is solved by continued fractions.
Coccoz* noted that the sum and difference of any two of the three
numbers 2399057, 2288168 and 1873432 are squares, and gave a general
solution depending on a function of degree 20 jTlollc18].
« ApplicatioD of the angular anal, to uidcter, prob, degree 2, N. Y., 1S48, p. 60.
• Math. QosbL Educ. Time», 14r 1871, 95^6,
- Mtfheeie, 10,1890,174г-5.
" Amer. Math. MontbJy, lv 1894V 96,136,169.
• /Ы^ 280, 325.
H Mattu Qsiest Educ. Time», 61, 1894, 115-6.
"Ibid., 65,1896,115.
31 Noov. Ann. Math., C), 15, 1896,288-290.
11 Math. Quest Educ. Times, 68,1898, 53.
"JttLeQ189638
"ЧЛПшЛгайоп, Jufy 20^ 1901. a. Geraidin,-Euler."

Chap. XV] LlNBAK FUNCTIONS MADE SQUARES. 457
To find*5 three integers the difference of every two of which is a square.
Likewise7* for four integers. To make*7 z-$-y-\-z, x-\-y, y+zt z+x all
squares.
Several78 solved 3s+l = □, 7z+l = П.
Л. Cunningham79 found integers xu - • -, xr such that, if a given number
N be added to their sum s or to the sum of any r—1 of themr the results
are squares. From s-\-N=o*, s—Zi-\-N=a*, we gets*—a2—o\ (i — 1, • • -,r).
Then the initial condition can be written
We may assign any values to <r, a&i • ■>, <rr such that the left member is
positive and hence a sum of four squares.
A. Gerardin* treated the problem to find a number N which can be
separated into four parts such that the sum of any two parts ts a square.
We need only use a number N which is a sum of two squares in three ways.
Or we may employ the formula for N=(ai-\-b*)(mi+'pi) as a sum of two
squares and take т=Р—д*> n^2fg, whence
P. von Schaewen*1 remarked that the triple equality A) is not solvable
by the method of Fermat or by any known method and proved that there
is a solution x Ф0 if and only if a*(z*—l)*-b4bV — П has a solution other than
2 = 0,2 = 1. For de Billy's case a=2,6 = 3, the condition is (z2—l^+ft^ Q,
which has no rational solutions other than s=0r z = l, as proved by Euler14<
of Ch. XXIL Thus the triple equation has only the solution z=0.
E. Haentzschel*3 treated the following problem. Given elr е*, e*, find
a rational number s such that s—eu в—е^ s—ej shall be rational squares.
Their product ^/4 must be a square. The relation
t^4(s—ei)(s—et)(s—es)
is satisfied if s is Weierstrass' function £>(u) and v^p'(u). Hence the
problem is to find a rational value of p(u) such that also &\u) is rational.
The solution is effected by means of the relation between pBu) and p{u),
and shown to be equivalent to that by Euler77 for his case of rational etf
«2, €2 [cf. Haentzschel1* of Ch. V]. Here is treated at length the case
; ,
H. C. Pocklington53 noted that the first, second, fifth and tenth terms
of an arithmetical progression are not all squaresr unless the first is aero or
all are equal.
" Amer. Math. Monthly, 9, 1902, 113, 230.
л /Wd, 10,1903, 206-7»
77 Ibid., 141-3.
" Math. Quest. Eduo. Times, 8, 1905, 79-80.
'•/fttf., B)r 9,1906,30-1.
••Sphinx-Oedipe, 1907-8, 10-12.
" Bibliotheca Math., C), 9, 1908-9, 289-300.
* JahreAericht d. Dentschen MaUb-Vereioigucig, 22,1913, 278-284.
u Proc. Cambridge FhU. Soc., 17r 1014,117.

458 History of the Theory or NmcBEK» [Chap, xv
E. Haentzschel, A. Korselt, and P. von Schaewen*4 treated the ргоЫещ
to find 3 numbers in arithmetical progression the sum of any two of which
is a square (Diophantus III, 9).
A. Gerardin8* noted further cages of Euler's relation B):
13920* = G4-3*)A74-l)r 629S52- A4<-5<)A8'-1),
He and A. Cunningham*6 noted solutions of
E. Turriere*7 obtained a second solution from one of os+a' = □>
H. R. Katnick8* noted that z±n can be made squares if n is even.
M. Rignaux89 gave Rolle's18 solution in factored form, and also
In terms of any given solution are expressed two new solutions»
On linear functions made squares, see Genocchi44 of Ch. XIV,
ahresber. d, Deutechcn Math,-Vereinigung, 24, W15, 467471? 25, 191B, 1384), 139-145,
351-9,
» L'iDtermediaire dee math,, 22, 1915, 230-1 E0-1).
• Ibid., 75, 233-5.
•7 L'eDseigncmeut math,, 18, 1916, 423-4.
« Araer. Math. Monthly, 24, Ш17, 339-40.
M L'interm^dLaire dee math., 25, 1918, 129.

CHAPTER XVI.
TWO QUADRATIC FUNCTIONS OF ONE OR TWO UNKNOWNS
MADE SQUARES.
Congruent numbers k\ x2±k=*\I\ both solvable.
Diophantus, III, 22, found solutions of (zi-Hc2-hrjj+x<)*±x, = n [see
Ch. VI] and, in V, 9, found solutions of x\±{xi+Xi+xi)^ Q. In each
case he began with the fact that in any right triangle having the hypotenuse
h and legs a, b, the numbers h2±2ab are squares.
An anonymous Arab manuscript,1 written before 972, contains the
problem fof congruent numbers]: Given an integer к, to fuid a square x2
<juch that x2±k are both squares. The most convenient artifice to solve
this problem is stated to be the theorem that if z*-$-y2=z2t then
&2zb2xy^(x±yJ. [Hence 2xy is a congruent number if x, у are the legs
of a right triangle.] It is stated that, if the triangle is primitive and if
x2dbk are squares, the final digits of these squares are 1 or 9, with the express
statement that the digit is not 5 [squares of odd numbers end in 1, 5 or 9].
An example is given: Using the primitive right triangle with the sides 3r 4,
we get 2xy = 24, 52+24^72, ^-24 = Iй. A table gives the expression of the
odd numbers 3, • ■ •, 19 in various ways as sums of two relatively prime
parts a, b; also the sides 2ab, a*±63 of aright triangle, and к=2BаЬ)(аг~Ъ*);
finally, wand v in z2-b& — it2, z- — k=v*t wherez is the corresponding hypote-
hypotenuse. The table has 34 such k7s. Woepcke noted that H we delete their
square factors, we get the following 30 " primitive congruent numbers >f:
5 34 210 429 2730
221 546 3570
231 1155 4290
286 1254 5610
330 1785 7854
390 1995 10374.
Woepcke remarked (p. 352) that there is no indication that the Arabs
knew Diophantus prior to the translation by Aboul Wafa (|998), but they
may well have derived the problem of congruent numbers from the Hindus
who were early acquainted with the indeterminate analysis of Diophantus.
Mohammed Ben Alhocain,2 in an Arab manuscript of the tenth century,
stated that the principal object of the theory of rational right triangles is
to find a square which when increased or diminished by a certain number к
becomes a square. He proved geometrically Diophantus' result that if
x*-\-y1=z2 then z2±2zy = (x±yy> so that 2* is the required square. Againr
6
14
15
21
30
65
70
110
154
190
* Imperial Library of Paris. French trans], by F. Woepcke, Atti Accad. Pont» Nuovi Lincei,
14, 1860-1, 250-9 (Recherche» sur plumeui* oyv» Leonardo Pise, 1st part, III). Some
of the results in the MS. were cited by Woepcke, AnoaJI dl Mat., 3, 1S60, 206.
2 French tranel by F. Woepcke, Atti Aocad. Font» Nuovi Lincei, 14,1860-1, ЗЯЪЗ.
459

4G0 History of the Theory or Numbers. [Chap, xvi
start with any two numbers a, b and take k=^ah(a-\-b)l(a—Ъ). Then
Г а*+Ъ* 1* Га+Ь об 1*
Or we may form the right triangle with the legs a2—b*, 2a& and take as &
the double of their product.
Alkarkhi1 (beginning of the eleventh century), to make $+£* and £—f1
squares, began by solving the system t/+^ = Q, у— !?=□, Set p=2x+l,
so that jf+s^CI. Then j/-^=2x+l— x2 will be the square of 1-х if
я=2. Then x*=4f y=*5 and $ = 4/5 [since the initial system is satisfied if
f*l£=x*ly2. The method is stated to be useful in the solution of x*+rrvz=O,
x*—nx=O. Although this problem does not belong directly to the present
subject, it has been inserted here in view of the use by Leonardo of the
same method.
Leonardo Pisano4 mentioned about 1220 the problem, which had been
proposed to him by Johann Panormitanus of Palermo, to find a square
which when either increased or decreased by 5 gives a square. He stated
that the answer is the square of 3+|+£ Е-4Ц; for, its square increased
by 5 gives the square of 4-Д-, and decreased by 5 gives the square of
2+4+\ Cef41 He said that he would treat such questions ш a work to
be entitled "liber quadratorqm." The latter,5 dated 1225, opened with
a bare mention of this special problem, but later* took up the general
problem: To find a number which added to a square and subtracted from
the same square gives squares; or, what is equivalent, to find three squares
Xi, x\y x\ and a number (congruum) у such that
Since any square is the sum of consecutive odd numbers 1,3, • • •, beginning
with unity, у must equal the sum of those odd numbers which enter the
sum for x\ and not in x\, and again those m x\ and not in x\. He proposed
to determine у so that the number of consecutive odd numbers whose sum is
x}—x\ shall bear to the number makingupzj—x\ a given ratio ajb. Let firat
v ' Ъ a—b
To treat together7 the two cases separated by Leonardo, let s and t represent
a and Ъ when a+b is even, but represent 2a and 2b when a+b is odd. Set
m=s(a-b), n«<(o-b), u=npt
'Extra* du Fakhri, French traimL by F. Woepcke, Paris, 1853, B8), p. 85; вате in B7),
pp. 111-2.
4 At the beginning of hia Opusooli, published by B. Boncompagni in Tre Scriiti Inediti di L*
Pianno, Rome, 1854f 2, and in Scritti di Ъ. Pieauo, Rome, 2, 1862, 227.
1 Tre Scritti, 55, eeq. Scritti, II, 253-283- B. Boucoropagni, Oomptee Rcndus Pane, 40,
1856, 77d, and R. 6. McClenon, Amer. Math. Monthly, 26, 1919, 1-8, gave а ватгаагу
of the topic» treated in the liber quadrfttoram. Of. O. Terquem, Annali di Sc. Mat.
FiH., 7,1856, 140-7; Nouv. Ann, Math,, 15, 1856, BuU. BSbl. Hiet, 65-71. Xylander
wroi^y said that Leonardo borrowed from Diophantua (c£. IJbri,»4 II, 41),
*Irwenuxnvmerum,Trt>BcritUtp.83; Scritti, II, 265.
' A. Genocchi, Note atvftliticb^ ворга Tra Scritti . . . , Annali di Scienie Mat. e Fie., ft,
1855,275-8, a, Leonardo»* of Ch. Х1П,

СИЛ*. ХУП CONGBUENT NUMBERS. 461
^Ц of which are even. By A), $(a—b)<t(a+b), whence m<q. Now
v^7ttq is the sum of m consecutive odd numbers
C) q-{m-\)> ■", 9-3r q-1,
Similarly, u=np is the sum of n consecutive odd numbers equidistant
])y twos from p. Thus the numbers of terms in the sums for v and и have
the ratio m : n=8 : t=*a : b. There are (q—m)/2 odd numbers <q—m;
their sum t\ is (q—my[£. The sum г* of the odd numbers <q-\-m is
(g+wiJ/4. Between q—m and $-Ьти lie the m consecutive odd numbers C),
so that their sum is v. But
Thus the n odd numbers between p—n and p-f-n, whose sum is uf are the
я odd numbers which follow ?+m. Finally, the sum Zz of the odd numbers
<p+n is (p+n)*/4. Hence 2i+y—32r z2-f-t«=zi, while zu z2, 2a are squares;
further,
v—mq=$t(a~ h)(a-\-b)—np — u.
Thus the proposed problem is solved by taking8
y = v=-u, x\=zb x\=zb x\=Zz.
Nextr if the inequality sign in A) is reversed, we have only to inter-
interchange vi and q in the definitions B), which were used only to obtain
д-\-тп=р—n, y—14. As the latter hold also now, the preceding discussion
holds for the present case also. The case a : Ь=а-\-Ь : a—b is shown to
be impossible in integers.9
Leonardo10 gave several numerical examples. For a=5, 5=3, then
0=240, *i=7, x«=17, л,=23. For a=3 or 2, b=l, then y=2±, xt=l,
*,=5, x*^7* For a=5, 6=2, then y=840, xt = l, ^=29, x,=4L For
o=7, 6=5, then у=840, Xi=23, Xi=37,xi=47. Xotc11 that 24 is the least
congruent number for which the three squares x\ are integers; but with
fractions, we can find smaller as shown later.
For, Leonardo11 provied that if a and Ъ are relatively prime and if а-\-Ъ
is even then ab(a-\-b)(a~Ъ) is divisible by 24 and stated1* that a similar
proof holds if a and 6 are not relatively prime. He proved also that, if
one of a and 6 is even and the other odd, 2a-2&(a-b&)(a—b) is divisible by 24.
Thus he was able to state14 that any congruent number is a multiple of 24.
The product16 of 24 by any square A8 is a congruent number and the
corresponding squares are the products of those for 24 by A2. We also get
congruent numbers by multiplying 24 by a sum of squares l*+22+3*-b * • ■
* B. Boncompagni, Annali dl Sc. Mat. e Fts., 6, l£55, 135, quoted Leonardo1* solution to be
у=4оЬ(д1—Ь*), 2*=а*-{-&*, xh Х1—2аЬ±{Ь*-чх*). But this corresponds only to the
case t =2o,<=26.
' Tre Scritti, 96; ScriUi, 11, 271. Genoochi/pp. 292-3.
11 Tre Scritti, 88-92; Scritti, II, 268-70. Genoocfai/ pp. 278-9.
Jl Tre Scritti, 90-93; Scritti, II, 269^270. Genocchi,* pp. 280-1.
u Si duo maneri, Tre Scrittir 80; Scritti, П, 264.
u Tre Scritti, 82; Scritti, II, 265.
M Tre Scrifcti, 92; Scritti, 11, 270. G«noochV pp. 273-4.
u Quotient enim 24, Tre Scntti, 93; Seritti, II, 270. Gextocchi,7 p. 283, p.
254.

462 History or the Theory of Numbers. [Chap. XVI
---. For example,
To find1* a congruent number whose fifth part is a square, take a—5
and determine b so that b, а-\-Ъ, a—b are all squares, say g2, h2, №, respec-
respectively. Then б^+й*. Either g^l, k=2f whereas a+b = 5-f-l is not а
square, or 0=2, Jb = l, whence 4аЬ(а*-ЬР) =720 is the desired congruent
number. Returning to the earlier problem to make x-±5 both squares,
and using the values a=&f b—4, just found, we have s = 10, £—8, and,
by B), m=10, q=72, whence z*^(82/2J, x2=41. Since 720=5-12*, we
reduce the numbers in the ratio 1 : 12 and get the solution z=41/12.
Leonardo1* affirmed that no square can be a congruent number. This
proposition is of special historical importance since it implies that the
area of a rational right triangle is never a square and that the difference of
two biquadrates is not a square. Leonardo stated without proof18 the
lemma that if a congruent number were a square there would exist integers
a, h for which a : Ъ=а+Ъ : a—& (proved impossible earlier).
Leonardo1* noted that many numbers are not congruent; but any
number is a congruent if the quotient of any congruent number by it is a
square, A number is congruent if it equals one of the four numbers af b,
a-\-by a—b, and if the remaining three are squares. For example, 16r 9,
16+9 are squares, so that 16—9=7 is a congruent number. To make
x*±x both squares, let A; be a congruent number and д*—к~Р,
then we have the solution x—g2/k since
To makeX^imX both squares, we set X=mx and are led to the preceding
problem, whence Х=7пд*/к. Leonardo considered the example with fc-%24,
g~5. Cf. Alkarkhi,3 and Ch- XVIII.
Luca Faciuolo30 reproduced part of Leonardo's Liber Quadratorum; be
gave as the first five " congruent^ " numbers 24, 120r 336, 720, 1320, their
corresponding squares (" congruo "il) being 5*, 132, 25*r 41* 612. From
n and лН-l he derived the congruent number 2n(rt-bl)i2(n+n-bl)}, the cor-
corresponding square being {ri*-\-(n+lJ\2. He made x*±b fractional squares
forb=5, 7, IS; and solved ^-HO- □> z*-ll = EJ. He gave a table of 52
congruent numbers, of which only** 14 are primitive, the latter being all
in the table in the Arab MS.1 (via,, the first six and 65, 70, 154, 210, 231,
330, 390, 546); the Arab had the advantage of excluding values a, b not
* Volo invenire, Tre Scritti, 95; Scritti, II, 271. Geaocchi,' p. 2&8.
J' Tre Scritti, 98; Scritti, II, 272. a. Ou XXII.
" For a proof, with л historical discussion, e&e Genocchi,7 pp. 293-310 (pp. 131-2). Cf. F.
Woepcke, Jour, de Math., 20,1855,56; extract in Comptea Rendus Paris, 40,1855,781.
i" Tre Scritti, 98; Scritti, II, 272. Geuoechi,' pp. 31(ЬЗР 345-6.
rt Luce de Burgo Summa de anthmetica geometria, Venice, 1494; ed. 2t Toecolano, 1523,
ff. 14-18.
л Thus interchanging Leonardo^ two terms, Cf. BibL Math., C),3,1902,144. Also noted
by BoncoinpagnL1
« F. Woepcker Antiali di Mat., 3, I860, 20G; Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 259.

Свар- XVI} CONGSTTENT NUMBERS. 463
relatively prime» The dependence of the work of Paciuolo upon that of
Leonardo was pointed out in detail by B. Boncompagni** and by G. Iibri.M
F. Ghaligai2* also borrowed [Xibri,*4 III, 14У] from Leonardo; he gave
5*-[-24=72, 5l—24^1, stating that 24 is the least congruent number.
To find another, start with 1 and 3; add and double the sum, getting 8;
multiply by 3-1, getting 16; multiply by 1X3, getting 48; its double 96
is a congruent number; in fact, l+32 = 10 and 102—96=2*, 10*+96=14*.
F, Feliciano** gave the same congruent numbers and rule as had
Paciuolo.20 He gave sl=6i as the solution of xl±6= П.
P. Forcadel de Beziers" employed right triangles with one leg less by
unity than the hypotenuse A, citing h—5, 13, 25, 41, 61. Their squares are
" congrus '* numbers, the corresponding " congruens " being 24r 120, 336r
720, 1320 [double the products of the two legs]. He gavera for n = l, 2r 3r
4, 5, the congrus D«2-H)* and corresponding congruens 8n{4w*—1).
N. Tartaglia* quoted two rules of Leonardo, as given by Luea Paciuolo,
for forming congruent numbers, one rule by use of two consecutive numbers,
the other by use of* (а2-\-ЬгJ±4аЬ(а?—fr2) = EL
G. Gosselin80 treated (f. 75 verso) the problem: Given a square 100,
to find the congruent number. Separate the double 20 of the side into two
parts 2L and 20—2Z> whose product equals the product of two other
numbers of difference 20, say Lf 20+L. Thus L=4 and 8X12-4X24
is the required congruent number 96, Conversely, given a congruent
number, to find the square (f. 77, verso). 'This is the problem which
Luca, Pisano, Tartaglia, Cardan and Forcadelus found so difficult, in investi-
investigating which they consumed not a little oil; nevertheless they did not
succeed and it remained unsolved up to the present; let us now explain
that difficult thing." Given the congruent number 96, to find the square Q
such that 96-f-Q is the required square. Hencethesuml92-J-<2of the latter
and 96 must be a square. Thus the difference of two squares is 96—4
>24=6-16=8 12. But 3(8+12)=-10 is excluded since 100 =j=192+Q, while
£D+24) = 14 and 14*=192-bQ gives Q-4r yielding the answer 96-bQ=100.
Beha-EddinM A547-1622) listed, among the seven problems remaining
unsolved from former times, as Prob. 2 that to make a£+10 and s*—10
both squares. As noted by Nes&elmann, it is Impossible.
* АппнЙ di Sc. Mat. e Fie , 6, 1855, 135-154, ""
"ffiet. Sc, Math, en Italie, ed. % HaUe, 1865, ITr 39; III, 137-140,265-271.
**Summ& de Aiiibmettca, Florence, 1521, f. 60; Practica d'arithmetic*, Florence, 1552,
1548, f> 61 left.
H Iibrodi Arithmetic* A Geometru epeculatiuft k. praticale: Francesco Feliciano . . , Inti-
tulato Seal* GriinaJdeffi, Venice, 1526, etc., Verona, 1563, etc>, ff. 3-5 (unnumbered
pgp)
n L'aritbmeiiflqTe, 1,1556, Paris, ff. S, 9.
** The related right triangle has the aidee 4nt 4nf-1,4nf+l.
11 La Seconds, Parte dd Qeneial Tnttato di nuineri et misnie, Venice, I556r ff. 143-в.
" The fizuJ factor, given as а±Ъх w&b corrected by the tnuisbtor, G. GoeBelio, 1578, 91.
** De Arte Troigri*^ ecu de occulta parte ишп.} Pbris» 1577.
и Еявеп* der RecfaenktiDflt von Mohamtned Beba-eddia ben А1Ьп<»яй» яд» Amul, orabiBch u.
deotacb von G. H. *". NqmJnrnnn, Berim, 1843, P- 65. *V^ncb tnuvL by Arbtide Marrer
fThrl^t al НшЬ, <** ^вомж da Gblcai de Beba-eddin Mohammai ben *l-Hcwn al-
A^nw^iK^ Nouv. Ann. МаЦц 5, 1846, 313; ed. 2, corrected and with new notes, Rome,
1854.

464 History op the Theory of Numbers, [Chap, xyi
Fermat2 of Ch. XXII proved that the difference of two biquadrates is
never a square. Hence no congruent number is a square»
L« Euler*3 noted (as had Leonardo) that p2=b5<f are both squares for
p^4i, g = 12; p*±7q7 both squares for p=337, g-120. He made p*±aq*
squares also for a=6, 14, 15, 30. The method is that used by him7* for
concordant numbers.
P. Cossali34 undertook to reconstruct Leonardo's liber Quadratorum,
then believed to be lost. A sufficient (adverse) report will be found under
Genocchi,35 Woepcke" and Boncompagni."
A. Genocchi86 stated that Cossali*4 was wrong in believing that
Leonardo's method of making x*±a both squares is only special. While
indirect, it is general and succeeds when the problem is solvable. In fact,
it coincides exactly with the formulas obtained by Euler76 after complicated
calculations. This coincidence escaped Cossali, who filled many pages
with useless calculations without discovering the general solution.
F. Woepcke* noted that of the Q26 distinct] congruent numbers in the
table of 29 1шея by Cossali,34 p. 126, only 12 are primitive, including all but
65 and 154 of those noted under Luca Paciuolo.20
B. Boncompagni37 disagreed with the explanation by Cossali,34 p. 132,
of Leonardo's method. The latter had remarked that A will be a con-
congruent number if its quotient by a given congruent number hi is a
square q*. According to Cossali's interpretation, q is rational only when
(Л1+2)BА1+2)C&1+4) is a rational square; while a more plausible inter-
interpretation leads always to a rational q.
" L. Pisanus " w made n2±13 and n* all rational squares. Since
are the squares of 2d*-b4d+3 and 2d*+4d+l, take d=2 and we get
13-25±36-a. In (a2+by±4aK«B-b0 = <«2±2ab-P)8, take a=ct\
o=s*. Then
Take с = 13, £=25, s* = 36. But A3-25J—362 is the product of the squares
found before. Hence (c2i4+s4)Vi4^2s2(c^4-s4)J is the required square n2.
J. Hartley19 took z2+13-(x+y)*, s*-13 = (s-yzJ, and from the two
rational values of x got y*=13(z—l)f[z(z+l)}. The latter is a square for
z = (f+&)/Brs) if r2-N?=13, 2rs-D. Take r=-Z-gt, s = 2-<. Then
13 gives ^D-6д)/{^+1). Take g=2, whence r=l/5, *=18/5,
»А1й*иц 2,1770, $ 226; Erench tranal., Lyon, 2,1774, p. 291; T)pera Отшя, A), I, 459^
«Oneine, traaporto in Italia, primi progr^^ in eeaa defl'algcbra, 1, 1797, 115-172, Cf. G.
Ubn, Hiatoire dee Sc. Math, en Ibule, ed. 2, III, 1865,139,140, 265.
» Compttt Rendue Paris, 40,1855, 775-8.
"ЛШ Accad, Pont. Nuovi lined, 14, ШО-1, 259.
» Ann&li di Sc. Mat. e Fie., 6,1855, Ш-151.
"Ladies' Diary, 1803, p. 41, Quart. 1099; and Priae РгоЫ 1118,1804, pp. 44Л; ЬеуЬоигаЪ
M&tb. Quart. L. D., 4, 1817, 10-Ut 31-33. The Prize problem etated that there are
tational equaree 2*, tf jsuoh that x*i;13 are sqiiajnee, and 13у* is the area of a right triangle
whoae sides are integers; 13 is л ват of two eqn&ra, double the product of whose roots
и a square^ and if the latter square be added to and subtracted from 13 the results
are square.
» Tbe Diary Companion, Supplement to the Indies' Diary, London, 1803, 45.

- xvi] Сожжхгетчт Numbers. 465
P. Barlow40 proved by descent that 1 and 2 are not congruent numbers.
J. Cunliffe41 noted that if, when n is given, a rational v can be found for
which л+tf2 and n—v* are rational squares, we can deduce a rational x
for which z*+» and x1—n are rational squares. Take (a+ЬJ and {а-ЬУ
as the latter. ТЪеп х*=аг+Ъ*> n=2ab. To satisfy the former, take
a=(p*-q*)!Br),h=pq!r. Then (p2-^)pg=nr2. Takep=n,$=t* Then
n2^v*=Qy which holds if rizbv* are squares. Application is made to the
case n= 13 by expressing 13 as a sum of two rational squares in two ways.
" Umbra '■« noted that x*+n^a\ z*-n=b? can be solved if
For, 2x*^a*+b* is known to hold if a, b = Bpq±p*zFq*)/rt x {
Then п = §(а*-Ь2) = 4рд(р2-$!)/г2. Taking p=c*, g=d*, we have
whence г is rational since c2—d*-0. Similarly, х*±7&= П are solvable if
n^{&—<&)!&, ^Н-^^П, or if л is double the sum of two squares the
double of whose difference is a square.
A. Genocchi4* noted that the problem to make x*±fof both squares is
equivalent to the single equation x4—h*qA=n. By the direct, but laborious,
method of Fermat (on Diophantua VI, 26), used by Lagrange (see papers
37-il, 54 of Ch. XXII), Genocchi treated the example h=5 far enough
to reach the special solution due to Leonardo.4 The direct solution of
x*±k=n leads to 4mn(mz—n2)^=hgi or the problem to form a rational
right triangle with a given area. The absence of a treatment of the latter
leaves an evident lacuna in Diophantus VI, 6-11 (V, 8 deduced a new solu-
solution from one). The method by Euler" is identical with that of Leonardo,
Genocchi (pp. 206-9) proved that an integer у is of the form 4mn(m*—n2)
in only a finite number of ways- To two solutions x of x*±y = O, each x
a sum of two squares, correspond distinct values of y. From one solution
(pp. 251-3) of zs±fc = D, we readily get others. Cf- Young1" of Ch. XIX.
Genoechi44 proved that r*+4e4, 2г*+28*} г4—s* are congruent numbers;
also re+6rV+e4 and ^{r4—6r*s4-*4) if one of the integers r, 8 is even and
the other odd. No prime $m+Z ш a congruent number.
Genocchi* proved that the double of a prime 8A+5 is not a congruent
number.
Matthew Collins*1 proved that the only congruent numbers < 20 are
5, 6, 7, 13, 14, 16; that a prime «=4n+3 is not a congruent number if,
«ТЬеаду Ы Namboft, London, 1811,100, 114. ~^
«T. Leyboura'3 Mfttb. Qmt. from Ladies' Diary, 3, 1817, 368-71.
« The GeatbmV М*Л. Сотршш, Uodoo, 4, No. 21,1818, 750-2.
«АамЛ*8с Mat. еБЬ^в, 1S55, 129-434, 291-2.
* П CWnta* Rivieta di Scl Let ed Artit Toiino, 6, 1855, 677-9. Genoochi,? p. 299 for the
ЬЮ
« A Tract cm tbe р<шД*; «dmprwJiTi саам of quftdratio duplicate equalities . . ^,
185S,«>pp. A*etr.iftBikbhAj»oc. Reports for 1855,1856,11,2-5; and in The Lady'a
and Geatkwwi'B Diary, London, 1857r U2-6.

466 History or the Theory op Numbers. [Chap, xvi
for m<a/2, m*-2 is not divisible by a (examples: a«» 11, 19, 43). To
treat x*zh5^ = n, we add and subtract and get 2x*=f+w1y Щ*=г*-и>\
Set z=z'+w\ w=z'-w'y where z\ v>' are relatively prime. Thus
whence г'=тг—n2, wf = 2mn, x=m*+n\ and 2*м'=5у*/2, whence y = 2tf,
mn(m2—n2) — 5y'7* If л is divisible by 5, n=5q2> m=Jjf^> m-\-n=r2$
т—п=8*, leading to a pair like the initial equations, so that this case is
excluded. If m=5p?f we get n^g1, min^r2, s*, whence 5p4-gfs=r*,
5p*—д*=з2. As the latter are satisfied by p=l, 9=2, whence m=5, n=4,
we get the solution [Leonardo's] x=^41, y = 12, z*=49t w=31. In general,
given a solution of
then X«»(*4+w*)/2, Г=2хугю make
and hence give a solution of X*-\-abY2= П, X*—аЬЗ^^П, For example,
if a = 5, 6^n^l, we have 5х2±у*=П, holding for x= 1, y=2, whence
X=41, Г=12 satisfy Х*±5Р=П.
F. Woe реке47 found 12 congruent numbers associated with the given
one 2xy, where j*+yz=z*7 viz., txy zy, х^-уР, ^H-x1, z?+y*, (*?
In fact, x=ai—biy y = 2ab. In 2xy replace a by z and Ь by x and drop the
square factor 4(z2-x*) =4y*; we get xz=aA—Ъ\ But if we replace Ь by y,
we get yz. In a4—b\ takea=a:, b=y, and drop the square factor xI-h]/2=s2;
we get x2—y2. Double the product of the latter by the congruent number
2xy is a congruent number; etc. He computed the above 12 functions for
each right triangle in the Arab manuscript.'
Woepcke* treated the problem, proposed to him by Boncompagni:
Given a congruent number k, to find a congruent number К such that
the product kK of the two is another congruent number. If fc is formed from
a, b, where 2a2—Ъ*=<?7 then
If k is formed from two numbers of ratio r, where
and К is formed from two numbers of the ratio
p~ 2(r-l) '
then kK is a congruent number formed from two numbers of the ratio
2. For, then
Ш di Mat., 3, 1800, 206-15. Same in Atti Armri Pont. NiK>vi lined, 14, 1860-1.
250-C7.
«AnmUi di Mat., 4, 1861, 2i7-55.

Chap. XVI] CONGRUENT NUMBERS. 4C7
Like results hold if we take (cf. Lucas61)
0r+
2Br+l)
If kK is a congruent number and hence equal to 4«0(a2—/З2);?2/^, we nay set
Thus if also К and hence К' is a congruent number, then k is the double
of a leg of a right triangle whose second leg is a congruent number.
If kK=K1 is a relation between three congruent numbers, the last
formulas show that с — 2Л£ and &i = 2Xa are solutions of the system
where ф=4\К', $ = \Jc. Соптегее1у, if one of these equations can be
solved, kK' and hence kK is a congruent number.
To find congruent numbers К, Кг such that kK=Kb where k is a given
congruent number, take as Kt in turn the 12 types in the ear'ier paper,47
each type multiplied by an arbitrary rational square. Give Кг the form
2—0*)р*/д*7 and equate the latter to kK. Hence
so that the leg <**—fl2 of a rational triangle is a congruent number and the
other leg 2<x& is h\% But this solves kK=Kx for K.
G. Le Secq. Destournelles'1* proved the impossibility in integers of the
pair
The equation obtained by adding these may be written
The terms on tbe right may be aseumed relatively prime. Thus
Z-U q'-ff2
2 ~°^' V" 2 '
or vice versa, where a, ff are odd relatively prime integers. Substituting
either set into 2уг—г2—и*, we get
Thus m2±n'=2fc2, 2l\ so that
But these are like the initial equations with &<x?
A. Gcnocchi50 stated that x'ifc are not both rational squares when Л
is a prime 8m+3 or the product of two such primes, or the double of a
prime 8т+5, or the double of the product of two such primes.
" Coagr&) Sc. de France, Eodex, 40,1, 187*, 167-182; Jornal de Mftth. e Aat.v 3, 1881.
60 Comptes Rendue Puis, 78, 1874, 433-5, RepnaUd, Sphmx-Oedipe, 4V 1909, 161-3.

468 History of the Theohy of Numbers. CChap. xvi
E. Lucas51 noted that a is a congruent number if and only if x*—а*у*=г*
is solvable; then a=0, dbl (mod 5) if xy is not divisible by 5. A congruent
number does not end in 2, 3, 7 or 8 when у is not divisible by 5. We are
led to congruent numbers by the problem to find three squares in arith-
arithmetical progression whose common difference is the product of a by a
square. The equations (pp. 184-6)
D) - х*-Ъу*=ч\ f+V=o?
were studied, but not completely solved, by Leonardo,4"le Paciuolo,* Euler,u
Collins,46 and Genocchi/ p. 289. We may assume that z, y, w, v are rela-
relatively prime, so that x and v are odd, у even. Hence in view of the first
equation we may set
x—u=10r2, x-h«=2^, y=2rs (r, $ relatively prime).
By the second equation D), (^Н-З^I-^4-^, whence
Adding the first two of these we get
Since the factors on the left are relatively prime, we find after considering
residues modulo 5 that the only two admissible cases are p*—g^ify*,
p*—2tf=±h\ For the upper sign, the evident solution p—3, <Z=2,
g=k=lt leads to Leonardo's solution x=4I, # = 12, u=31, t>=49 of D).
For the lower sign, we get the system q*-5g2=p?, д*+5р*=Л*, like D);
hence from one solution we get the second:
X=tiV+5tiyf U=tfi&-UW, V=*u4-2x4, Y=2xyuv,
which differ only in form from the formulas by GenocchL Lucas solved
(pp. 191-3) the equation to which Woepcke48 was led:
9a4 - 2Oalb - 2a^ +20ab*+9b4 = c2.
This may be written #+44а*&*=9с*, where ^Эа'-Юаб-ЭЬ2. Thus
3cd=d=V> ЗсТс^-ггЛ ab=pq.
Set Ь=щ, p = ma. From px— llq*=±d we get a quadratic for m with
a rational root if 13aydb(a4+llg*) -z\ For the upper sign,
If we take the factors of the left member to be r4 and 125a*, and add, we get
(г2-Ш2)'-4а2=44в4.
Call the factors of the left member ±2u\ T22»4; adding, we get
which is like the initial quartic, but with smaller values of the unknowns.
A like result is proved in the remaining admissible cases. The system"
я'=ЬоУ = □ is treated (pp. 18(M) by the method used for the generalization
given in the next paper. From Leonardo's solution x = 5, у=2, is deduced
1512
Bidl ВЫ. Storia 8c. Mat., 10, 1877, 170-193.
Abo in Nouv. Ann. Math., B)v 15, Ш6, 46^70.

Chap, XV!} CONGRUENT NlJMBEHS. 469
Lucas* noted that if there exist relatively prime solutions of
x2—Ay2=w2, zz-\-Ayz=v2,
then Л is of the form X^(X2—p2). For, by addition,
where X, /i are relatively prime and one is even. Hence
w, v=X2-
He next showed how to derive a second solution from one, given that
Л is a congruent number resolved into its prime factors. If a, fi are two
integers whose product is A, the second equation gives
, , уф
Substitute the resulting vy x into the first given equation; then
The two factors of the left member equal ±2/3?£4, ±4$ft4F where
gh=f. For the upper sign, we add and get
The two factors equal <*ip* and cmf, where а^а*—a, pq=e. Taking
ai = cti=px = l, we have a system like the proposed. Hence a solution
x, yt u7 v leads to the second solution
For the lower sign above, we obtain a complicated set of formulas giving
a new solution from one. The formulas are said to give all solutions when
A =6 and for the problem x*±(x-\-2) = П of Beha-Eddin." By combining
Lucas7 result (ibid., p. 433) with the results of Fermat and Genocchi,4*
Lucas concluded (p. 514) that xy(x2—y2)— Az2 has no rational solution if
A = l, 2, pt 2qt where p and q are primes of the respective forms 8n-|-3,
8n+5.
B, Gunther" treated z*-\-a^y2, x2—a—z2 by setting
z-y=m(z-z), z+y=4lz+x)f
which determine у and z in terms of x and m. Substituting these into one
of the proposed equations, we get x as a function of m\
4m—4m1'
Set m=a$P. Then z is rational if 1— a?pA= П, Hence we seek among the
rational solutions of 1—а*{* = ч* those values of £ which are squares. If
such exist, a is a congruent number, otherwise not. We can not go further
with the general solution of the system since the character of a decides
whether or not such a biquadratic root of the Pell equation exists.
*Nouv. Ann» Math., B), 17, 1878, 446.
" Prag SiUungAerichte, 1878, 289-94.

470 History or the Theory of Numbers. [Chap. XVI
S. Roberts'* proved the known result that if х*±Ру* are squares then
P* ie of the form tabiat-h*), where *=4 or 1 according as a, b are of
different or like parity. He stated that the values of P which are primes
or the doubles of primes are all obtained by tbe rule of Leonardo which
makes three of а, Ъ, а±Ь squares, and carried further the analysis of
Genocchi." Inadmissible values о! Р are primes 8fc+3 or doubles of
primes 8feH-5- He proved that various classes of primes P are excluded,
all being such that z*~2Py2 = — 1 has no solution.
A, Desboves5* started with a congruent number V(X2—/Д changed X
to s!, /i to y*t absorbed the factor xV into the term F* of Х*^аК* = П,
and obtained the congruent numbers4—y4. Since a4+# —b4+c4 is solvable
and hence also
we can find an infinitude of numbers which are differences of two bi-
quadr&tes and whose product is such a difference, and hence an infinitude
of solutions of Boncompagni's4* problem to find two congruent numbers
whose product is a congruent number.
A. Genocchi" proved that the following numbers are not congruent:
a prime 8fc+3 or the product of two such primes; the double of a prime
8fc+5 or the double of the product of two such primes.
Genocchi** stated his44 results and that no congruent number is a
product of a square by a prime 8m+3, or by double a prime 8m-\-5t or by
a product of two primes 8m4-3, or by double the product of two primes
87П+5.
G. Heppel* found a such that 10F+fl=A01+*J, 10P-a={101-0*
by taking l=k+c, whence 2&*=202с-2*с-Л Since с is a factor of 2k\
but not of fc, с=4. Thus к = 18, a = 3960.
M. Jenkins6* found an integer a for which (m*+n*J±a= (h±tJ. Then
a=2ht, (m2+n2)*=k2+t\ One solution of the latter is k=m*-n?, t=2mn.
G. B. Mathews*1 discussed x*±;a= П. From х2+а=?(х+т)г, we get x.
Then j*->a=N/{4m2), where N=a2-Gam*+m\ Set N^(a-m*k/l)*.
Then m*=fa, J=2l{k-2l)l{k*-P). Take a=/fr*, where Ъ is arbitrary.
Then m=fb and x is found.
R. Aiyar** noted that, if A2±1B are squares, A and В are expressible
in one and but one way in the forms A=\{mi-\-nr)t B=\zmn(mt-n2O
where m and n are relatively prime and one is even.
A. Cunningham and R. W. D. Christie" solved x*-y*=y1-w*=czi>
where с is given, as c=65, by use of x2—2yz= —uF. [Hence у2—с£=ур,
«Proc. Lond. Math. Soc,, 11, 187&-8O, 3644.
и Адаос, franc-, ^* 1880, 242.
" Memorie di Mat. e Fie. Soc. Ital. Sc., C), 4,18S2r No. 3.
»Nouv. Ann. Math., C), 2, 1883, 309-10.
*» Malb. Quest. Educ, Tiroee, 40, 1884, 119,
" Ibid., 41, 1884, 65-6.
«/W*, 107-8,
" Math. Quest. Educ. Timee, 65, 1896, 100.
tt Math. Quest. Educ Tunee, B), 13,1908, 77-9.

Chap. XVI] CoNGRUXNT NUMBERS. 471
G. Bisconcini" determined the numbers А=±Г8(г*—«*) which are
products of powers of three primes. К 2 and 3 are the only prime factors
of Л, then A =24.
R. D. Carmichael" proved that the system g*+n*^m*, т*+пг=рг has
no positive integral solutions [whence тг+пг^р\ т2—«*=$*]-
H. B- Mathieu" asked if x*+A = u\ х*—А=& are completely solved
by the identity
L. Aubry* replied that all solutions of 2хг—u*+tP are given by
The case l-l9 r=a±b, s = b, gives the above identity, which with
give all relatively prime solutions» [Cf. Ob XIV.]
G. Metrod" treated х*4-у=и*, хг-у^&. For их=а*~Ъ\ etcv
Hence (ul+v1y+(u1-vt)t^2xt; 4, v=a*-W±2ab, у=4аЬ(аг-Ъ*).
J. Maurin and A, Cunningham" noted that from one solution of
х2-пу*=г\ х*+пуг=&1 we get a second solution X=x4+nV, Y=2zyzt.
A, Gerardin^ listed the values <1000 of h for which x*±hy*= D for
x<3722. He noted (pp. 57-9) the solutions
x=
L. Bastien71 listed the 25 values <100 of h for which x*±hy2 are not
squares, and stated (besides Genocchi's*0 results) the following cases of im-
impossibility: h the double of a prime 16m+9; h a prime 8w*+l=02+fc*,
with g+k a quadratic non-residue of h (as for 17, 73, 89, 97).
T. Ono*2 noted that х*±5у* are squares for x=41, y=12 [Leonardo4]
and х = ЗШ161, у=1494696,
G. Candido78 noted that, from two sets of solutions (x», y») of the system
*2= П, we get a third set by Euler's identity
E. Turriere7* noted that if x, у, г are the rational coordinates of a point
Af on the quartic space curve хг+а=у\ х*+Ъ=г\ the osculating plane at
M
аГ,^|Г1909, 157-170.
•* Amer. Math. Monthly, 20, 1913, 21^
M I/intermediatre dee math., 20, 1913, 2.
"Ibid., 2П-». Practically ваше by Wtlecb,
« Sphim-Oedipe, 8, 1913r 130-L
" Ulnt«rm6diair« dee math., 21,19H, 20-21,
»/ЬЙ.,22, 1916» 62-3.
*/Ш>, 231-2.
И7
tf.f 23» 1«в, 111-2.
* L'eneeigoement m»tb, 17, 1916, 315^324.

472 History of the Theory of Numbers. [Chap, xvi
and meets the curve at a new point Mi whose coordinates are rational and
easily found. Thus if we employ Leonardo's solution z=41/12, #=49/12,
*^31/12 when a=5, Ъ= —5, we obtain in succession an infinitude of rational
solutions. Or we may find the points with rational coordinates, on the
hyperbola y*—x*=a, by setting x+y=u, y—x=aju7 and identifying their
abscissas with those of the analogous points x=(&— b)!Bv), y = (v*-\-b)/Bv)
on z2—х*=Ъ, obtaining the condition uv{u—v) = av—bu. The tangent at
a rational point (v, v) meets the cubic at a new rational point. Finally,
x*-\-a^y2, x*—a=zL have the solutions yy z=x (cos 0±sin 0); x*=a/sm 26;
bence the rational solutions are given by those rational values of tan 0/2 for
which sin 29 is a rational square; there are none if a^l or 2.
For congruent numbers of order n, вее papers 200-1, 210, 222 Qf Oh.
XXIII, and 320 of Ch. XXII.
Concordant forms: x*+my*, z*+ny* both squares.
Related Problems.
Diophantus, II, 15, required x, m, n such that x*+m and x2+n are
squares, given the вит m+n. He took m=4s+4, n=6x+9, m+«^20,
whence x=7IW. In II, 16, (x+2I—m and (z+2)*-n are squares if
m=4s+4, n=2i-|-3; for m-|-n-20, z = 13!&. The same problems occur
in 1П, 23, 24.
Diophantus, II, 17, required xy m, n such that x*+m and z2+n are
squares, given the ratio m/n. He tookx-3, m/n=3, n^(#+3)*—9. The
condition is 3*+3((v+3)*-9} -3^-|-18t/-h9=n, say By-3J, whence
jf=30.
Certain Arab writers1' * of the tenth century treated the special problem
to make x*+k and x1—к both squares, taking к as given, unlike Diophantus.
Rafael Bombelli75 divided 40 into two parts C0 and 10) such that if
each be subtracted from the same square C0J) the remainders are squares.
L, Euler* treated the problem, equivalent to Diophantus II, 17: If a
and Ь are given integers, find 2, p, q, r, s such that
A) ^Н-в^-Л V*+bz<?=s\
Eliminating 2, we get ^^(br2—as2)/(b—a)* Since the latter is a square for
r=e, set r=8+(b—a)t. Then р*=&+2Ы+Ъ(Ь-а)Р. Set j>=&+txjy.
Equate to t the numerator of the resulting fraction for t/s. Thus
i-2sp-2ty*, 8=Ъ{Ъ-а)у*-х\ p = (x-byJ~aby2,
г=(Ъ-а)Bху-Ъу*)-х*, q*z
Simplifications arise if we set x = v-\-by; then
Thus we take v and у arbitrary, and choose g2 to be the greatest square
factor of the final expression. It is shown (§ 230) that y^+g2 and p*-\-Zq*
are not both squares. Cf. Euler.w
"L'Aleeibra Ope», Bologna, 1579, 461.
Alb St. Pteteirfwig, 2t 1770, «225-230; French tnmel., Lyon, 2, 1774, pp. 28&-Э02;
i A), 1р45в-4б4. Cf. Eukr» of Ch. XV.

. XVI] CONCORBAMT FORMS, 473
Eukr" called jp+my1 and x*-\-ny? concordant forms if they can both be
made squares by choice of integers z, у each not zero; otherwise, discordant
forms. He treated the problem: Given an integer rn, to find all integers n
for which the two forms are concordant. Set m=ti*y where one factor may
be unity. Thmzt+my2 = (iip2+*tf)tiIx = Mpp>*-*tf),V=2P9- To make
j^+ny2^!^, we must take n = (wl-xi)lDp*qi)) where x Ъш the preceding
value. Then both factors w±x must be even. Set рф=г*&, where 2г*
divides w+x, and 2s3 divides u>—x> the respective quotients bemg/and g.
Then n =fg. Hence we consider xy r, s as known and seek /, g such that
/тг—д&=х- The latter is satisfied by f=№±<rx, g=hi*±px, where p/«r
is the convergent preceding r2/^ for the continued fraction for the latter.
A table (g 10) gives the values of p, a for г*^1&, г^г1. For то=1, Euler
found (§ 12) those values numerically < 100 of n which result from small
values of r, s. It is known that z'iy2 are discordant; also x*+yp, x*+2y*.
Proof is given (§§15-19) that х»+^,х*-|-Зу»are discordant; also (§§20-23,
§31) that х*+уг=г\ zI+4y*=02 are impossible* Hence
and i»2—4у*=х*, &—Ъу2=*гг are discordant. But я^+у2, x*+7y* are squares
for x=3, y-4. In papers 109, 110, 113-4 of Ch. XVI it is noted that
х*±уг and ^±4y2 are not both squares; also, a^dby2 and jp^FSy*.
Euler78 satisfied ai^-l-my2^ D as in the last paper,77 and noted that then
x*+ny* = {n<p-pq*+2Mj&fy if п=Мф(?+М{мР-*?)у where M is arbi-
arbitrary. If M=N+v№>n = {Ntffv){Nq*+ti).
Euler7* noted that x*+aby* is a square for x = t(a$?-bq2)) y=
that x2+cdy*=\3 for х^^сг2-^?), y=2ifrz. Hence set
By the values of x,
Set 0 = £т}. Hence must d(^-|-b^(«a/l+dak1) = П. But this condition
was not discussed.
Euler® had previously treated the more special problem to find all
integers N such that -A2+J5* and A2+NB2 are both squares for ЛБФ0.
Take A=x*~y2> B=2xy. Then
The last gives for N an expression which is an integer ii г=х*-\-2ахгу2±у\
According as the upper or lower sign is chosen, we have
To investigate the rational a'& for which the first N is integral, when x and
у are integral, set a=a/(gV), x—pq, y=rs, where a is an integer, while py
77 Mem. AcacL So. St. Fetersix, 8,1817-8 A780), 3; Comm. Aritb., П, 406.
71 Open poetuma, 1,1862, 253 (about 1769).
n Ibid., 256 (about 1782).
*> Not» Acto Ac»d, Ptetrop., 11,1793 A777), 78; Comm, Arith., П, 190-7.

474 Histoby of the Theory of Numbers. [Chaf. xvi
etc., may become unity. Then, И a=s=l, we have iV
If p=7t g=5, tf divides рЧ-l, and J\T=2r4-50. If a=-l, « = 1, then
JV^Cp'-lX^-eW and $* divides p*-l if p=3, g=2, etc. A list is
given of the resulting N's numerically <100; those ^50 and >0are7,10,
II, 17, 20, 22,23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50. But the problem is
not proved impossible for the omitted values of N.
Euler*1 made aV+ by and aV+^ both squares by taking
Ъу~ 2pq 9 bx~ 2rs
By division, we get sVv*- Hence it suffices to make the quotient of
pq(?2—q*) by r*(r*—«*) a square, a problem* which had been frequently
treated, but not completely solved. The first of three special methods is
to take s=qtr=p+q] then we are to make (p—g)/(p+2^) = D, which is
the case if p=ul-f 2P, q=ul— f2; the resulting solution is
а=Ш, Ь=2(ч1~Р), x=tBu*+F), y = u(u*+2P).
To obtain the general solution, we may take s=q without loss of generality,
since it is only a question of ratios. Then
r(r»-fl*) ' * p-nr
Set p=rv. Then (t^-n)/(v~n) = □ =(v-z)* if
(«+22)^-2B71+2)^4-^(^-1) =0.
From a given solution z, v, we get a second solution
а,=2у
Thus v=0t z= 1 leads to the second solution
, 3n
Replace л by (*/иг; we get tfie above special solution. He investigated the
third solution t/', z'\ and also started with v=G, z = — l; z=0, t»=±l;
p=oo. Further, he treated the general condition for n=4 and n—1/4.
In conclusion, he found a, -, d such that
are all squares. For/=^~3w2, f?=2i«, we have
Then a solution is a=2g, 6=2Л, c=/+^ d=f—gt and the three quartics
are the squares of P+7g\ 2(P^fg-\-2tf).
C. F. Degen*2 treated &+my2=p*, &+ny*=q*. We may set
" Mem. Acad. Sc, St. FeU»bouTgr П, 1830 A780), 12; Comm. Arith., II, 425^37.
♦Euler,*7 eeq.p and Euler" erf Ch. IV, РеШи" and Euler» of Ch. XV, Eulet11-11 of Ch.
XVIII, Eulei« of C*. XXIL
te Mem. prteent^e acad. к. St. Pgtersbourg per divers eavamj, 1, 1831 A823), 29-38,

Chaf.xvi} Concordant Forks* 475
To avoid fractions set a=2; then y* ** 4 (m+«)(£—«*}- Set m-\-n**fgf
$*~и*=4/дг*. Hence t=Jz?-\-gf u=fz?—g, y=$fyzf and we obtain "our
fundamental solution "
bet k1 be the maximum square dividing ffm=&*£, and set
Then ?+L=A\ Now fg±2kA must be a square /V, say B1. Thus
y=4gB. Hence to exhibit the simplest solution, set* m+*=fg=k*L, and
let Ф be any factor of L such that £*+£— Л2 is satisfied [identically] by
Let В = кС. Then/р±2Ы =В* becomes
If the latter can be solved, we have х=(Ыф—ф){2> у=С>ятеех :у=£:С.
It is proved (p. 33) that x*-\-my* and хг-\-щ? are concordant if
since they equal {тп+2т+\)г and (тп-\-2п-\-1)г for x=*mn—1, y=2P
also, if fn-hn=2Q*, since Шеу equal CiwH-nJ and (Зк+я»)* far x=m—
4Q
M. Collins" proved that ^+1^ = П, ^-haj^^D are impossible for
Ka<20, except for a=7, 10, 11, 17; also, ^-^=ПГ я?-о^ = П for
1<а<13, excepta^7, XL If we know sohitione of
then X=x*w*—yVand F=2a^w are solutions of
С. Н, Brooks and S. Watson" found that ^+jf* and x*+Ay* can be
simultaneously squares only for the following 41 positive integers A^IQO:
1, 7, 10, 11, 17, 20, 22r 23, 24, 27, 30, 31, 34, 41, 42, 45, 4&, 50, 52, 67, 58,
59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 83, 86, 90, 92, 93, 94, 97, 99, 100.
Set xfj/^Vj »*-f 1 = (Н-«)*, »Ч-Л^(р-р«)а. The twa rational values of v
give пг=(А-\-р)/(р*-\-р), where p may be any positive or aegative integer
or fraction for which А ш integral,
S. Bilk86 gave a theorem said to include all the theorems by Collins.41
The equations x*+Ay*= П, ^+5^= П, «ted as (F), am satisfied if
7п Л
bvjew of the latter, tBkep=fgtq=hkfr~fh,*~gk. Then the former holds
11 Bat the «utbor bad щтеМг «* mFL
** The I^dys and Cfentlemn'» Dkiy, ЬожЫц 1807, 61-3, QdMfi. IUI1.
11 The Lady'» Ы CkntlcB^n'B Пиву. 18Ю, ЯВ*

476 History or тнв Theory of Numbers.
if (FO mtf-nh'^NV, m-iAff-TTiBf^NF, or if (Ft) mF+rrlBk*=Nh\
nf1-hmr1Afc1=iV^. Hence the solution of (F) can be derived from the
solution of (Fi) or (F0- Giving suitable values to mt nt Nt A> Bf we can
readily derive all of Collins' formulas from (FJ and (Ft).
A. Genocchi11 stated that &-\-h and x*-\-k are not both squares if (i)
h—1 , к a prime or square of a prime 8mdb3 provided the odd prime factors
of fe—1 are all of the form 4»+3; (ii) A—% & a prime 8m+3 or double of a
prime 8m+5, provided the odd prime factors of jfc—2 are all of the form
8n+7; (ш) h a prime 8m±3, Jk a prime 8m+7, provided the odd prime
factors of A—Jfc are аЛ of the form 4n+3 and quadratic; non-residues of k;
(iv) h a prime 8т-|-3, ife^A2, provided the odd prime factors of A—1 are all
of the form An+3; (v) A a prime, k^hp, when* A and p are primes one of the
form 8m+3 and the other of the form 8m-\-7, provided the prime factors
of p—1 other than 2 and A are all of the form 4n+3 and quadratic non-resi-
non-residues of A.
A. Genocehi* treated A) by the method of Diophantus: Set r=mx-\-pf
з=пх—p. Then tejI=(rx—р*)Ь/а,яо that the second equation A) becomes
p being given in the problem of Diophantus П, 17. In Euler^^ problem, p
ia unknown; the first of the preceding equations determines p in terms of
m,ntx; thent^^amB3?{m-\-n)!(ari+bm)<, Bysettingn=b^a:—2(m-\-ofyf
we get formulae dmved from Eufcr's by changing p, I, m into — pt yf r.
Genocchi noted that the pneeent probkm is equivalent to that of solving
y1—z* :&—y*^a : b, treated fully by Leonaido Pisano.* For, A) ^ves
r*—p2 : г2—р*^а :Ь, and conversely, if we set г*—р(*=агд*. Genocchi
proved (pp. 9-23) Uiat the Byetem a?-hu=Q, x3-hi = D is impossible m
rational numbers foe a=l and 6 a jmme8t±3 such thatb —1 has no prime
divisor 4^+1 (as 5=3, 5, 13, 19,29,37,43); for a=2 and & a prime S*+3
such tfiat every divisor of Ъ-2 ia of the formSl+7 (as ^^163, 331, 449);
a=2, b=*2A, A aprime8A-|-5 such that A—1 has no odd prime divisor not
of the form 8t+7 (as A^5, 29, 197, 317); «=Л, 6-ЛВ, wbere Л, Б are
primes, one of the fofm 8&-I-7 and the other Sfc+3, eucb that A is a quadratic
residue of В when ,A=8AH-7, 5 — 1 has no odd prime divisor 4£-\-l not a
quadratic residue ЫЛ, and, ш case A=%k+Z, {B—1)/2 is divisible by A if
a quadratic residue of Л (*вА=3,В=7; Л=7,В=-Зог19; Л=-11,В=23)^
a a prime 8&-|~3, every odd ртоде divisor of «— I bring of the form 4i-|-3,
Ь=а*; а=1, Ь=±8 or the negative of a prime №±3 or the square of a
prime 8fe±3, no prime <fevisor©f b—1 being of the form -У-1-1 in the third
case*
The following three papers relate to the system ^-ИР=2^ i4-2u2=3w*.
E. Lucas88 treated the equivalent system 2v*—u*=P, 2р54<«^-3ю$ and
showed how to get new eolations from one. [OL i]
* Compte» EtmduB Рлжщ 78,1S74,4SS-4L
n Memorie di Mat. e Fbjl 8oc. ХЫкв» ^
*Ny. Ann, Math., B), 16,1877, 40M

Chap. XVI] CONCOBDANT FORMS. 477
G. C, Gerono89 took t—1, without loss of generality. Since и is odd,
1, the first condition gives t^=A^-h(fc+lJ* He proved that also
(\-1у. Using the [improved] theorem of de Jonquieres2* of
Ch. XVII, we get i?=5 (excluded) or 1.
T. Pepin90 noted that Lucas88 did not treat all possible cases, whereas
the omitted cases add new solutions. By using a somewhat different
method, we get all solutions by a single set of formulas. We may limit to
relatively prime solutions t, и and take v and w positive. By the first
equation,
(or at b relatively prime, a odd. By the second given condition,
Comparing the values of t and of u, we get equations equivalent to
аг+4аЬ -W=c*-\-2cd-2d*> 4ab=3cd.
Thus a—oX, Ъ=3£/i, с—оф, d=4Xfi. Inserting these into the difference of
the two preceding equations, we get a quadratic giving
a
Since the radical must be rational, 30*-4A2 = :t72, 2X2—3^=±5*. The
upper signs are excluded modulo 3. Hence 2\*=уг-\-&у Ъ(Р=у*+2&,
a pair like the given pair. Hence a solution vt wt tf и leads to a second solu-
solution
Vi^aV+36/ДО, w1=«20K+32^V, tlf ul = (av^Gnwy-72tA*wll
where ^ :a=vw±tu : 2(9^—81^). Starting from v=w=t=u=l, we get
v/<x=Q or 1, the second giving vL=37, 1^1=33, ^=47, ttx=23; ete. It is
proved that we get all solutions in this way.
To find61 two squares whose sum is double a square and difference is 10
times a square, take x9 y=2pq±(p*-qr). Then
t
pqpq) (pt
J. H. Drummond and W. P. King?2 proved that 2зГ-у* = D, 2у*-з*= П
imply x*=y\
A. Gerajdmw noted that х7+пу* and nz2+y* are squares if
^7, x=3, j/=, y
Several writers94 gave solutions of the last problem.
R. Goonnaghtigh* made Sx*+Pyz and Stf+Px1 botfi squares.
L. Aubry*6* proved that 2у*-\-иг and Зуг-\-и* are not both squares.
» Nouv. Ann. Math., B), 17,1878, 381-3. ^-^^^^
■• Atti Accad. Pi>nt. Nuovi lined, 32, 1878-9, 281-292.
B Math. Quest. Educ. Тйпвя, 63,1895, 64.
«Am». Math. Monthly, 6, IS99, 47-8,151-5.
•» L'interm&iiaire dv rnatb, 22, 1915, 128.
М/Ьп1 23 1916 63i 2057
184^6.
26, Ш9, 84-5. For u «

478 History of the Theory of Numbers. [Chap, xvi
X*-\-y AND X-\-y* BOTH SQUARES.
Diophantus, II, 21, took y=2x+l. Then х*+у = П. Let
be the square of 2x-2. Then s
АШагкЫ9* (beginning of eleventh century), after repeating this solution,
added the condition s*-Hf = □, takings—Зг, y=4z.
Rafael Bombelli" set fl=4(x+l). Let s+2/?-16V+33x+16 be the
square of 4z-6. Then s^20/81.
W. Emerson*8 treated the problem.
L. Euler" set x*+y=(p-xJ, y*-\-x=($-yJ, whence
Euler had first inserted the value of 3/ from s*+y=p2 into #3+:г, obtaining
(pj-^i+jj^ Q, which he stated would be difficult to solve.
R. Adrain1* noted that Euler's last condition is satisfied if we take
Again, for p-\-z=v, it becomes
if x
The equivalent problem хг—у = П,у*—х— П was solved101 as by Euler."
J. W. West102 noted that Euler's" condition is satisfied if p*-x*^y,
x*=%y-\-l. Solve the quadratic in x obtained by eliminating y>
C. A. lAjsant,1* after recalling Euler's* solution in rational numbers,
remarked that the system is evidently impossible in positive integers, since
y>xhy theifirst equation and x>y by the second. Similarly for negative
solutions.
A. Auric1" noted that Euler's solution is not general, since his problem
is equivalent to the solution in integers of the homogeneous system
3*-\-uy=z*f w+lf~**, which can be solved for x, у after giving arbitrary
values to z, tt и (by factoring г*—Р).
L. Aubry** and G. Quijano10* proved the impossibility of integral solu-
solutions.
«Ertrait du FeWiit, Rreiichjtau»L by F. Wbepcke, Pkrie, 1853, 88-9.
97 L Vlgebm opera, Botogrw, 1579, 487.
«• A T№&tue Ы Algebra London, 17W, 1808, p. 239.
»Al«ebm,2,m0Tait.23ft; fttadi tranrf^Ljon, 2,1774,335-6. Opera Omnift, A)^1^482.
^TbeMatL.Coci^poiHknt.NewYork,^ 1807, 11-13,
^ Ibe Lftdie^ and Geniiemen'e Divy (ed, M Noab), N. Т., 2,1821, 45.
^ Math. QimvL Ednc. Тпвлц 67, 1887, 01.
» Noqy. Ann. MfttlL, D), 15,1015,106-8.
, ,, » p p. 23ft,
M n*L, 23,1 Ив, 87-8.

. XVI] 3^+^-1 AND Х2-у* — 1 BOTH SQUARES. 479
У7— 1 AND %* — y2— 1 BOTH SQUARES.
Bhascara1<ff (born 1114) gave sets of values of x, у for which
are both squares:
8a2—1 v* 1
VS X=+V x=^+a » = 1 *
Bhascara1" treated this problem and the similar one on x*±y*+l, as
being due to an ancient author. To find two squares whose sum and
difference increased by unity are squares, call the desired squares [^=[]4A2
and [>2=]5fc*-l, the latter being a square for k^l or 37. For decrease
by unity, use 4A* and 5#+l, a square fork=4 or 72.
Having chosen the coefficient 4, the other coefficient E) is to be deter-
determined so that when 4 is added or subtracted we get a square. Thus 2-4
is the difference of two squares. Taking 2 as the difference of their roots,
we get the roots to be 1 and 3, whence 5=4+P=32—4. Similarly, taking
36 as the first coefficient, we must make 72 a difference of two squares.
Taking 6 as the difference of their roots, we get 45 as the second coefficient;
taking 4, we get 85.
J. Cunlffie108" solved ^=Кс*+еР)+1, y'^J^-cP) by taking c=d+nt
y=rn, whence n=2d/Br*—1) by the second condition. Take tf-*Br*—1),
x^U+\. The first condition is satisfied if Df*+l —p)s=2t. For t=2r*,
we get ВЫвсага'э final answer*
E. Clere10* treated the same pair z2-\-y*~ 1=2*, хг—у2—l=u2. By
subtraction, 2jf=z2—ti*. Let y=pq and set z-\-u=2q*t z—u^p7 Qthus
limiting to integral solutions]. Substituting the resulting values of z, и
into the proposed first equation, we get 4tfI=4+4g*+p<, which is a square
if p =£*. Thus we have the special solution
A. Genocchi1" proved that all rational solutions are given by
where p, q are relatively prime integers, q odd; r an integral divisor of
д^(р4-\-4^*) and r=p (mod 2). We may give any rational values to g, p,
q, г and, without loss of generality, replace r by gr. Then y=4pgr/rf,
^(p'+^+r^/d, where d=p<+4tf<-r». If we set r=2f> p = l, we get
у =8g*, x=$q*-\-l; tf we set r=pr—2f> p = —1/2, we get also the first set
by Bhascara,
i" LfUv&tf (кпйи), И 59-6L Algebra with arith. and mensuration, from the Sanscrit of
Brahmesupt* and Bhieeafs, tmnst. by Cofebiooke, London, 1817, p. 27. Iiiawati or
a treatise od aritb. and евош. by Bhaac&m АяЬягущ tnuL by John Taylor, Bombay,
1816, 36.
i" Vfjvg&nita (algebra), | lW; Colebrooke/^ pp. 257-4.
»•* New Seriea Math. Repository (ed.p T. Leybourn), 2,1809, l^ 199.
^ Nouv. Ann. Math., 9> 1850, lld-8.
0, 1851,80-85.

480 History of the Theohy of Numbers, [Chap» xyi
T. Pepm111 found that all rational solutions are given by
dx=mtp-\-n\ dy=4mn8i, dz=2mn{s*+2P), du=2mn(v*-2P),
where d=mtp—n2q, while тп, n are relatively prime, also s, t> Further,
pq. To obtain integral solutions, take
^/5
Various writers112 gave solutions.
J. H. Drummond11* took
=2re*, whence x=2n\ y=2n.
E. B. Escott1" asked if x*-\-y*-lt x2-y*+l are squares when xy$Ot
lor integral values other than x = 13, y=ll; in other words, if
4mn(?n*—л2)+1 = П (тиифО, тлФ**) has a solution other than m=Z, n=2.
Several replies112 show there is an infinitude of solutions.
R. P. Paranjpye1" gave Bhascara's1*7 third solution. Suppose in
2j/*^z*—t1 that the common factor of yt z,t is a square. Since the difference
of two squares is divisible by 8, we may set *+f=4£2, z—i=fl^f y^2fa
Then s*=l-0*+*=4F+V+l. Assume that ч4=Ч\ whence £=2^,
**-\-%fxy+hy*, x*-\-2gxy-\-ky* both squahes.
Beha-EddmK beted as the last of seven problems remaining unsolved
from former times that to make x*±{x+2) both squares. His translator,
Nesselmann (pp. 7245), discussed the problem.
A. Mane117 found only the solution x^—17/16 and concluded that
the problem is impossible in positive integers.
A. Genocchi11* called the squares {p+q)* and (р—дУ, whence x2 =^p2-\-q*>
x+2=2pq. By elinunating xf Dp*— l)q*~8pj-(p2-4)=^0. By taking
the first or third coefficient zero, we get x= — 2, —17/16, 34/15.
E. Lucas11* solved completely the corresponding homogeneous equations
x*+xy-\-2y*=u\ x*-xy-2y>=v*f
where x> y> u, v may be assumed relatively prime. Adding, we see that the
sum of the squares of (u±t>)/2 is x1, whence
Substitute the resulting values of u7 vf x into the equation obtained by
subtracting the proposed equations, we get 2у2-\-ху=4г8(г*—s1), whence
y=J(-z±O> where
A)
m Nouv, Ann. Math., B)x 14V1875, 63.
"* Math. Viator, 2, 1887, 66-70.
"»Amer. Math. Monthly. 9, 1902, 232.
ш L'intenrtdiaite dee m&th., 12, 1905, 76.
Ш/КЛ,2ОТ-2П; 13,1906,25, Cf. Zen" of Ch. XIX»
«Jour. Indian Math. Club, Madras, 1,1909, 18S-9.
nT Noov- Ann. NUth., 5,1846, 323.
Ui AnxiaU A Sc. Mot. e Ffe, 6, I855r 132, 303-4.
"' Noot. Ann. Math.t B), 15,1876, 359-365, Same in Bull. BibL Sioria Sc. Mat., 10,1877,
86191

Chap. XVI] x*-\-2fxy-\-hy\ xt-\-2gxy+kyt Both Squares. 481
Hence the product of r'+lon?-*2** is 252rV; call the factors
and add and subtract. Thus
For the upper sign> one solution of A) leads to two new solutions:
so that the proposed pair has the solutions
For the lower sign, the problem is reduced to the earlier case.
A. Gerardin1* used the known solution of u2+v>=2xt:
u=2m*-P, v=2m*+l*-4lmt x=2m*-\-F-2lm.
It remains to make 8u*—7afe (x-\-4yJ a square:
Then x=34, |/ = 15, «=46, t>=14 [Genocchi]. It Is stated that we have
also у =—32.
L. Euler121 solved x2+2fxy+hy*=P*, x*-\-2gxy-\-hy*=Q*. Subtract and
set P-Q=U~0)V, whence PfQ=2x-\-y(h-k)l(f-g). Squaring and
adding, we get 2F?-|-2Q2; equating to the value obtained by adding the
proposed equations, we get
x :y^(f-gy-2(h+k)(f~g¥+(h-ky :Ц/-д){р~д*-Н+к).
N. Fuss122 made Jl=x1-\-2axy-\-y'1 and /2=:^-f-2bx$/-|-jf both squares, say
p* and q\ Then р*—дР~2(а—Ь)ху. Hence х=4(а-\-Ь), у = (а—Ъу-+4 is a
particular solution since
To find n such that х*±2пху-\-уг are squares, say (p±q)\ we have
^4-^=^+5», nxy=pq-
Set p=oxy> ^^n/^. Then n^a^-h^
A. S. Werebrusow1* reduced the system
H. В. Mathieu gave the solutions ^=15, y=-8; ^^1768, ^=2415,
of a*+j«--nf xl4-^4-j/2=D- L. Aubry126 gave a general discussion.
Adrain,m Genoechitug etc., of Ch. ХХП proved that x*±xy+y* are
not both Bquaies.
ppe, 190fh7v 162; A^oo. fmnr?., 1906» 17.
Opera poetuxria, I, 1862, 254 (about 1777). Nova Aeta Acad. Petrop,, 13, 1795-6 A778),
45; Comm. Arith., П, 292.
■*M&n. Acad. So.St. ШвпОхищ, 9, Щ24 A820), Ш-160.
» Math. Soc. Moscow, 26r 1098, 497-543; Fortodiritte, 39,1906, 259.
» L'intermediaire dee math., 17,1910, 219.
»»Ibid., 283-5.

482 History or the Theory of Numbers» [Chaf. xvi
TWO FUNCTIONS OF ONE UNKNOWN MADE SQUARES»
C. G. Bachet^6 treated the double equality
by factoring the difference N into \ and 4JV, where 4JV is the double of
V4iV*, and equating the squares of %DN^\) to the given left members*
whence, in either case, -JiV = 65/04» If the second equation is changed to
4JV*—JV—1 = П, use the factors 1 and 4JV.
For JV2-12 = D, ¥#-12-111, use the factors N and JV-13/2 of the
difference, so that their sum shall contain the double of VjV^
Fot 4Ar2-JV-4 = D, ^VH-ieiV-D, use the factors 4 and 4JV+1 of
the difference.
For AT3-6144AT+1048576=D, JV+64-П, first multiply the latter by
163S4.
Fermat1*7 treated many double and triple equalities»
J. L. Lagrange12* considered briefly the system
A) а+Ъх +ca*=U, «+0x+tx'=D.
If o,-\-bf-\-cp=g27 the general solution of the first is
Then the product of the second quadratic by (m2—c)? is a quartic function
of m. There is no known rule to make the latter a square» If a=«=0,
set x-=*lfy\ we are led to the simple problem by-\-c= D, Py-\-y = D.
IL Adrain129 treated ax*-j-b = D, сз^+d- D, given аг*+Ь = е2, by setting
х=т-\-у. Then ах'г-\-Ь=е2-\-2агу-\-ау2=(гу—eJ determines у rationally
in z. For this value of y, the second condition becomes Q — □, wheie Q is
a quartic hi z\ but no treatment is given. Next, consider A) for the case
in which с and 7 are squares; by multiplication by squares, we may assume
that the coefficients of x2 are equal and proceed as in the following example.
For x*-x+7=A\ s*-7z+l=B2, we have 6s+6=A*~B*. Take
2x-\-2=A+B, Z=A-B.
Inserting x+5/2 for A in the first given equation, we get x = 1/8.
Several13* solved l-8a = D, x-4z2+4^D by inserting x = (I-a*)/8
into the second condition. Two answers are
x = 19740/177241, 72165/578888.
W. Welmin130* employed the elliptic function Ф(Х) obtained by the in-
inversion of the integral
B) X
1M Diopbanti AlcxaDdrioi Aritb., 1621, 439-440. Comment on Diop. VI, 24 (p. 177 above).
ш Oeuvree, Ш, 329--376, French tranel» of J. dc ВШу'в Inventum Novum. See de Billy** of
Cb. IV; Feramt9"'! and Оадплт« оГ СЬ. XV; Format3" of Ch. XXI; Fermat" of Ch.
XXIL
** Additions to Eulerb Algebra, 2,1774, 557-9. Eulcrb Opera Omnia, U)p It 596; Oeavrea
de Lagrange, VII, 115^7. Extracts by СовдаК," 1QS-113.
m The Math. Correspondent, New York, 1, 1804, 238-240.
ш Math. Miscellany, Flushing, N. Y., 1, 1836, 67-72»
*•■ Annales Univ* Warsaw, 1913, 1-17 (in Russian).

Chap. XVI] PAIRS OF QUADRATIC FUNCTIONS MADE SQUARES. 483
If X be chosen so that ф(\), [аф\\)-\-Ь\и2 and (^(XJ+dI'* take rational
values, rational solutions of the pair of equations ах*-\-Ъ = \3, cx2-\-d=\3,
&Ге%1 = ф(\)9Х2=фB\),зь=ф(\-\-2\)) — >. In order that there be an in-
infinity of solutions, it is necessary that the integral B) have an irrational
ratio to the same integral extended from x to <*> ш
M. Rignaux111 proved that 2y*-\-\ and3y*-|-l are both squares only when
Miscellaneous pairs of quadratic functions made squares.
Diophantus, II, 31, made xy±(x-\-y) squares» Since 22+32±2-2-3 is
a square, take xy = B*+&)x*, x+y=2>2>3x\ whence у = 13х, 14х^12з2.
Paul Halcke132 gave three ways of solving the problem.
L. Aubry, Welsch and E. Fauquerabergue1" proved that the problem is
impossible in integers.
Diophaiitus, II, 26, found two numbers A2a;2 and 7xz) such that the
square (lfo2) of their sum minus either number gives a square. Hence
This problem was treated by J. H. Rahn and J. Pell,134 and the latter
treated (p. 102} the corresponding problem (Diophantus, III, 3) for three
numbers.
Bhascara1*5 made 7y*-\-$z2 and 7y*-$zz+l both squares. Treating the
first by the method of the " affected square " (Ch. XII) with 8z* as the
additive quantity and 2z as the least root, we get 7BzJ+8z*=FzJ. For
y—2zt the second expression becomes 20z*-|-l and is a square for 2=2 or 36.
W. Emerson13* made xy-\-x and xy+y squares.
Fr. Buchner1*7 made xy—x and xy—y squares by taking y=p*x-\-l.
Then xy-y = (px-mJ it x = (mz+l)lBmp-p*+l).
S. Tebay138 made х*-\-сху+у*±а squares. Let x*-\-cxy-\-y*-\-a= {y-\-p)~
determine y. Then х*-\-сху-\-у*—a = D if xA-\—• ={x*—cpx-\-qJt which
gives x.
Several139 proved that P+Q=R\ P*+Q*=S* imply that P*+Q* is a
sum of two squares:
Also PQ is divisible by 12. To find140 all integral solutions Pt Q, set
Q^Pqj-p. Then P+Q^R2 gives P, while P*+Ql=s*P* if p*+g*=pV and
hence if p=m2—n*, q=2mn.
№ L'intermeduure dee math>, 25. 1918, 44-5.
ш Deliciae Matbematicae, oder Math. Sinaen-Confect, Hambui^i 1719, 245-6.
L'i h 8 ft
m L'interm&iiaire dee math., 18, lftll, 71-2, 285-6; 20, 1913, 249»
lu Rahn'e Algebra, Zurich, 1659, 110. An Introduction to Algebra, trenal, by T. Braocker
. . . Augmented by D. P[eil], London. 1668.100.
ш VijargRmta, 5 187; Colebrooke,!07 p. 252,
ш A Treatiae of Algebra London, 1764, 1808, p. 379.
™ Bdtrag eur Aufi. tmbest. Aufg, 2 Or,, Progr Elbuigr 1838»
U6 Math. Queet. Educ. Tinrte, 44, 1836, 62-3,
l* Ibid.t 54> 1891, 38.
w Ibid., 60, 1894, 128. Cf. Teflhet111 of CJh. XXI.

484 History of the Theory of Numbers, [Сйар. xvi
A. C. L. Wilkinson141 made 2y*—z? and 2z*—у*-|-1 squares x1 and
?utx=a+b, z^a-Ъ. Then sI+z*=2y* gives а*+Ъ*=у*;
Ъ=2ктп. Take тк=9, n=7. Then «1 = 193Bifc)i-hl. By the continued
fraction for Vl93, u=6224323426849, k=224018302020.
A- Gerardin and R, Goonna^itig^i14J treated
<y-x)*+x~A*t (у-хУ-у=В*.
Add and set t=x—y; then 2P+t=At-\-Bt) an easy problem.
ш Jour. Indian Math. Club» 2,19lOt 193.
»> L'intemwkliaire dee math., 22, 1915,193; 24f 1917,

CHAPTER XVIL
SYSTEMS OF TWO EQUATIONS OF DEGREE TWO.
TWO QUADRATIC EQUATIONS IN TWO UNKNOWNS.
Beha-Eddin1 A547-1622) included (ae ProK 3) among the 7 problems
remaining unsolved from former times that to make x*-\-y=10, tf-\-x=5.
Nesselmann noted that there is no rational solution. Marre p* 323)
noted that it leads to x4—2Qx*+x-\-95=Q, having no rational solution.
Cataldi2 required x, у when a* + j/z and xy/(x — yI have given values,
and treated separately the case in which the values are 20 and 1.
Fermat1 treated the problem to find in how many ways a given number
m is the difference of two numbers whose product is a square. If m=2kp*qb,
where p and q are odd primes, the number of ways is 2db-\-a-\-bw If there
is a third odd prime r with the exponent c, the number of ways is
±abc-\-2ab-\-2ac-\-2bc-\-a-\-b-\-c; etc.
s*+jf=n, then x^y or x=l-y. For the latter,
yyy&y
A. Martin6 found the rational solutions of х-\-у=х*-\-у*=О by setting
x=02, y = bz, z = (a+b)/(al-\-h*)J where a'+fc^Q, the last being satisfied
in the usual way, M> Brierley* took y^rx and found s=(r+l)/(H+l)#
Then take r=3/4, whence r*+1 = D >
J, Hammond,7 to divide a product N of two unknown primes x, у into
two parts P, Q, each > 1, such that PQz= - \ (mod N)y tabulated for each m
A<wi^15) all solutions щ щ, P, Q, N, x, у of P+Q~N, PQ+l=mN,
whence P=m-\-n> Q=m-\-nu N=2m+n-\-ni, nn1 = m*—1.
Problems of Heron and Planudb; generalizations.
Heron of Alexandria8 (first century B.C.) treated the two problems:
(I) Find two rectangles such that the area of the first is three times the
area of the second [and the perimeter of the second is three times the
perimeter of the first]. It is stated that the sides of the first are 3*-2=54
and 54-1 = 53; those of the second, 3E3+54)-3=318 and 3; the areas
are 2862 and 954 [semi-perimeters 107, 321],
1 Eaaeos dcr ВдеЬе&кшшЬ yod Mohammed BehA-eddin beo АПюавып лив Amul, едаЫвсЬ u.
deutech von G. H. F. Nwelmann, Berlin, 1843, p. 55. French transit by A. Mvre.
Nouv. Ann. MatJi., 5, 1846,313. Of. Oenocchi, Aniu£ di Sc. Mat, e Fie., 6,1855, 287.
1 Nuova Algebra Proportionale, BoIqsds, 1619, 51 pp. (chidJy on cubca and cube roote,
4!M3)
)
, П, 216; letter from Fermat to Meraenne, Dec. 25,164a
4 The Gentleman's Math. Союрашоп, London, 4, No. 20,1817,643-4.
* Matb Qoeet- Eduo, Tim», «2, 18Э5, 70,
'/Ы, 67,1897,72.
7 Math- Queet, and Solution*. 1,1916,18-19-
«lib» Gwponicus (ed., F. Hultflch), 218-ft. H. Soh6ne, Heronia Opeia, Ш, Leipiig, 1903,
4S5

486 History of the Theory of Numbers. [Chap, xvii
(II) Find two rectangles of equal perimeters such that the area of the
second is 4 times the area of the first. The sum of two sides of the first
rectangle is taken to be 4*—1 = 63, and one side 4—1=3, so that the other
is 63-3=60. For the second rectangle one side is 4*—1 = 15 and the other
63-15^48, The areas are 180, 720,
Pappus of Alexandria9 (end of third century) discussed the simpler
(determinate) problem: Given a parallelogram, find a second whose sides
have a given ratio to the sides of the first, while the areas have a given ratio.
Maximus Plamide10 (about 1260-1310) discussed the problem to find
two rectangles of equal perimeters such that their areas have a given
ratio b : 1. The solution is given in words; expressed algebraically, it
states that the sides of one are b—1 and bP-b, those of the other Ь2-1
G. Valla 1 applied the last rule for 6 = 3, 4,
M, Cantor12 noted that the general [but see Zeuthen"] solution of
Flanude's problem is as follows: sides of first a and b(b-\-l)at sides of second
(b)d
P. Tannery" discussed the generalization of Heron's two problems:
(!) a(z-\-y)=u-\-v, xy^buv,
and stated that the general solution, obtained by setting a=pq, a2b — l = rs,
$b = mn, is
u==af!q, v=a{x-\ry)—u> x^abu+a^ur, y = abu-\-Pms.
For a=b} Heron gave x = 2a3, y=2a3 — 1, u=at v=2aBa*—1); for a = l,
B) x-F-l, y=V(b-l), u=b-lt »=b(V-l).
♦ Ad. Steen" discussed the rational solutions of Planude's problem.
H, G. Zeuthcn1* noted that, to obtain Heron's solution of A) for a=b,
it suffices to assume that u=a, whence by the first equation у is a multiple
za of a. Then x-\-y-\-\-zy xy^a?z> Eliminating г, we get
which holds H x-a3 = a3—1, у—а*=а3ш Next, for a^l, try v=bx, y=Vu.
Then the first equation A) gives Q>-l)x = (¥-l)u) which is satisBedby B).
If we replace the common factor b— 1 in B) by a, we get Cantor's solution,
which is however not the general one. If we use vlx=m in place of the
earlier vjx = b, we get y = mbu by (h) and find that the general solution of
■Sammlung, Bueh III, Раррия ausgabc (cd., HulUcb), Berlin, 1875, 1877, 1878, 126. Cf.
M. CaDtor, Geschicbte Math., cd, 3, 1,1907, i54,
10 Computation (Reehenbuch, Livre de CalcuJ). Greek text by C. I. Gcrhardt, Halfe, 1865,
pp. 46, 47. German tranal. (iDadequak) by H. Waeechkc, Halle, 1878- M. Cantor,
Gttchichte Malb., ed. 3, 1,1907, 513.
11 Dc Expetendia et fugiecdie rebua opue, Aldus, 1501, Liber IV (= Arithmetic**, Ш), Cap. 13,
n I>ic Romiscben Agnmenaoren, 1875, 62-3, 194-6.
u I/Arith» des Greca danft Heron d'AIexaodrie, M6m. soc< ec. pbys. et nat. Bordeaux, B), 4,
1882,192.
y^if for M*th., E)F П, 139-147,
M&Ui., C3), 8, 1907-8,118-120, 127-4.

System x^2y*-l, s*=2z2-L 487
(I) (ora=b is
G. Lemaire1* and E, B. Escott17 gave the solution
of Planude's problem. It becomes B) for с=Ъ* — 1. Escott gave two
particular solutions of the problem to find two parallelopipeds with equal
sums of sides, equal surfaces, and with volumes in a given ratio q:
х+у+г*=а+Ъ+с, xy-\-yz-\-zx^ab-\-bc-\-ca7 xyz—qabc.
See papers 438-440 of Ch. XXL
IT. Biniw gave two solutions in integers of the last problem and nine
sets of solutions of Planude's problem, each involving a parameter. He
rationalized the discriminant of the quadratic with the roots x, y, satisfying
A) for a = 1.
System x=2#2-1, s*=2a2-l.
Fermat19 stated that x = 7 m the only integral solution, excluding of
course the evident solution x = ±L Cf.pp. 56, 57 of Vol. I of this History,
E. Lucas* wrote 3=2^-10*, w=±l. Then x2 = By*+w*y-2{2yw)\
Multiply the latter by -l = P-2-R Thus x*=2r*-#, where
In view of the proposed second condition, set s = zfcl, whence
becomes
Also т — {у^ЫJ-\-у2, since iy=±l. Thus r and r2 are sums of squares of
consecutive integers and hence r=5, z = 7, by papers 26-30»
T, Pepin2; treated 2y2(y2— 1) =г2~1, obtained by eliminating x. For у
odd, у = apt z±l^2a% zTl =8p% whence «2jS*-l-8uA, ±l = az/b
so that
Thus «^±4^—mA, ^2rFA=4n/:, where то, n are integers making mn^L If
?n=n^+l» the lower sign is excluded and the upper gives 2h—0l-\ra2i
8* =/?-«*, whence «*jS*-l=8A* becomes «4-272-l, 7^(a2-i32)/2. The
case m~n= —1 leads to the same relation. This Pell equation has no
integral solutions except « = ±1, 7=^0. Next, let у be even, y=2u* Then
*-J2» L-Dw« -lJ, г
w L'iutenuediairc dee math., 14,1907, 287.
177Ы„ 15, 1908, 11-13-
"Ш., 15,1908, 14^18-
i'Oe\ivifflF U, 434, 441; letters to Carcavi, Aug., 1659, Sept., 1659. a. C. Henry, Bull,
Bibl. Storia Sc. Mut. e FSs,, 12, 1879, 700; 17, 1884, 342, 879P letter from Carcavi to
Huygene, ScpL 13, 1659 (extract from letter from Fermat).
10 Nouv, Ann. Math., B), 18, 1879F 75-6. ffia u, ж, у are replaced by aj, y, w.
31 Atti Accad. Pont- Nuovi LinceiF 36, 1883-3, 23-33.

488 Histoky of the Theory of Numbers, [Chap, xvii
/, g relatively prime integers; the upper sign is excluded by use of modulus 4>
Restricting to positive integers, we have u=ct&, /=«*, д=*0*> Thus
(а*+2$2У = 1+2B^)*, the discussion of which as a Pell equation leads to the
condition mA—2n4 = ±l. The upper sign is excluded as it leads to
l+c* = d*. For the lower sign, т*=п? = 1, as noted in his next paper {ibid.,
p. 35).
A, Genocchi22 treated z2^yA-\-(y2—IJ, obtained by eliminating z. For
у odd, y*—l = (P—Q1)/2, y2=fg, where /, g are odd and relatively prime.
Thus/=m2, g-n2 and 2(тк-\-1) = (т*-\-п2)\ whence m*^n2 = l, x = lt by
known theorems on 2r*+2s<^n. For у even, y2—l =P-(f, y*=%fg, where
/, g are relatively prime, and/is even. Thus f- 2л2, g=&* and p2 —1+8*%
wherep-2<*2-|-/32. Hencep±l=2m\ p=Fl = 4n4. Thusm^l=2n\ which
is impossible unless m2 —1, whence x= — 1 or 7.
Genocchi23 cited his** paper of Ch. XVI in which he proved that
2г*+2$*ФП, whence р4-Ьа4Ф2П, so that Pepin's'1 condition тА-\-1=2{п*У
requires тя2^1.
S. ReaHs24 gave a discussion quite similar to that by Genocchi.22
E. Turriere26 treated the system s=2y*-l, x* = 2z2-L
A NUMBER AND ITS SQUARE BOTH SUMS OF TWO CONSECUTIVE SQUARES»
E. de Jonquieres25 gave a proof, valid only when у is a prime, that у
and y* are both sums of two consecutive squares only when у=Б. Like
errors invalidate his27 result that if a number and its square are both
expressible in the form z*-\-t{x+iy7 then $«=1, % 4, 5, 7, or 9. Cf. Lucas,20
and papers 89, 90 of Ch. XVI,
T- Pepin2* reduced the problem to a certain quartic which he did not
solve completely» If y=P*-\-P\, all decompositions of y1 into a sum of
two relatively prime squares are given by yz={P2—P?)*+BPPiJ. Taking
2PPi and ±(P2—P]) as consecutive integers, de Jonquieres assumed that
P and Pi must be consecutive. While this condition is necessary if у
is a power of a prime or the double of such a power, it is in general not
necessary,
E. Catalan25 asked for numbers 2x expressible as a sum of squares of
two consecutive odd numbers, while BxJ is a sum of squares of two con-
consecutive even numbers, citing the case 2x —10» G. C. Gcrono30 proved that
2z = 10 is the only solution of the equivalent system
» Nouv. Ann. Math., C), 2, 1883,306-10,
u BulL BibL Stori» 8c. Mat., 16, 1883, 211-2,
«Ibid., p. 213, Repnxluced, Sphinx-Oedipc, 4, 1909,176-6.
» L'enaeignenient math., 19, 1917, 243-4,
* Nouv. Ann- Malb, B), 17,1378,21&-20,241-7,289-310; B), 18,1879,4S4-5, a. Meyl"
ofCh. IV.
»1Ш., B), 17,1878, 4i9-24F 433-46. Cf. Аягос. fran^. av. ee.t 7, 1878, 4(M9»
" AtU Accad. Pont. Nuovi Lined, 32,18784), 295-8.
" Nouv. Ann. Math., B), 17,1878, 518.
"JMG21

Сват. XVII] MISCELLANEOUS SYSTEMS OF TwO EQUATIONS» 489
E, Lionnet*1 stated that 1 and 5 are the only sums of squares of two
consecutive integers whose product is a sum of such squares; 1 and 5 are
the only primes x, yt each a sum of squares of two consecutive integers,
such that x7 and y* are such sums of squares» Similarly, 1, 13 and their
biquadrates arc sums of squares of consecutive integers. Cf. Lionnet314 of
Ch. XXII.
Miscellaneous systems of two equations.
Bha'scara30 of Ch. XII gave a solution of the system x2-\-y2+xy = z2,
(х+у)г+1 = П. On systems of two equations involving sums of squares,
see papers 108, 176 of Ch. VI; 97, 259 of Ch. VII.
" Umbra " *2 found numbers ax, Ъх> ex, - - - whose sum added to or sub-
subtracted from the sum of their squares gives a square. Sets — a-\rb-\-c-\-
Choose a, b, ■ • ■ so that the sum of their squares is a square q2 (by setting
q=a-\-m and finding a). Hence qVztsx are to be squares. Take t = $jq*.
Then x2±tx are to be squares. Determine x by x*-\rtx = {k—zJ. Then
z*-tz=n \l k*-2kt-$=U = {n-k)\ which gives fr.
R. F. Muirhead,33 to find pairs of quadratic equations x2 — pz-\-q = 0,
x2—qx-\-p = 0f all of whose roots are integers ^0, found all integral solutions
^0 Of а-\-$ = а'р', а'+0' = ар. Set r-(a-I)(/J-l), r'= (а'- 1)($'-1),
whence r+r'=^2. It is shown that a+0. Hence either r=-0, r' = 2, a1 =2,
$'=3f « = l, /3=5? or r^2, t'=0, or r=r' = l, <*-=<*'=$ = $'-2. He solved
also the pairs 0±л = <*'0', Р' — а' = аР*
A. Cunningham** solved S1^Si=S3, where
by multiplying by 2-103 and setting a^lOWy+r. Thus
a\—a\ = al - a\ = a\ - a\.
But if four integral squares are in A, P., they are known to be equal.
M. Rignaux** gave integral solutions of the two systems
xy+2t=* П, xz—yt=U\ xy-\-zt=xz — yl=uA,
A. Boutin" proved that x*-2y2 = l,y*-3z* = 1 imply y2^4.
» Nouv. Ann. Math., B),^Tl88l, 514.
" The Gentieman*H Math. Companion, London, 4, No. 20, 1817, €73-5,
« Math. Qucet. Educ, Times, 70p 1899, $4r4.
»1Ш.ГB), Ю, 1906,29.
* L'ioterm^diaira dee math., 25, 1918, 113-6.
*lbid.tW, ШЭ, 123.

CHAPTER XVIIL
THREE OR MORE QUADRATIC FUNCTIONS OF ONE OR TWO
UNKNOWNS MADE SQUARES.
a2x-\-dx, bV-\-ex, •■■ made squares»
J. Cimliffe1 took tP+mv^id-vy, whence v=d*!Bd-\-m). Then
if d?-\-2dn-\-mn = (q~d)\ which gives d. Then v*+pv=n if
№-mnJ+4p(q+n)(q*~mn)+4mp(q+nJ=n = (tf-2
whence $ = {p—m—n)!2.
W. Wright* equated the numerators and denominators of the two values
of d given by d*+2dn+mn~(q-dJ and d?-\-2dp+mp = (t-d)\ Thus
{f—mn=f—pm> n-\-q = p-\-L By division, q-\-t= — m. Hence
q=(p-m-n)/2.
A. B. Evans3 made k?x2—kx a square for k=at b, с. From
we get x. Then frx*-bx = О, tf^-cz- □ if
Subtract and set bcBa—m)=y+z, rn(ab—ac)=y—z. Substitute the re-
resulting у into a№(?~abc2d=y'2\ we get
Aabc(ab—be -\-пс)
&ab2e— (ab+bc—acJ"
3. Matteson4 solved d?-\-2dn-\-mn=A\ d?+2dp-\-mp = B2 by taking
2d-\-m=A-\-Bt n—p = A—B> Inserting the resulting value of A into the
first of the initial equations, we get d rationally. An equal value of d is
obtained by use of B. It is stated that if m, n, p be any three of the numbers
2016, 3000, 3696, 4056 (or any three of certain 13 numbers of 6 or 7 digits),
the six expressions tfdtmv, ifztnv, v*±pv are all squares when f = 652.
D. S, Hart5 found three squares such that each increased by its root shall
be a square. Let as, bx, ex be the roots. Take а2хш*-\-ах=т?х2. For the
resulting value of x, b2x2-\-bx and c2x2-\-cx are squares if а№—a*b-\-abm?
and aV—a?c-\-acm* are squares» Since this is the case when m=a, set
m*= a-\-n. Multiply the resulting expj-essions by c2 and b2 respectively»
Then shall
abc2n2-\-2a2bcin-\-a1bzci= □ =A>,
lThe Math. Repository («L, Leyboum), London, 3, 1804, 07. The Gentleman's Math.
Companion, London, 3, No, 14, 18ll, 300^2* Same in Math, Quest. Educ. Timee,
14, 1871, 54; 24, 1876, Ж
> The Gentleman's Math. Companion, 5, No. 24, 1821, 55-60; 5, No. 26, 1823, 214,
1 Math. Quest* Edac. Times, 14, 1871, 55-6.
4 The AnalyHt, Dee Momest 2, 1875, 46-9.
ЧШ., 3,1876,81-3.
41I

492 History of the Theory of Ni^bers.
Factoring the difference, we set a(c — Ь)п^А—B, 2dbc+hcn — A+B. In-
Insert the resulting value of A into the equation involving A\ We find that
{(ас-\-Ъс-аЬJ-4аЬ<?}*
The initial squares will be in A. P. if we take
а = 2тб-г2-\-^ b=r*-\-s2, c
whence o=l, b = 5, c = 7 if r=-2, a=l. Then x«= 151321/7863240, a result
found by J. D. Williams* by starting with the squares z*, 25x*, 49з*. For
r=4, « = 3, we get a=17, b = 25, c=31, x— —X, where
X=(86457l)V110H044931800,
and hence a solution of а?Х*-аХ=П, - >, с?Х*~сХ=П [Perkins23 of
Ch. XIVJ
Hart7 made кгхг-\-кх=П for k=a\ h\ —-. Divide by k2 and set
a = l/a', ••-. Then z2-\-ax, з?+Ъх, ••• are to be squares» Set x=22>
Then 22+o, 2?-hb, . •. are to be squares. Suppose that гг is a sum of two
squares in the required number of ways: 22=-rrii-\-7ii=jfi-{-q!= - •■, and
set a = 2mn, b = 2pg, ••.. Then 22-ha = (w-h^J, ^-h& = (p-h3J, •••.
J» Matteson* gave the solutions by Hart5-7 with amplifications.
G, B.M.Zerr'solved the systemx1-hy^z2-hti^-□,xI-t^=ze~i/I= D,
ako the system
P» von Schaewen10 made 4x2—2жх ^Н-Зх, 4х*+5х all squares» Setting
x = l/Dxi), we are to make 1—2xl} 1+3^, H-5a?! all squares Qvon
Schaewen81 of Ch. XV].
On three squares which increased or decreased by their roots give
squares, see papers 12, 12a, 21, 26, 52-54 of Ch. XIV. For two squares,
papers 3, 19 of Ch. XVI; 32 of Ch. XVII»
Thkee linear akd quadratic functions of two unknowns
SQUARES.
Brahmegupta2 of Ch, XV made x-\-y, x~y and xy-\-\ all squares.
To find two numbers whose product is a square and product plus the
square of either is a square, J. Hampson11 took Ъ*а and a as the numbers.
It remains to make ЬЧ-1 = П = (Ь—c)s, say, which gives b. R. Mallock
1 Algebra, BcetOD, 1840. 413.
7 Math. Queet. Educ. Times, 39, 1883, 47-9.
'Collection of Diophantine Problems with Solntione (edM A. Mwtin)» Washington, D. C,
1888, pp. 10-20.
•Acuer. Math. Monthly, 15, 1908t 17-18. Епопеош eoiutian in J. D. Williami' Algcbr»,
l832t 419.
ii Archiv Math. Phys., <3), 17,1911, 249-250.
u Ladi<* Di&xy, 1763, p. 34, Queet. 491; Leyboum'e Math. Quest. L. D., 2,1817, 209.

XVIH3 Quadratic Functions Made Squares, 493
took two perpendicular segments AC and CD; let CB be the altitude of
g ACD. Then AB and DB measure the required numbers»
T» Thompson12 divided a given square a* into two parts
4ra+ 4rs+l
such that each plus the square of the other is a square. Take s=r-\-l.
Then the sum of fractions is a* if 2r-\-l=ar1f whence r = (l — a)/Ba),
J. Whitley18 took х*-\-у = {х-\-ьУ, yz-\-x=(y-\-zyf which give x, у in
terms of v>z pSuler" of Ch. XVI]» Taket>=l-z» Then x-\-y=a2 gives
(l)/Bu)
J. Cunliffe14 found two numbers whose sum increased or decreased by
their difference or difference of their squares give squares» He took x and
1-х as the numbers» Since either difference is 1— 2xf 2—2x and 2x are
to be squares» Take 2x=4rc2, n = s—1/2» Then
2-2^=14-48-4^= n=Brs-1)*
gives 8>
W. Wright and Winward" took x and у as the numbers required in the
last problem. Then 2x, 2y>x-\-y^(x2—yr) are to be squares. 8etx-\-y-p,
x—y—q* Then p±g and p±pq are to be squares. Take p-\-pq=nl.
Then p-pq=D if l-g*= П = 0--rq)\ whence g = 2r/(r^-hl)» Set
Then prfcg-D if (r^-hl)(^=h2r)-a. Now r2+l-D i^ r-
Take « =2, whence r =3/4. Take m=P/2. Then m2d=2r - □ if Р*±6 = □»
Set р2_|_б=(ЗЯ-Р)*, which gives P» Set й = *+2» Then р2-6=П
if 4-| +М4=П = B+36*+3**)*, whence i=47/6. B» Gompertz took
x-|^y=p^ 1+s—y=lfp and by a long discussion obtained the preceding
numerical answer»
"Jesuiticue" w imposed the further condition that x-\-y— D. Thus
—r2, 2a;—p2, 2^=^ l-\-x—y=m*1 l—x-\-y=rilf whence p2
7i1=2, Take p=m, q=n} whence r=l- Then т*-\-п* = 2 if
Several1* solved easily the problem to find two positive rational numbers
such that each and the sum s of their squares exceed their product by
squares, and the problem when 9 is replaced by V».
FotTR QUADRATIC FUNCTIONS OF TWO UNKNOWNS MADE SQUARES»
L. Euler18 made AB±At AB±B all squares. Set А^х\гл B=y\z\
then xy±xz, xy±yz are to be squares. Since а*-|-Ь2±2аЬ — П, set
xz=2cd, уг=2аЬ.
u 1Ъе GentJeman's Diary, or Math. Bepoeitory, Nq, 65, 1795. A, Davis' cd,, London, 3,
1814,229-30,
uIbid.t No. 68,1808, 36-7, Quest» 917»
u Шха? Diary, 1810, p, 40, Quort» 1203; Leybourn's M. Queat. L. D., 4,1S17, 122-4»
u The Gentleman's Math. Companion, London, 3, No. 16,1813,421-4,
u Udiefl' Diary, 1839,41-42, Quest» 163&
1T Math» Quart. Educ» Times, 5,1860,60-L
» NoviComm. Ac»d, Petrop., 19,1774r 112; Comm. Arith.! II, 53-63; Op» Om., A)F Ш, 33S.

494 History of the Theory of Numbeks. [Chap, xvill
Then
, 4abcd ?^
2 ~V+6*' z~z2 ~2ab ' z " 2cd '
The problem to choose a, - - •, d so that a2-{-b? = c*+d* and so that the
expression for z* shall be a square was treated by Euler in §§ 3-17. In
§ 18, he began by setting (in accord with the above)
2b '
2a/ *
Then A±l^(a±bJfBab), Bifcl = {cifcd)YBc<J). Hence the conditions are
4abcd ' Aabcd
Make the numerators the squares of r*+$* and j^-bg2 by setting
To make the common denominator a square, we have the condition
which is satisfied if we have two rational right triangles the ratio of whose
areas is a square [cf. Euler81 of Ch. XVI]. The above ratio is aj& for
p = S«, q=2&—«, r=30, 8 = 2a—fi and for seven similar sets. The case
a = 9, 0=4 gives p=27, g= — 1, r = 12, s-14. By a table (p. 60) of values
of xy(xt—y2)> we get right triangles of equal areas 2-3-5-7 for x=5, у = 2;
г = 6, у = 1; x=8, у = 7; also two of equal area for
Euler19 made the four expressions AB±A±B all squares» Set A =xfz,
B=yjz> Then xyzkz{x-\-y) and xyzbz(x—y) shall be squares» This'will
be the case if
xy = a2+b -c2+tf, z(z+y)=2ab, z{x-y)=2cd,
whence
ah+cd ab-cd . <№—&&
Since x^ shall be a sum of two squares in two ways, set
Then
"Novi Comm, Acad. Petrop., I5r 1770, 29; Mem.r 11, 1830 (l78O)r 31; Comm. Arith., 1,
414; II, 43S. The simpler solution here reproduced is given [n the second of these two
papeiti, and is practically the ваше as that in Euler's poethumoue paper, Comm. Arith.r
Ilr 586-7; Opera postuma, lr 1862, 137-9, In two letters to L&gr&nge (Oeuvres, XlVr
214, 219)t Jan, and March l770r Euler (Opera postumb, lr 1862r 573-4) gave diecusmone
occurring in the first and third of these papers. On the related problem to find p, <?, r, 9
such that \pqr8(jnt—8l){ql—T4) = \JtaeeEuiert Opera postuma, If 487-^90 (about 1766J.
The eecond letter ie quoted in rintermediaire dee math.» 2it 1914, 129-131, and in
Sphinx-Oedipey 7» I912r 57-8. Fim paper in Opera Omuia, {l)t III, 148»

Свар. XVT1I] FOUR QuADHATlC FUNCTIONS MADE SQUARES. 495
The final expression is a square if and only if
By special assumptions, Euler was led to the values
Then
The condition is thus (а+30)(«+70)(л*+02) = □, which is treated by the
usual methods. From «^2, 0=1 and a= -17,1=1, we get the solutions
Я-!^:292 5-29* _J?j532_ 37 13г 532
Euler2" made z-bydba2, x-{-i/±^3 all squares» Replace x7 у by x\z> yfz.
Then 0r-b2/)zd=32, (х+у)г±у* arc to be squares- This will be the ease if
x*=2AB, y2=2CD, (x-\-y)z=A*+B2=CS-\-D*.
The final equality holds if
A=ac+bd, B = ad-bcy C=ad+bc7 D=ac-bd.
The first two conditions hold \[x = Af7 y = Cg7 2B=AP7 2D = Cg2. By the
latter,
a 2c+df g
b~2d-cf*' b
which are equal if
Hence the problem will be solved if we make R^O. Set 0 = 1. Since
#=□ for/=l (which makes x = y), we take/=l+*. Then R is the square
of 5+2*+13*75 if «=60/11. Dropping the common factor 13 in x, yr we
get
l, Acad. Sc. St. Peterisbourg, 11, 1830 A780), 45; Comm. Arith., II, 447,

CHAPTER XIX.
SYSTEMS OF THREE OR MORE EQUATIONS OF DEGREE TWO IN
THREE OR MORE UNKNOWNS,
X*-\-Z2, y*+Z2 ALL SQUARES,
Paul Halcke1 gave the solution x=44, i; -240, г=117.
N. Saunderaon2 satisfied &+& = □ by expressing г2 as a product of two
factors aw and z*/(aw) and taking half their difference as x. Similarly for
y*+z*= □. Take a2+¥^&* Then
Equate the sum of the last two terms to zero. Hence w^czlBdb). To
obtain integers, let z=4abc~ Then z-a^P-c2), з/ = ЬDай-с*).
6-4, we get s-U7, j/=44, z=240.
L. Euler3 made the last two sums squares by taking
z 2p '
Then the first gum will be a square if
First, let g-l=p+L Then must 2р*-\-Ър*+&р*—4р-|-4= □. Since 4 is
a square, the condition is satisfied in the usual manner if p= — 24. Next,
-l = 2(p+l) leads to the solution p=48/31, and $-l=4{p-l)/3 to
/ For
both (p+1J and (p—iy may be cancelled and the condition becomes
say the square of tp*+(t*+l)p—L Hence p^ — 4//{*2+l), where * is arbi-
arbitrary. If x=a, у = Ъ, z = c is one solution of our problem, x = ab, y = bc,
z=ac is another.
Euler* madeS-A2, S-B\ * • - squares, where S=Az+B*-{ . Thus
S is to be expressed as a C2 in several ways, the most general way being
-J. B:
if S=x2-\-y2 is one way. For three numbers A =xf By Cy take as С the
1 Deliciae MatheirmLicae, Oder Math. Sinnen-Confeet, Hamburg, 1719, 265.
' The Elements of Algebi», 2,1740, 429^431.
•AJgebr», 2, 1770, art, 238; French famnrf., 2, 1774, pp. 327-335. Opem OmnEa, A), I,
477-S2. a. Fuffl* and Scbwerintf11 of Ch. V.
* Novi Comm. Acad. Petrop^, 17, 1772, 24; Comm, Aiith., I, 467; Op. Omnia, A), III, 201.
33 497

498 History of the Theory of Numbers» [Chap. XIX
function derived from В by replacing/ by (/+l)/(/—1), viz»,
Then x*+B*+C>=S=xt+y2 gives */2/=8/{/2-1)/(Я+1J. Take у equal
to the denominator and multiply all the numbers by/2-bl- Hence
For/=2, we get 240, 117, 44. E. B. Escott5 also gave the last solution.
Euler* set x2—4irnnpq> y = mp—nqf z=np — mq. Then
-s*)} we get
The resulting expression for s2/4 is a quartic function of r which is the
square of wmr2 — (ra2+ra2)r$-|-mns2 if r = ±mn7 s^rrf+n*. Then
which, apart from signs, equal the products of n* by Euler's4 values when
f=m/n. The simplest solution arises from m=2, n = l: x=44, ^ = 240,
г = 117, whence x2-\-y*f etc., are the squares of 244, 125, 267.
From the sum of the roots of three squares, the sum of any two of
which is a square, subtract the area of a right triangle; the remainder is a
square which if decreased by the sides of the triangle yields remainders
which are squares in arithmetical progression. L. Blakeley7 took 44x,
1Ш, 240z as the roots of the required squares, the sum of 442, 1172, 240*
by twos being known to be squares; also let 3y, Ayt by be the sides of the
right triangle of area 6/. Then 401x-%2= D =<x2. Also, a2-3j/ = ^,
a*-4i/=c2, a?-5y=d>. Take y = r2-2dr. Then
The product of the last two is a square if d=r/6» Then d?=T2j3Qf
c2 = 25r2/36, 52=49r736 are in A. P.
P. Barlow8 noted that the first part of this question is satisfied if the
roots of the squares are 575г/48,485^/44 and z (from J. Bonnycastle's Algebra,
p. 148)» Next, we need a square which if diminished by each side of a
right triangle the remainders are three squares in A» P., whence the sides
of the triangle are in A. P., and hence proportional to 3, 4, 5» Let the
squares be (a2+2ab-b2)\ (tf+b2J, (Ъ2-\-2аЪ-а2J, with the common dif-
difference $=4аЪ(а2-Ъ*)> Thus let 35, 45, 55 be the sides. Then
if 8а3&-8Ьал^12а^, i. e., Ba—2b)(a-\-b)=3ah, which holds if a = 26.
h., 3, 1901, 103-4,
• Poetb. paper, Comm. Aritfa., 2,1849y 650; Opera postuma, t, 1862,103-4*
'Ladiee1 Diary, 1806, p. 43, Queet. H31; Leybourn'e Math. Quest. L. D., 4y 1817, 45-6,
9 The Diary Companion, Supplement to Ladies* Diary, London, 1805, 45-6,

Chap, XIX] *2+У2, &+*\ У*-\-*2 All SqUAKES. 409
J. Cunliffe9 set x2-\-y'1 = (x+y-ayj x2+zq = (x+z-b)\ From the re-
resulting two values of x we get
_(a*-d?)y-ad(a-d)
Z 2d+*2d ' a °*
Then ^+*2=D if 4<*у+4<*(аг-2л4)^+ >> *+as^(a-dJ= D. Let it be
the square of 2cfr/-+(a2 — 2ас£)з/ — ad(a — d). Then
"Calculator0 first solved &+y*=a\ x2+z2 = b7. Take &«л>-а,
z=^-s»; then a2-y2=b2 — z2 gives r = Bra—2sy)/(ri-^)f whence h7 z are
known. To satisfy x2-\-y*=аг take
л = (г»_^)(от2+пв^ t/ = (rJ-s2)B/?rn), x-(r2-s2)Gn2-7i2).
Then z = (r1+&*)~2mn—2rs(m<i-\-n2). Then j/2+z2 becomes a quartic ш m
which is equated to the square of m2—mn{r2+$*)/(r$) -n2t whence
та : n=4rs : r2+5^. Taking п=т2-{-8'г7 we have
wMch equal the products of s* by Euler's4 values for/=r/$. Cf. Euler»5
S. Ward11 took x*-\-y* = a*, x*+z2 = (m+n)\ y2+z2=(m-nJ. Then
4x2 = 2a2+8mn> 4y2=2a2-8mn, Az2=±m2-\-±n2-2a'1.
Let 2a2 = 7tt2-hl67i2» Then the first two expressions are squares and the
third becomes 3™2-12n2=CX Take т=пр, 3p2-l2=p(p-2J. Thus
et/l+g. The quartic is the square of 3-2$-|-£if if q= —16/3» Then
= 240, у=44, г = 117р which appear to be the least numbers.
W. Lenhart1* took x = (p2-l)/Bp), y=2q!(q*-l)f z = l. Then
if
provided p2-\-q* = 5= l-\-4. As usual,
e + l or3. For 5-2, p = 11/5, 5 = 2/5.
C. Gill13 obtained Euler's6 result by setting
6—a cos А+г sin A, !/=2 cos A—a sm A
and с, тЧо be the analogous functions of B. Then а2
» New Seriea Math. Repoeitaty (ed.? Leyboura), London, lr 1806r II, 3», Abo in Math.
Repository, 3r 2804, 5.
1Э The Gentleman'» Math. Companion, London, 4, No. 19, 18l6r 626-7* Same with altered
lettering, 8. Billsr The Mathematician, London, 3, lS50r 200-1.
u J. R. Youngre Algebra, Amer> ed.r 1832, 338-9.
*■ Math. Miecellanyr 2r 1839, 132. Reproduced in Math. Magazine, 2r 1898, 215-6; Sphinx-
Oedipe, 8r ШЗГ 84.
11 Application Ы Angular Analysis . . l(N. Y.t 1848; Reproduced in Math. Quest. Educ.
Times, 17, 1872,32-3.

500 History of the Theoby of Numbers. [Chap, xix
Take Л+В^90°. Then z*+if = a2 if г = 2аып2А. Take еоЦА=ф,
а«(г>+«*>3-
C. L. A. Kunze" set x=2mn, у=т9-п*,г=т?а[Ь-п*Ь/а. Then
Take а, Ъ to be Ieg3 of a rational right triangle with hypotenuse A, and
set n^mh!{2b). Then p'+s^-AW/a*. Multiplying the resulting x, y7 z
by 4ab?fm*t we get
The last solution was obtained abo by taking
x-2mnf y=mn*—m, z=
Then the first two conditions are satisfied, while
if m*+n*=4. Take m=2a{h, п=2Ь/Л, а^+Ь^А2, and multiply x, p, z
by A3/2. We get the former solution.
Judge Scott1* took x2+^=(]/-wiJJ x*+z*={z-n)\ which determine
y, z. Take w/^=0>J-32)/Cp*+g2)t nla=2pqHp1^-qi)i whence w*+rc*^s*.
Then ^H-z^D if e2x4-4m2nV+mt»V^n=sV, say, whence a:=e/2.
Take 8 = l&pq(p4—g4). We get Euler's* answer. The latter was ob-
obtained also by A. Martin (ibid,), who satisfied и*-уг=*иг*-2* by taking
i*=a(rl-sJ), t/^6(r*-$*), ««с^+^-гЬгв, ^=Ь(г2+«г)-2агз. Then
14*—^*= □ =a^ if a^pS-j-g^ ^ = 2pg. There remains the condition
^+2l = a. Divide by 4^8^ and take wi=(r?+^)/{r*). Then a quartic
in p is to be a square, say (p*—mpq—<f)\ whence plq=A/m.
С Chabanel1" used the devices of Diophantus for a similar problem.
Set
(y/t+yitJ, Vt+# = (bli+hlt)\ Since
which is a square for t=zl(a4— /5*) since т^+а^^-НЭ2J and
Multiplying the initial x, у, ^ by 2(a^-h^) V2«/3, we get
far
X, g
where the upper signs give X, У. For a^2, P= 1, we get Halcke's1 solution.
u Ueber cinlee Aufg. Dioph. analysis» Weimar, 1802r pp. 7-9»
"Math.Quert.Educ.Timee, 17, 1872,82-3. a. Martin».
» Nouv. Ann. Math., B), 13, 1874, 289-292.

Chap. XIX] &+у\ 32+2*, y*+Z* ALL SQUARES. 501
X Neubergir satisfied Euler's* condition p2+^-4-hl/p2+l/g2-urI by
the special values to = 2/(pq)t pJ+g*^4. The latter holds if
4rs
Hence ^+^=f2,a2+z2 =
C. Leudesdorf18 solved the equivalent system 2(«2+у*-и>2) =*?,
Я^+иг1—^) = у* 2(^+10*—«2)=£2 by use of trigonometric functions (cf.
Gill13). G. Heppel repeated NeubergV* solution.
J. Matteson" obtained Euler's4 result by the method of Euler.1
A. Martin30 varied Scott's15 method by making the first two terms of the
quartic in x cancel (giving x = 2mn/s), instead of the last two.
K. Schwermg21 proceeded as had Neuberg17 with X, tt in place of p, q.
To connect the result with elliptic functions, set ЯA2
Jo
Then p is a weff-know» elliptic function. By the addition theorem,
Hence if ^(w) and $'{u) are rational, also ^Bu), ^Cv), - • •, fBu), - * *
are rational. Thus one solution p, q yields an infinitude of solutions. The
relation of the same problem to Abel's theorem is considered on p. 11.
Several writers22 gave solutions.
*R Ferrari18 gave an infinitude of solutions»
R. F. DavisM gave Neuberg's17 solution.
A» Martin14* gave another derivation of Euler's* result.
H. Olson"* proved tliat, if a^+j/2^*, x'+zI=i?i, y2+22=M)*, the product
xyzuvw is divisible by 3*-44*5l.
• M. Rignaux24* stated that all solutions of s2+t/2^ П, etc», are given by
and noted four solutions, involving parameters, of the final condition.
I'Ntev. Corresp* MaOi., 1,1874-5, 199-202^ ^^
11 Math. Quart. Educ. Times, Ut 1881r 95-6.
!• Collectioa of Diophantbe Piobiema . . . r ed4 Martinr Waehiogton, D. C,r 1888, 2L
» Math. Mag&Kioer 2r Шг 214-6.
" Qeonu Aufgaben mit rattonakn Loeimgen, Progr.r Dflren, 1898r 9.
» Amcr. Math. Monthly, 6, 1899, 123-5; Math. Queet Eduo. Time*, 68,1898, l(Hl B), 11,
1007, 26-7.
» SuppI, al Periodioo di Mat., 14,1910-11,138-140.
« Math. Queata., and Solutions, % 1916, 24-25; Math. Mag., 2,1898, 216.
»• Аш. Math, Monthly, 25P X918,305-6.
♦*»7Wd., 304-5.
'♦" L'intm&ddtaire dei math,, 25,1918,127.

502 History of the Theoky of Numbers. [Chap. XIX
The preceding problem is evidently equivalent to that of finding a
rectangular parallelopiped whose edges and diagonals of faces are all
rational. If we add the condition that also a diagonal of the solid shall be
rational, we have a problem which H. Brocard25 attempted to prove im-
impossible by means of the terminal digits. P. Tannery28 noted that the
proof is insufficient since it supposes that the numbers in question are
relatively prime in pairs.
V. M. Spunar** noted that the last problem is impossible.
A. Muldiopadhyay** proved it impossible pf the edges be relatively
prime integers]. The solutions of x2+y2=O are known to be x=2k7
y=№-\. Similarly, у = 21,z = P-l; z = 2m, x = m*-l. Then
requires x=2n, $+l=n2— 1, whereas n*—P—2 has no integral solutions.
M. Rignaux29 remarked that the problem is difficult and not yet solved.
He satisfied three of the conditions, but not the fourth»
A. Traneon30 stated falsely that a tetrahedron with six integral edges
cannot have among its solid angles a tri-rectangular trieder, and stated
that one can find, in an infinitude of ways a tretahedron OABC with
integral values of the three edges meeting at 0, and of the areas of the four
faces, while the three face angles at О are right angles. C. Chabanel31
and C. Moreau31 gave the solution
0A=4xyz, OB=2y(x*+y*-z*)t OC=2x(z2+y2-z*)t
area АВС=2ху(х*-\-у*-г*)(х*+у*-\-г*).
Four squares whose sums by threes are squares.
L. Euler32 applied his4 method to A = x7 Bf С and the following D, but
was led to a condition difficult to treat and abandoned that method. Next,
take A=y, Band С as in Euler,4 and Z)= {2pz-(p1-\)y}f(p*+l)> Then
Since S=x2+y21 the condition 8^y2+B2+C2+D2 gives 1/+1У*-2дху
Inserting the value of Z), we get
Take R=gp*+2p+g. Then p=-gt 4дх/у=2(&+1) or 4. Using the
u L'intermediaire dee math.r 2r 1895, 174-5.
»/Ш., 3r I8fl6r227.
'7 Amer, Math. Monthly, 24r 10l7p 393.
28 Math. Quest. Educ. Times, 41, 1884, 60.
" L'interroediairc dee math,y 26r 1919, 55-57.
™Nouv. Ann. Math>r B), 13r 1874. 64; correction^ 200,
IJ Ibid., 340-3.
**Novi Comm. Acad. Petrop,, 17, 1772,24; Comm. Arith.p 1,467-72; Opera Ommft» A),
JIIr 203. Second method reproduced by Martin, Math- Mag<T 2,1898r 217-8,

Снлр. xix] Squares Whose Sums by Thbees abe Squakes. 503
latter value 4 and dropping the common factor x, we get
A-a В Ч-9(.
Using the former value and taking y — 2g> g=mjn> and multiplying A}
4fmn*7 D
In his second method (§§ 56-60), Euler denoted the squares by i^ ^2,
2, ^2. Let a, a be two numbers for which a*-\-cc2=A\ Let
The first two lead to a single condition and the last two to a single one:
By adding these two equations, we get z=avz/(ay)> The first of the two
becomes
ax
у V*-y2 »
To make V^ rational, set v=*y(l+s), *lR^y\A+2a*slA+<*&). Of the
resulting two solutions, one is complicatedr while the other (given by
xjy = l) is
To obtain a simpler solution in which the numbers arc distinct, take two
numbers \ & such that №+&*^В2, and set av* = pM, ау* = ЪМ. Then
ч[Ё=ВМ> But aPl(ab) must be the square of v/y; take it to be тп2/п2.
Thus
v m xAbmzbaBn z_amx
у п' у afin — abn ' у an у
Taking e = 21, a = 20, b=35, /3-12, we get Д=29, В = 37, m=3r n = 5.
For the lower sign, s/y = 3/8. Hence v-168, x = 105r 2/=28O, ^=60.
Finallyr he noted the solution
Р+Я\ U?)W?)
M. S. O'Riordana* developed the idea underlying Euler's first solution.
Let 8=А*+В2+С*+О\ S-A2^a*, ■■■>S-D2=b\ To obtain a number
w The Gentleman^ Math» Companion, London, 2, No. 12r 1809г 18Б-7; Math- Repository
(ed., Leybourn), New Series» 6r II, 1835r 1^. Reproduced in Math. Magazine, 2, 1898.
218-9.

504 Hi&tory of the Theory of Numbers. [Свдр. xix
S which is a sum of two squares in four ways, employ
Hence S = A2+<xi=B2+p = C2+y2=D*+P if
A = eti+ -fF+, В = eF^ -/E_, С=eE- -/FL, D=eE+ +/F+,
It remains to satisfy the condition S = 2AZ or, if we prefer, A2+D2=yz—B2,
viz.,
Divide by P and set =fw. Thus
v?(ac+bdJ+(ad-lny~2w(a
The roots w are rational if the discriminant
is a square. Takea=mb, c = nd, mn^r(n—l). Then shall
r(n+l)(rn+n—r)(rn—n—r) = D.
Taken = 2r. Then shall 2r2-3r = DI as is the case for r=Zs2j{2s2-\). For
e = l, we get Euler's solution 168, 105, 280, 60. Removing the restriction
n = 2r, let (nr+n—r)k^(nr—n—r)L Then shall n(n+l)(n-l)e= D,
ll—k). Take n=e+x. There results the answer a = l+k,
В. Gompertz14 employed x2^ уг, г2,
Then x2+y2-\-z2 and t^-J-^-J-г1 are squares. Also, tf+vP+z* and x2+w*+y2
are squares if
Then
A=n. Set z=tyt pq/y*=b. Then /. = 0 if
Set t= 1-J-P- The condition becomes v4^ = D = B-|-Дг±^)г and holds
if A =2-frV2, *=zbfr74-l. For g=2, r=l, we get Scott's* solution.
C. Gill16 treated the problem to find n squares the sum of any л—1 of
which is a square. He* gave elsewhere his solution for n=b and remarked
that the smallest numbers given by his formulas are so very large as to
discourage any attempt to compute them. For n = 3, see GUI.1* The
method was adapted to the case n=4 by S. Bills.** If г2, уг, х2, v? are the
required squares, their sum shall equal
« The Guatfeman'e Math» Companion, 2, No. 12,1809,182-4. Reproduced (essentially) by
A. Martin, Matb, Mag., 2,1898r 216,
•Application of the angular analyeis . . ., New York, 1848,60-76.
» The Lady* and Gentleman's Diarvr London, 1850t 53-5> Quest. 1797.
» Math. Quert. Educ. Timee, 16,1872, 108^110.

р. xix] Тбвве Squares Whose Differences are Squares. 505
Take
b=a cos4-J-z sin.4, y=a sin A — z cos A,
and c,x\ dtw coirespondmg functions of angles B7 C. It remains only to
satisfy y2-\-x2+w2 = a7t viz»,
<j'B sin2 Д-l) -o*2 sin 2A+z2Z cos2 Д =0.
The discriminant must be a square, whence
A2 = 2S сов 24+22 cos (Д-В).
Take C = A+B—90°. Then fc2 = sin 2^-sin2tf. Take sin 24 = tan #/2.
Then u=sinl4/(l-J-sin24). The case cot 4/2=2 leads to the solution
[due to Euler>« §58]:
z -186120, у = 23838, X -102120, w = 32571.
Bills gave also 280, 105, 60, 168 and 1120, 3465, 1980, 672.
Judge Scott** found 639604, 3456000, 3750000, 832797 [due to Euler,"
§55],
S. Tebay39 gave the solution x\ ■ ■, u2, where
A. MartinMa gave a complete solution by the method of Tebay.39
Three squares whose differences are squabes.
Under Euler28 of Ch. XV are cited various papers on the related problem
to make x±y, x±z7 y±z all squares.
L. Euler40 made the differences of x2, y*> z1 squares by taking
z р2-Г Z q*-V
whence x*—z* and y1—z2 are squares. Also x2—y2- Q if
Each factor will be a square if
П~ 2ab > V 2af'
The product of the latter must be a square g2. Take a, b=f±g; c, d=*h±k.
Then must (^-^(A'-fc4) = D [cf. Euler28 of Ch. XV.]
J. Cunliffe4*1 treated the problem.
"Calculator" «took
Then zt-y2 and x2—z2 are the squares of 2mn{f+&2) and
** Math. Quest. Educ- TSm«r 16, I872r p. 1QS.
*»/6irf68 883^
, , 1913,81-2.
« Algebra, 2,1770, $§ 236-7; 2, 1774, pp. 320-7; Opera Omnla, (X), I, 473-7.
«■• The Math» Repoeitory (ed.r Leyboum), Londoo, 3, 1804r 5-10.
ti The GeDtlemac'e Math» Companiont London, 3, No. 14, 1311, 334-6-

506 History of the Theory of Numbeiis. [Chap, xix
if mjn- C^-i-8)/D?). Or we may use
A = rtfr
z+y B'
Take B=*{tn-smY to get w. Then A^r2n* if r^4te7<P-3s*). He42
later used the same z, but took z = 2mn{ir1^rsrO a = (r2+e2)(m2-n2), whence
22—32=0*. Set 6=a—rt?> i/=z-|-st>; then a3-!-^2 = b*+#2 gives и in terms
of a, r, s, г. Finally, y2-z2=O if a quartic in m is the square of (say)
m2—mn(r2— «*)/{«)H-Ti2, whence
J» Cunliffe43 obtained Calculator's41 first result by the same metbod.
S. Ward44 discussed Euler's" final condition. Set/=/'$, h = h'k,
which reduces to/V*'W"-2. The latter is a square if f>* ^)
mdr2+2e'2 = nilr=P-2,3 = 2L The value for f* is asquare if l{lz-2) = D.
Taking * = 2, we get х/г= -41/9, я/г = 185/153. Or we may treat P=D
by setting $=mp and treating (mfp* — l)(m2 — 1) — Q by the usual method
for quartics, one solution j> — 1 being known.
W. Lenhart4* took the roots of the three squares to be
The square of either the first or the second exceeds unity by a square-
Hence it remains only to make the difference of their squares a square,
viz., {vx+wy){vx—wy){vy+wx)(vy—trcr) = Q. Take v=ty+x, w^lx—y,
whence vy+wx = t{vz—wy)> Then shall t(vx+wy)(vy—wx)=-O, which
holds if
The second condition is satisfied if x=2ylt. Then the first becomes
4+3^D = B—pt)z, say, whence we get t and z = p2-3, y^2p,
v=(p*+iy+Si ю = 2(р2-г)р. Or we may take xz-yi+2lxy={x-j>yy,
whence s = p2-H, ^ = 2{p-J-0- Then ^(y2_x2+2x^/0^(^-rJ if
-
Then
r 4 *
Dividing the values of x and у by <*=(рг-И)/(8р) and those of v and
41 The Gentleman's Matb. СотраЫоа, London, 4, No, 19,1816, 628-31.
«Ibid., 5r No. 26, 1823, 262-4.
« J. R. Young^ Algebra, Amcr. ed,, 1832, 339-341.
« Math. MisoelUny, 2,1839, 129-132; French tamsl.r Sphinx-Oodipc, 8, 1913, 83-4,

Chap. xiX] Squabes, Sum or Any Two Less Third a Square. 507
by d/2, we have
Three squares, sum or any two less thibd a square.
L. Euler/1* gave four methods to solve
A) y2+z2-z>=p2, x^+e'-y'^g8, x2+yl-z2=r1*
(i) Let $=x2+y2+z2. Since s=p2-\-2z2, etc», 5 must be expressible in
three ways in the form a2-J-262, whence s must have at least three prime
factors of that form. Take m=ac±2bd, n = hczFad) u=mfzk2ng, v=
Then
т2+2^ = (а*+2Ь*)(с2+2<**), (m*+2n*)(f*+2g*) =**
Take u*+2i^-=s. By using the four combinations of signs, we get four sets
of values of uf v* As we need only three sets, omit that given by both
lower signs» Set
,2v p, q=f(ac+2bd)±2g(bc-ad)t r=f(ac-2bd)+2g{bc+ad),
*i y=f(bc-ad)=Fg(ac+2hd), z=f(bc+ad)-g(ac-2bd))
where the upper signs give p and x. Compute rf+y^+z2 and compare
with the earlier expression u2+2v? for 5; we get
C=-(bc+ad)(ac-2bd).
Taking F=0, we get с : d=2b — a ib—a or —a—2b : Ъ-$-ау iwid also
f ".g= —G : 2C> The same solution results also from G=0.
('0 By C), f\g= - (С ± V) : F} where
Let Q be the square of c2—4mcd+2d*. Then с : d=2wi2-J-l : 2m.
(iii) Use pf q, x, у given by B), but take
where a, & are such that a2-J-2)SJ=a2-|-2fe2» Hence we now get new values
for F, G, С in C). For F=0, we get
с : d=— a-2b :/?-J-a or — a-J-2b :0-a.
He deduced the following simple solution of the problem: Start with any
two integers m and nr m odd, and set
s = m?+2n\ L=m2-2n2f u=2mnf
or take s, t, и such that s2=/2-h2u2; we have the solution
where p=
* Poelh* paper, Co шт. Arith.r II, 603-16; Opera рое turn a t lt 1Й62,105-118. Freoch transl,
in Sphinx-Oedipe, 1906-7,

508 History of the Theokt of Numbers» ccha
(iv) The first two equations (I) are satisfied if z = mn(A - B), y+z = 2m*A,
y—x—2n2By р=тп{А-\-В), q = mn(a2—2ab-b2)y where A = a(a+b)> and
B = b(a-b). The third equation A) becomes
Set m=f+g, n-f-g, r = (A+B)F+4(A-B)fg-(A+B)g2. Then
A. M. Legendre17 noted that the last two conditions A) are evidently
satisfied if
— as2, z=r*-rs-s2.
Then the first condition becomes r4—4rV+s4 = Q. Set r = s{2-J-0) and
make the quartic function of ф the square of 1+$ф+аф2. The case a = l
gives ф=— 23/4, r=15, s=4, whence «=241, y = 2№, z = 149, which is
apparently the least solution.
J. Cunliffe" noted that A) give x*=^(gi+r*)l etc., whence
Hence, if we set x=y+pi\ q—p+<rv, we get v=B<rp—4py)/Bp2—a2). To
satisfy 2y*—p*=r\ set
y = D(m2+n2), p = D(n*-m*+2mn), r = D(m2-7i2+2mn)J D=2^-^.
The resulting value of iffi+q2) will equal the square of
where A^2p^-\-2p<r-^-a2l if
m : n=4pV2-h4p
Taking p — <r — I, he obtained, as his least answer, a = 149, ^ = 269, 2=241»
D. S. Hart49 noted that A) are equivalent to 2^+2^= □,
2g*+2p2=CL The first is satisfied if r = p2-2<r2I q=p2+lp*+2v*. Set
p = i_^r, a = p2-J-2p<r-h2<r2. Then the last two conditions of the problem
become 2J*-HM-Hr*= □, 2Г-5-4г^4а2-П. Equating the latter to
Ba-UJ, we get I in terms of *, r, a. Then the former becomes a quartic
in t. S. Bills satisfied the first two of Hart's conditions by taking
The third condition leads to a quartic.
G, B. M. Zerr50 took a;^, у2гг and z2 as the squares and set
(A) rf+^-l-(!+«)■, ^-j^+1-a-tt)».
Since ^^^-hu2, take ^=n(p2—g1), u=2npq, whence x = 7i(p2-J-g2), Take
* ТЫопв dee nombreer 1798, «1-2; ed. 2r 1808t 434; ed. 3, 1830, Пг Х27; German tranel.
by Maser, 2,18&3,124.
« The Gentleman^ Diary, Loadon, No. 62, l802r 4l-2r Queet» 823. Math. Repository (ed,,
Leybourn), 3,1804, 97.
'■ Math, Queet. Educ. Times, 20, 1874, S4-6.
» Amer Math. MoDthly, 10, 1903, 207-8. Cf. papene 114-5 of Ch, XVI.

Chap. xiX} SQUARES, Sum of Any Two Less Third a Square. 509
y=2mn—1. Then the first condition (A) is satisfied if
n = mjz, z = m2
There remains the condition
which is satisfied if m = p*—<f, pi+qi—4;pi(f = (p--2rq2y, whence
For r=13, p = 15g/4 and the numbers are proportional to Legendre's.
Further *ets of three or more linear functions of three or more
squakes made squares*
Leonardo Pisano," to make x*-\-y2, 2*+ff+z\ z*+y7+z*+w\ - > all
squares, took the first square x2 to be 9. Then the second, y2, is the sum
16 of all odd numbers 1, 3, 5, 7 preceding 9, whence 9+16 = □ = 25. As
the third square take the sum 144 of all odd numbers < 25 whence
144+25=П = Ш. As the fourth square take 1+3+••- + 167=7050
whence 7056+169 = 0=7225. As the fifth square take
1+3+ —+7223 = 13046444.
Leonardo noted (p. 279) that, since 7225 is the square of 85, not a prime,
we can get several values for the fifth square. Besides that given above
we may take the sum of all odd numbers ^ 7225/5—5 — 1 and get the
square 720s, or the sum of all odd ^ 7225/25-25-1 and get 1322. A.
Genocehi62 noted that a fourth solution was omitted, viz-, the sum 204^
of all odd ^7225/17-17-1.
F. Felicia-no53 gave only 9, 16, 144.
N. Tartaglia" obtained 25, 144, 7056 by Leonardo's method.
J. de Billy* found the squares 9, 1/100, B3/15)* such that if 15 is added
to the sum of any two of them there results a square. £Due to Diophantus,
V, 30; cf. Fermat* of Ch. XVJ
L. Euler,*6 p. 604, stated that it is not possible to find four squares
such that if each be subtracted from the sum of the remaining three the
difference is always a square.
H. Faure66 proved the last theorem by use of the lemma that
2x2+2y2+2xy=z7 is impossible in integers»
Euler" noted five sets of solutions, like p = 89, g=191, r = 329, of
11 Scritti, II, 254, note od margin; 279. T*e Seritti, 57, 112»
"Annali di Sc. Mat. e Fis.r Q, 1855, 355-6.
« Libro di Arith . . . Scaia Grimaldem, Venice, 1526, f. 5»
и La Seooada Parte Gea, Trattato Numeri et Misure, Venice, 1556, f. 142 left.
u Diophantva Geometri», Рапн, 1бвОг 117-8,
» Nwjv. Ann. Math., 16,1857, 342-4.
17 Opera poatuma, I, 1862r 259-60 (about 1782).

510 History op the Theory of Numbers. [Chap, xix
To make x2+y*+2z\ ^-h^-h^2, y2+z2+2x* all squares, A. M.
Legendre5» set y=x+2p, z=x+2q. Then я8+у*+2з*=4Сг+,/)* for
Bf—p—2q)x~рг-5-2^—/*. Equating this to the value found similarly
from х*+г?-\-2у2 = О, he was led to the values
Substitute these Into y2+z2+2x2 and set p/q = 1-J-0. Then shall
The particular solution 6=208 gives z = 18719, ^ = 62609, 2 = 18929.
T^ Pepin5* noted that also в=— 1 and —2 (whence x=y = 7> 2=23;
x=y —2 — 1) and applied his first formulas (Ch. XXII) with Xi=0,
x*= —1, arj= —2 and found $= — 8/15, whence з : t/ : 2 = 77 : 77 : 253.
C. Gill and W. Wright» made x*-J-2/2-M*+i>2, x2+y2-z2+v>, x7-y2
-h^-h^2,2/2-h^—x2-J-i^ squares. To satisfy the second and third conditions,
take 2vx=y2—г2, say 2v=y+zt x=y—z. The fourth condition holds if
y2-\-l0yz+z2= C\ = (y—p)*t which gives y. Clearing of denominators, we
now have
Then the first condition leads to a quartic in p; equating it to Cp2—2pz+dJf
To find four squares the double of whose sum is a square, and double
the difference between the sum of any three and the fourth is a square,
they took (x+yJf (x—y)*, v*, г\ Then two conditions are satisfied if
v+z=4x, v—z = y, and the solution follows readily.
The solutions of the system 2x*+2y*—3z2= П, etc., and the system
x2-\-2(y2— z2) = O> etc., offer no special interest.
To find three numbers such that the square of each plus the product
of the same number and the sum or difference of the remaining two gives a
square, several61 used the numbers а2, Ъ2, с*. Then the conditions reduce
totf+^+c^n, аг+<*-Ь*=П,а3+&-<?--=П. То satisfy the first two,
take b?=2ac. Equate the third to (cn-aJ. Take n= —3/4.
A. GcYardin" treated the system N^Ptf-k* (P=rc+1, n+2, •--,
«).
E. Fauquembergue*3 made the four functions x2±hy2, u2±.hy2 squares»
II. Holden**1 showed that
u Throne dcB DOmbnee, 1798, 460-1; ed, 2r 1808, 433-4; ed. 3,1830r II, 125; German tranaE.
by Maeer, II, 122. J. Cunlifie, New Serica of Math. Repo»itoiy (ed.t V. Leybourn),
1, 1806, 1, 189-191, used the same method with 2j> - 2q, - 2p replaced by m, n, and
obtained an equivalent result.
" Atti Accad. Pont. Nuovi Lineei, 30, Ш6-7Г 219-20.
и The Gentleman's Math. Companion, London, 5, No. 30, 1827, 579-83.
« Ladiee* Dtary, 18ЗЗ, 38-39, Quart. 1547.
л LJinterm&iiaire de& math,, 23, 1916, 88-93. He gave 139 examples.
л Ibid., 24, 1917, 3S-9.
*■ Meeeenger of Math., 48r 1918, 77-87, I6G-179.

Chap. XIX] QUADRATIC FoRltS IN X, y; Xt z; yf Z МА1Ж SQUARES. 511
can usually be made squares by values of x, yt z which are rational functions
of a parameter к whenever the auxiliary equation &ур*+гуас?—аAг* can be
similarly solved. For, if rational functions pi, gL, rx of к satisfy the latter,
a linear relation ptf+qiy+nz=0 implies А = П. Similarly, if rational
functions pi, $2, r2 of m satisfy the auxiliary equation, then т2Х+ргу+Яг£=0
implies Д = П. Solving the two linear equations, we obtain xf y, z as
quadratic functions of m and k. For these values, С becomes a quartic
function of m whose first and last coefficients are squares of functions of к,
so that С can be made a square. For a=0^2, 7=1, we have the problem
of a rational triangle with rational medians. EulerV* equations A) are
treated by this method and by & related method. In the second paper he
used the method to make px*-h$Y—ps*, pyz-\-q2z2—pw\ р2г+£и>2—рх*,
pvfi+q2^—py* all squares.
On 2s2-j-2y2— z2=D, etc., see triangles with rational medians, Ch. V.
Quadratic forms in x and у, x and г, у and z, made squares.
J. Cunliffe*4 found rational numbers x, y, z such that
D) x*-xy+y2, &-xz+z\ tf-yz+z*
are squares, by equating the first and second to the squares of ±a—xf
4b—x} whence
_16u2~y*=l№-g*
X ^8a-y Sb-z '
Equate the denominators. Thusy=5a—3b, z = 5fc—3a» Then
if a : b=n2-49 : Un-94. J. Whitley equated the first two functions D)
to the squares of x—ny and x—mz; hence take
Set p=2n—1, д = п*— 1, v=v?—n-\-\. Then v*=p*—pq+q*. Equating
to the square of pm2—qm+vf we get m rationally.
To find rational numbers such that85
E) rf+atf+y1, $>+xz+z\
are squares, equate the first and second to the squares of x+y—m and
x+z—n. We get x and z in terms of y. The third condition leads to a
quartic in y} which is made a square as usual.
Lowry6*" made a^x*+axy+by*, Р^хР+а^+Ь^2, y=y*+a>iyz+b$z*
squares. Set r=n{aYn-\-2m)l s = m*—bin2, />=^«(a«4-2^), a = ^—bu2. Take
y/x=Pl(Tfzjx^r/s. Thenaa*lx2=(vz+auv+bu'*y. Similarly^^a. Since
« The Gentleman's Math. OompanioD, LodcIod. 3, No. 14,1811, 310-11.
■7Ш.,4Г No. 21, 1818, 7*7-60; J. Cunliffe, Leyboum's Math. Repoaitory, New Ser,, 2,
1809,1, 93-5. Cf. Ch. V.1^
«- New Series of Math- Repository (ed., T. Leybouro), 3, 1814, I, 153-164-

512 Hisromr of the Theohy op Numbers. [Сна*, xix
z[y = r<r/(sp)t у = П if
F) ^v-h^spo'-i-^cr2 = a,
or kfV-f \-ки*=П, к^Ъ&ь—аагЬп+аЧ2. This quartic is made a
square in a special way for special values of wand n for which k= П. For
the case bvjb=n^<F, make <r=s by taking u^dn, v=m. Then A) be-
becomes р*+О1Гр+^г*= □ = (/»—re/02, if 2et+atf=rlnt e*—bil2-pjnt which
are linear in m and n.
An anonymous writer** gave an elegant solution for the case
h=bi = b* = l. Take x=nR, y=m?—n*, z=nS, where R=an±2m,
S=azn+2m. Then ?
Also, 0-n*(p2+aipy+ff')' if K=p*—q\ S^arf+Zpq. Comparing the two
expressions for R and the two for S, we get m and n as fractions whose
common denominator is 2(а*^а), which may be omitted since x, y, z are
homogeneous in m and n. For a = oj, use the lower sign»
J. Whitley and W. Rutherford" equated р*х2+ху+у* and p*z*+xz+x*
to the squares of px+y—a and pz+x—Ъ, finding x and у in terms of z.
Then pV-hyz-H^ □ if a quartic in г is a square.
W. Lenhart57 took x^abc, y=-bdf} z=cfn in E)» By Lagrange's Addi-
Addition, § 90, to Euler's Algebra (Lagrange63 of Ch. XX), the resulting func-
functions are squares if
To solve the equations in the first column, set p+q-afp—q—cJ2p-^-q = dlrf
q=rf. From the two values for 2p and the two for q, we get c=a—2rf,
d=rBa—Tf). Similarly, by the equations in the second column, b=a—2sf,
n-sBa—$f)> By the two in the third column, 2p2 = d-\-b=tc—njt>
2qt = d—h^2njL Eliminating t, we get Cd+b)(d-b)=4tnc> Inserting
the earlier values of c, d> fe, n, we get
{(h)(h)/H()J()/[Br/)(a^/)
The final factor will occur also on the left if 2e-J-3r*=J$Fr-hl). Then
12Ч5M* /12*C2b2l
Next, for D), equate the last two functions to A2 and B2. Their differences
are equal if A-\-B*=2{x+y—z)t А—В = Ь{х-у). Insert the resulting value
of В into з/2-0г+г*=Я*. Thus a?-Cx2-J-102^-h3y2)/i8(x-hy)!. Finally,
x2-xy+yi= □ if x=p*-~q\ y=2j»q-tf.
T. Strong {p, 301) equated (х+уУ—Аху, (x+zJ-Bxz, (y+zy-Dyz
to the squares of x-\-y--a, x+z—b, y-\-z—c. By the first two we get у
and z in terms of x. Then the third condition states that two quadratic
functions of x are equal. We may squate the constant terms or the coeffi-
coefficients of x2 and get x rationally.
•"New Series of the Math. Repository (ed4 T\ Leybaum), 3, 1814, I, 151-3.
modified solution by A. Martbt The Analyst, Dea Moinee, 6, 1878 124-5,
»Indies' Diaryt 1834, 37-8, Quest. 1560.
« Math, Miacellany, Flufihing, N. Y., 1, 1836, 299-301.

Chap. XIX3 xy+a, xz+a, yz+a All Squares. 513
N. Vernon (p. 302) equated the first and second functions E) to the
squares of (r*-xy)jBr), and (s2~xz)[B8). Then x+y- {i*+xy)JBr)f etc.,
which give x} у in terms of z. Then the third function becomes a quartic
in z which is made a square as usual.
D. S. Hart" noted that x2+zy+y*= D if x = m2 — n2, з/ =
Thenx*+xz+z*=n \1г=хBщ+<?)Ц$2-<?)ш Take ^ = 2, n = lt pk
Then y*+yz+z*=D if r=7g/4,/> = 9g/4. Hence an answer is 195, 325, 264.
A. Martin and A. B. Evans** took z2+axy-\-y* = {mx—уJ to get xjy<
Then x^+axz+z2 and tf+ayz+z* are made squares by known methods.
Several writers7* made the functions E) squares. It. F. Davis71 noted
the solutions 7, 8, -15 and 435, 4669, 1656.
N. G. S. Aiyar*2 solved а^-Ьзд-Ьу^с2, etc., by geometry, algebra and
trigonometry, without attention to rational values.
A. Gerardin78 assumed that a solution of а2+а&+(Р = А* is known and
sought a solution of
by setting B=x+u or B—a—zp/q, or x = t—a—P, obtaining a quartic
function of t which is made a square in three ways. There is found a solu-
solution in positive integers by functions of the sixth degree.
E. Turriere74 considered the system
Ах2+Вху+Су*=П, Dy2+Eyz+Fz*=nf Gz2+Hzx+Ix* = П,
under the assumption that each has a set of rational solutions, say x0, ya
for the first. Solving the first with у —y$ =* Z(x —Xq) , where Z is a parameter,
we get x and у rationally in terms of Z. Similarly, zjy is rational in
X=(z— z^Ibf^уду *nd x!z ш Y=(x~~Z2)/(y—y*)- The condition that
the product of the values of yfx, zfy, x/z be unity is of the sixth degree in
X, F, Z. The problem is thus reduced to finding the rational points on a
certain sextic surface.
M. Rignaux,7** to treat the last system, would use a solution z=Xq, у=у0
of the first equation, where xcf yQ are quadratic functions of two param-
parameters га, n; Likewise a solution x=xu z—Ziol the third equation in terms of
parameters pf q. Hence take х=ХоХ1} у=у&и z^z&i. The given second
equation becomes a quartic in mt n and is solvable in known special cases.
xy+a, xz+a, yz+a all squahes.
Diophantus, III, 12, 13 and IV, 20, asked for three numbers such that
the product of any two increased by a given number a shall be a square.
For a = 12, he found 2, 2, 1/8; for a=—10, complicated fractions; for
a = l, x, z+2, 4x4-4. In V, 27, the numbers themselves are to be squares.
« Malh. Queet Educ. Times, 20, 1874, 59-60.
■»/Ьм/.,21, 1874,45-6.
71 The Math, Victor, 1,1880,105-6, 129-30; Atoer. Math. Monthly, 1,13&4, 208 for D).
к Math. Queet. Educ. Timee, 11,1907, 25.
» Jour, of Indian Math. Club, 2, l910r 24-25-
и Nouv. AnD, Math., D), 16, 1916, 62-74,
»Ibid., D), 18, 1918, 4349. For such a system, see Ch. V, p. 223.
""Lpioterm6diairedeemath.,25, 1918, 132-3.
34

514 History of the Theory of Numbers. [Chap, xix
F. Vieta7* generalized the method of Diophantus 111, 12 [13]. Let A
be the second number. Then the first is (Я2—а)/Л, and the third is
(D2—a)jA. Hence must
— — н-а-а.
A A
We can make B2—a=F*> D2—a=G1 in an infinitude of ways» Then
F4P+aA* is to be a square, say (FG-HAJ. Hence A=2HFG!{H2-a).
С G- Bachet76, who doubted that Diophantus had a general solution,
used the canon: Subtract the given number from each of two squares and
divide the remainders by the difference of the roots of these squares;
then the quotients and the difference of the roots are three numbers giving
a solution. For a=6, take ЛГ+3 and 2iV-|-3 as the roots of the squares;
then N, N+S+d/N and 4ЛГ+12+3/ЛГ give a solution.
DeSluse77 took an arbitrary square b2 and set d=b2—a,xy=x2+2xb+d,
whence xy+a-{x+h)\ Similarly, we can set z^xc2le2-^-2bcfe-\-dlx7 whence
хг+а = (хс/е+ЪJ. Let yz+a be the square of (cx+cb)!e+b+d!x> Thus
e + ё* е* + е '
When b* is replaced by d+a> this reduces to 2 = c/e, so that the required
numbers are x, x+2b+dlx, 4x+4b+d/x. For a negative, a=—A, call the
numbers x, y=x+A/x, г=хЪ*!с1+А/х. Then xy-A=x*, xz-A=x2h2j<?t
yz-A = (xb!c+A!xJ if bfe=2.
N. Saunderson78 (blind from infancy) gave the solution
x= , y = , z—r—8 or 2x+2y—(t—$),
T —S T $
where r and з exceed Va and r>s. For a = l, a solution is
X, y
V. Ricatti79 treated the problem.
L. Euler7*1 set ху+а=р*, z=x+yzt2p, whence ,
yz+a-(y±p)\ For a = Y2, p=4, then x=y=2, z = 12. For a = 12, p=5,
then x = l7 y = 13, z=i or 24. In art. 231, he noted that for a=l the
general solution is
treated АБ-1-р2, AC-\=q\ BC-\=r\ Thus
|bZeteticaf 15&1, V, 7[8J, Frnnclsni Vietae Opem mathematics, ed* Francbci 4 Schooten,
Lugd. Bat,, 1040, 78.
" Diophanti Aiex,, 1в21р 149Р 215.
77 Renati Frtnicieci Sludii, Me*)]abum, acceasit pars alt^ra de anaJysi el miscellanea, Leodli
ЕЬш-oniim, 1668P 177-8.
"The Elements of Algebra, 2, 1740, 300-5.
i* Inetitutionee analyticae a Vincentio Riccato, Bononiae, I, 1765, <H.
7*Algebra> 2y 1770, art. 232 (end of art, 233); 2y 1774, p. 305 (pp. 310-1); Opera Omni a,
(l)I465<46S)
■■ Poeth. paper, Coram. AriuV, II, 577-9; Opera poetumftj 1, 1862, 129-131.

Chap» XIX] PROBLEMS RELATED TO THE LAST ONE. 515
Set -ABC=m+n+t(nr—m). Then, for d=»(i*-
To obtain integral solutions, set Б = (рг-[-1)/А, C = {q2+l)jA. Then
is a square if A—n—p—q. Then £=А-|-С-|-2д. It remains to make
<f-\-l divisible by A, which requires that A = EL If A =5, q = 5u±2} then
C^5u2±4u-|-1, B=5vr+14;U+lO or 5и2+ви+2> Among other ways of
obtaining integral solutions, take AB=l+p2, (AC-l)(BC-l) =
whence C=(A+B+2m)/Q7 where Q=AB-m\ Then
Hence we set Q=n*t whence m*-\-n2=p2-\-l> Take m=ap-\-a> n=ap—a,
where a*+c? = l; for example, a = (P-?)!(J2+tf), a^2fg!(P+tf). Then
For/=2p,g = l,C
where/i=4p^+L Next, we take/—Д, g=2p> In this way Euler obtained
С=(А-|-Б)М2:Ь2рМЛГ, where <M, Л0 = A, 1), Dp*+l, 4p2+3),--- are
given by a recurring series with the scale of relation 4p2-|-2, —1; he gave
the general terras.
J. Leslie*1 madexi/H-l, xz+l, yz-\-l squares by factoring {cf.
P» Cossali82 gave the result due to Saunderaon.™
Fr, Buehner83 treated xy+l=p2, xi+l=tff yz+l^r*. Then
V=m{p-l)=l(r-lO г=п(Я-1)=
Thus p, q, r and hence also x} y, z are functions of mT n, L
A. B. Evana?w to make xy — l7 etc., squares, took x=a?-\-b*, y=c?-\-d2,
г=е*+р7 E=-bc-ad, F=be-af, G=de-cf. Then xy-E2, xz-F2, yz-GP
are squares. Take e=a+c, f=b+d* Then F=E, G=—E> It remains
only to make ^=±1,
E. Bahier85 noted the answer a—1, a, 4a—1 and gave de Sluse's77 values
with x = l and SaundersonV* with the second z.
Problems related to the last one.
Diophantus, III, 17, 18 [19], treated the problem (which evidently
reduces to the last one): to find three numbers such that the product of
any two increased ^diminished] by the sum of those two gives a square»8*
■' Тпшй. Roy. Soc. Edinb>, 2P 1790, 209, Prob. XII.
"Origine, Trasporto in Italia . - , Algebray 1, 1797, 102.
и Beitrag zur Aufloe. unbeet. Atifg. 2 Gr.t Prog- Etbing, 1858, p. 9.
" Math. Queat. Educ. Time», 14,1871, 75-6; 29, 1878, 90-1.
" Recherche Methodique et Propri^t^a dea Triangles Rectangles en Nombres Entiera, Paris,
1916,198-^.
M In Diophaattid IV, 38, 40, the reeulta are to be given numbers, instead of equarefl. His
condition tliat each number rauat be 1 lead than a equare is not necessary, аз noted by
Stevin, Les Oeuvrea math, de Simon Stevin ... par A. Girard, I625y 589; 1634, 148.
Thus if the numbers are 14, 23, 39, an answer is 4r 2, 7.

516 History of the Theory of Numbers. [Chap, xix
Take у and 4y-|-3 as two of the numbers, which each increased by unity
have a ratio which is a square 1/4. From
we get j/=3/10. For the numbers 3/10, 42/10, x, the conditions are
By the usual method (Ch, XV), x-7/10. Cf. Nesselmann,95' pp. 142-4.
N, Saunderson87 found three numbers а, Ъ, с such that the product of any
two increased by t times their sum is a square. Since (a+t){b-\-t) ^n2+t2f
express n*+t2 as a product of two factors, say n-|-r, n—$, each >L Then
c=a+b+l±2n and
=r-s-t or
a
7"-$ Г — S
When * = 1, taker—s = l, whence a^r2, Ъ=$2, с=0 or 2a-|-2J>-}-2.
The same numbers are such that the product of any two increased by t
times the third is a square (Diophantus, III, 14).
P. Cossali82 noted that if the product of any two of x*y z\
2\x*+z*+{z-xy} be increased by (z-xJ times the sum of the two or by
(z—xJ times the third, we get a square. On adding 2(z—xy to each of
these three numbers, we get three numbers such that the product of any
two diminished by (z—x)* times either their sum or the third gives a square.
Diopnantus**, V, 3 Q4], required three numbers such that any one of
them or the product of any two of them increased [diminished] by a given
number a is a square. He quoted from his Porisms that if x-\-a=m2%
y+a=n'li xy+a — П, then m and n are consecutive numbers.ю Thus if
a=5 we take x^(z-|-3J—5T #^(*-|-4J—5 as two of the required numbers,
and 2(x-|-i/)-l=4z*-|-283-|-29asthe third. We are to make
say Bz-6J. Hencez = l/26,
For V, 4, Diophantus took a=6, x=z*+& and у = (z +1J+6 as two of
the numbers, and 2(x+y) — 1 as the third. The latter less G is
)
if z=17/28.
Diophantus' method shows that xy+a, xz-^-a, yz+a, x+a, y+a are all
squares if x=m2-at y = (m+l¥-a, s = Bm-|-lJ-4a. To make also
z+a=L!,say Bm-rJ, we have *tt = (r2+3a-l)/{4(l-K)i,
Fermat (Oeuvres, III, 250) gave a solution for the сазе a = l. In
the constant terms increased by unity give squares; further, xy+1, xz+1,
yz+l are squares. The "triple equation |7х-Ы^ П,у+1- O,z+1 = □ к
readily solved since the constant terms are squares (Ch. XV).
■> The Elements of Algebra, 2, 1740, 39Э-405; French tnuifil., Sphinx-Oedipe, 1908-9, 3-9.
«BuC this i» incorrect; m—n—=kl и a aufficieat but not necessary condition (отху-{-&= D.
In fact, by-eliminating^ yt we get «jV-q^H»1-O+e1^ D. While this is aatiaQed
if pf+n*—l = 2mnt wbeneem-ni^it can be Bailed us usual by aetting гд=л±1+р.

Chap. XIX] РвОЩГСТЭ ВТ TWOS PbUS UmiT MADE SQUARES. 517
Pkoduct of any two of fottr or five numbers increased by unity a
SQUARE,
Diophantus, IV, 21, required four numbers such that the product of
any two increased by unity ie & square. He took x, x+2, 4s-|-4 as the
first three (by IV, 20), and Cx+l)*-l as the product of the first and
fourth» Thus the fourth is 9x-|-6, The product of the second and fourth,
increased by unity, is 9x*+24s+13; let it equal Cs—4)!, whence д;=1/16.
The remaining conditions are now satisfied»
Rafael Bombelli» treated the problem for four numbers,
Fermat" tools 1, 3, 8 as the first three numbers. The conditions on the
fourth number x are x+l = Ht 3*+l = □ , Ss-hl^D, His method
(Fermat1*'ll of Ch. XV) of solving a " triple equation " gives x = 120.
L. Euler92 gave the solution а, Ь, с=а+Ъ+% d^4J(J+a)(l+6), where
a&+lBPfAadnoted the cases 3,8^1,120 and 3,8,21,2080. Heextendedthe
question to five numbers, by seeking z such that 1+аг, — •> 1+dz are all
squares. Denote the product of these four sums by P = 1 +рг+дг*-\-тъ*+&г?,
where therefore p=a+b+c+d, --, seabed. Let P be the square of
lfr^ where g=q!2-p*/8. Then
For brevity set a+b+l=ff dfA=L Tben
NowA-/(ab+l)+teb, 4k*=±kf+4kab(f+T). Hence
The denominator ^—5 of 2 is fortunately the square of (s—1)/2, Thus
and P is a square. Eulcr stated that each factor l+az} etc., is then a
square. Taking a-1, 6=3, we have 1=2, c=8, d=120, p=132, g=1475,
r-4224T s=2880, г=777480/2879*, and the ten expressions ob+1, --,
йг+1 are the squares of
2,3, и, 5,19,31, m$> mi mi 1ж>
To obtain smaller (but fractional) numbers, set a—1/2, 6-5/2. "Then
c=6, d=48, г-44880/128881.
A. M. Legendre83 verified Euler's preceding assertion that 1+az, etc.,
are squares by noting that a, 6, c, d are the roots of
M LTalgcbia opera, Bologna, 1579У р. &43.
*Geuvree,III,251.
* Оривс. алд!., lr 1783, 329; Comm. Arith., П, 45. Results stated ш a tetter to Iagrange,
Sept. 24,1773 (Oeuyree, XIV, 235-H»; Euler's Opera poetua», 1, 1862,584^6.
» Theorie des nombree, ed. 3, 2P1830, H2-4; Maser'e tiazul^ 2,1893,138.

518 History of the Theory of Numbers. [Cbaf. xix
and showing that when ri is replaced by its value from the preceding
equation, (s-lL£z-M) becomes B£*-p£-5-lJ.
С. О. Boije af Gennas94 gave the solution
r, s(rs+2), (>+l)(r*-|-r-|-2),
For r = l, 5=2, we get 1, 3, 15, 528.
J, Кпгггад took as the four numbers
The product of the second and third, increased by unity, is
\аЬп+(а+Ъ)Г+[1+4аЬ-(а+ЪУ}
and is a square if the final part is zero, whence fc=a=bl. The product of
the second and fourth, increased by unity, is then the square of l if
The coefficient of p is unity if g = an-|-l. G. H. F. Nessehnann11^ took
Ъ=а+1, р=а+2.
С. С. Cross5* gave the set due to Boijc94 with r, s replaced by m7 n— 1.
He and others failed to find five such numbers. He97 later took the fifth
number to equal the first one m, the only new condition being m?-\-l = U,
for example, i»=(fc*-l)/BJfc).
M. A. Gruber98 noted a special case of Euler's*2 five numbers»
A, Gerardm" obtained special solutions by recurring series.
Fermat100 treated the problem to find four numbers such that the product
of any two increased by the sum of those two gives a square. He made
use of three squares such that the product of any two increased by the sum
of the same two gives a square. Stating that there is an infinitude of such
sets of three squares, he cited 4, 3504384/d, 2019241/d, where d** 203401.
However, he actually used the squares 25/9, 64/9,196/9, of Diophantus V, 5,
which have the additional property that the product of any two increased
by the third gives a square. Taking these three squares as three of our
numbers and x as the fourth, we are to satisfy
This "triple equation17 with squares as constant terms is readily solved.
T. L. Heath101 found x to be the ratio of two numbers each of 21 digits,
L. Euler102 gave a more general treatment of the latter problem. Let
А, Б, C, D denote the numbers increased by unity. Then AB—1> ••-,
CD-I are to be squares. Take AB=p2+l,
u Nouv. Ann. Math., B), 19, 1880, 278-9; E, Тдкэд, ThSorie dee nombresy 189ly 129.
<»Dte AuflosuDgdcr Gleichungi1—c^ = l, 18. Jabreabericht OberreabchiJe, 1889, 31*
950 Zeitscbr. Math. Fbya., Hiat,-Nt, Abt., 37, I892y 167»
* Am«\ Math. МопШу, 5, 1898, 301-2.
"/Wd.,6y 1899,85-87.
*ЧШ,, 122^3.
м Lpmterm6diairc dee math., 23,1»l6y 14^15.
loe Oeirvree, III, 242-^3. A apedal саде of our main problem eincesy+*4-y~(:E + l)(tf-bl) —1.
«<• Kophanius of Aiexandri», Ы. 2, lftlOy p. 163,
ie Posth. paper, Comm» Arith., 11, 579-582; Opera poetumay 1^ 1862, Ш-*.

Chap.xixj Products by Twos Plus Unity Made Squabes. 510
Then five of the conditions are satisfied. There remains С/)—1 = П.
Replacing A+B by its value (A2+p*+l)/A, we see that the condition
becomes A4+2A*k-\ Н-(р*-НJ-П, where k = (a+b)p+*+p. The
quartic is the square of А2+Ак—p*-~ 1 if
This solution is of course not general. For instance, if л=0=О, а*=1,
b= — 1, then the preceding A is zero, whereas we may obtain solutions as
follows. We have, in this case, C=A+B+2p, D=A+B-2p> Then
Thusg=(r2+3)/Br). AkoA+B^2p+s if p = )/( ,
5 = 15/4. Then g = 7/4, p=-2/3, Л=С=13/12, Я=4/3, /) = 15/4. For
r=2, s = 7/2, we get
J()/
that all the resulting solutions are fractional. He cited the solution
A-D = l, В=2, С-5, and asked if there are other integral solutions»
Product of any two of four numbers increased by n a square,
C. G. BachctlW proposed the problem and took «=3, From (JV+2)*
and (i\T-|-6J subtract 3 and divide the remainders by the difference 4 of
the roots of the squares; we get
As the third number, he took
с=2(а
Hence by a general canon, ab+3, «c-f-3, 5c+3 are squares. Take the fourth
number to be d=4. Then a<2-|-3 and bd+3 are squares» Finally,
cd+3 = BN-10)* if Л^-5/8.
He gave also a second method of solution.
Fermat1M remarked that it is easy to deduce a solution from
Diophantus88 V, 3. As three of the numbers take solutions xh x2, x2 of
the latter problem. As the fourth number, take x+lm We then have a
"triple equation'1 XiV+Xi+n = D, whose constant terms Xi+n are squares,
and hence easily solved (Ch, XV)»
P, lacobo de Billy105 took тг=4, Н as the first number, and ft+2,
2tf-|-2, ЗД-|-2 as the roots of the squares obtained when Tt is one factor»
Thus the remaining-three numbers are .ft-H, 4Л+8, 9tf-}-12. Then
(#H-4)(9flH-12)H-4isthesquareof3fl-Siffi=l/8. The other two condi-
conditions are seen to be satisfied.
N. Saunderson87 (p. 398) took any number a> Vn, subtracted 4a2—3n
from any larger square б2, and called d the quotient obtained on dividing
IW D[oph. Alex., 1621, 150. ^^ ~"~
i«Oeuvr»,ni,254.
m DJophantvB Geometr», Paris, 1660, 100.

520 Histoby of the Theory or Numbers. [Chap. XIX
the remainder by 4a-|-25. Then an answer is given by
dy e = (a*-n)ld, f=d+e+2a, 0=3e-|-/+2a-
Thus forn=3, take c=2, b=3. Thend^l/7, e^7,/-78/7, 0=253/7,
L. Euler1» called the numbers A, By C\ D. Set AB~p2-n. Equate
the product of AC+n and BC+n to (Cx+nJ; then
n(A+B-2x)
L
Hence (x2—AB)jn к to be a square y2, whence
Similarly,
D
In CD+n^ O, replace A+B by (Az+p2-n)!A. Hence
is to be a square. It can be made the square of A*—A(x-\-v) — (p*—n) by
choice of a rational A. To simplify the formulae, Euler took v= —x,
z~y* Then the condition becomes
and is satisfied if у=2. It remains only to satisfy p2 — x-—3n. Set
p=x-L Thenz = (*H3n)/B0,p=Cn-l*)/B0. To secure homogeneity,
mt x, v = {Znu*±V)l{2tu). Then
2p«« ' 2//u ' '
To find four numbers such that the product of any two increased by the
sum of the four is a square> we have only to take mA, - - -, wD, where
7П — (A+B+C+D)jn, while A, -•»,/),« are the numbers given by the
preceding solution, Euler gave two solutions in integers: 15, 175, 310,
475 and 36, 96, 264, 504, Since n may be negative, we obtain four numbers
the product of any two of which decreased by the sum of the four is a
square. A solution in integers is 8, 24, 44, 80»
E. Bahier,8* pp. 199-208, employed the numbers of Saunderson,78
taking his two values of z as two of the four numbers. There remains
only the condition (r+$J—3a= □, which is satisfied by expressing 3a as a
difference of two squares»
Other products of numbers in paitis increased by linear functions
MADE SQUARES.
J. Collins107 made the six functions xy±vy xz±v7 yzd-.v squares, where
v=x+y+z. Take xy±v=(tzts)\ xz^bv-(r^qJ% yz±v=(pzkn)\ and A)
^ Т\& f- (a2—№)g,
l»Comm. Arith,, IIp 582-5 (poeth. paper); Opera postuma, 1, 1862, 134-7; Algebra,
l770r arU, 2334; 2, 1774, pp. 306^14; Opera Олпшц (l), I, 465-4.
197 The Gentleman's Matb. Comp&Dion, London, 2, No. 10r l807r 66-7.

Снаг. xixn Products by Twos Plus Linear Functions Squares» 521
$ = 2abg, r = (a7-cT)g, q=2acg7 p=(d?-a*)gt n=2adg. Then
X
To satisfy A), take a=p^fh^h\ b=f-h\ c-2/A+A8, d=f-\-2fk. For
four numbers, see Eulcr.106
J» Cunlifle108 made xy+z, etc», squares by taking y-x=2ny z=n2.
Then ху+г = (х+пУ, while яг-|-# and yz+x are linear functions of x and
may be equated to squares» S» Jones took y>-x — 1, z=s—4. "J, B»"
took y=t2x-v*i z^&x, whence sp-|-2=№. Then zz-|-y = (t3;—r}2 gives ж.
From yz+x = O, we get a quartic in r which is solved as usual.
W. Wright105 took xy-a=p2, yz-b=g2 and made p*-\-a and tf+b
squares» Then хг—с = П if D/y*—c = D, which is easily satisfied.
Cunliffe110 took xy-\-z=A\ xz+y=B*. Thus (у+гХж+и^Л'+Б8.
Hence set y+z=a*+b\ х+1=с2-|-с^7 A=ac+bd7 B=ad-bc. Also,
(y-z)(x-l)=A?-B2. Hence take |/-2=Л-5, х-1=А+В. By the
two values of x, we get 5 in terms of a, c, d. To get integral values of
by equate the denominator c—d to unity.
D, S. Hart111 made xy-\-z, etc», and xtf+x+y, etc», all squares by taking
x=n\ у = {п+\)% ^=4{пг+«+1).
К N. Barisicn112 treated the system
Subtract the first from the other two» Thus
ax=u*—F, bx^v*—P, av*—bu2 = (a—b)t2.
Set v—t-\-h7 u=l+L Discarding the denominator 2ha—2№, we have
L = bV-ah\ u=bP+ah?-2alh, v = ah2+bl2-2Uh, x~4!h{h-l){bl-ah).
Then y7 z can be found from Az—y=B* Set В^А-р+т^ z=q-\-p\ then
y=Aq-r> [Take g= -akjlj^ -Vja]. Then
x=4fg(a+g)(b+g), u
t^ftf-ab), z=g,
He113 elsewhere merely stated the latter solution,
" V. G» Tariste14 treated the ca«e a ^ 1, Ь - 2 of the last problem» Then
t?+F=2v?, whose general solution is и = ЦА7+Вг); v, t = \(A*-B*±2AB).
Several writers115 made xy+z, yz+%, xz+y all squares (Diophantus
III,H)»
Б, Bahier7fiS pp» 208-212, made xy — v, xz — v, yz—я squares the sum of two
of which equals the third.
1H The Gentleman's Math. Companion, London, 3, No. 17, 1814, 463-6.
**4bid.t 467-8.
"■ 1Ш., 5,No. 27,1824, 34^-53.
»■ Math. Quest. Educ» Times, 28, 1878, 67~S.
 Sphinx-Oedipe, 1907-8, 180-1,
™ Matheeis, C)y 9, 1909, 154-5.
w Umtennddiaire dea math.y 19, 1912, 38-9*
i^Zeitechr. Math^Phye., Hist.-lit. АЫ., 37y 1892, 138; Math. Quest. Educ, Timce, 25,1914,
4OP 102-4; Ащег. Math. Monthly, 24, 1Ш7, S8-89, 294,

522 History of the Theory or Numbers. [C«ap. xix
FXJBTHER EQUATIONS WHOSE QUADRATIC TERMS АДЕ BUMS OF PRODUCTS*
Bh&caralie (born 1114) treated the problem to make ioH-2, z+27 tf+2,
z-\-2 the squares of numbers in A. P., and utt+lS, xy-\-l&, yz+18 all
squares, such that the sum of the roots of the seven squares when increased
by 11 gives 13*, Since 13/2 is the square of 3, the roots of the first four
squares are yi y-\-$, #+6, y+9* Then the roots of гкг+18, etc», are found
to be y*+Zy-2, у*+9у+1Ь, y*+15i/-|-52. The sum of the roots plus 11
gives 3^2+31i/+95 = 132, у=2.
Diophantus, IV, 16, solved z(x+y)=a, y(x+z)=b, x{y+z) = c, when
a^35, b—32, c=27, by assuming that х = 15/г, y=20/z, whence z^5.
Rallier des 0urmeslir obtained 2xz=a+c—Ь, etc., by elimination from
Diophantus* equations» From yz = m} хг = п, xy=p follows y= Vpm/n> etc»
For a=2A7 6=45, c=49, we get m=10, n — 14, p — 35, whence x = 7, y=5,
z ~ 2. He gave also a solution by listing the pairs of complementary factors
of the smallest two, 24 and 45, of the three given numbers:
24-1-24 = 2 ■ 12=3-8=4-6, 45 = 1 45 = 3 15^5-9.
From each list select a pair of factors with a common sum, as 2-12, 5-9,
and select by trial one of a pair as one unknown and the cofactor as the sum
of the other two unknowns»
To find n numbers, given the product of each by the sum of all the others,
list the pairs of cofactors of each of the smallest n — 1 of the n given num-
numbers and select those pairs, one from each list, which have the same sum
(the sum of the unknowns). The smallest cofactor of each pair is one of
the smallest n — \ of the unknowns and their sum subtracted from the
total sum gives the largest unknown» For я = 5, use 180=4'45^94=^7-42,
418 = 11-38, 444 = 12*37; the unknowns are 4P 7, U, 12,
15-49- D-I-7-H1 + 12),
3. Jones11* took х(у-\-г) -a7xl> y{x-\-z) = b2} z=ax+b7 which give xt y, z.
Then z{x+y) = U if a*+2a-l^ П = (a-nJ and a2-2a-|-3 = D. The
Utter becomes a quartic in n which is a square if n — —2/3.
L. Euler119 developed a method to make various functions simultaneously
equal to squares. The method will be explained for his problem (§§ 31-34):
Given an integer «, find integers x, y, z such that xy-\-n, xz-\-ni yz-\-ni
xy+xz+уг+п are all squares» For any set of solutions of
and for any function P, P2— f is a square» Taking P—x+y—г, we find
that A{xy-^-7i) is a square. Taking P=x—y+z, we find that 4(xz+n) is
a square» Similarly, yz-\-n is a square» Taking P — x+y-\-zt we find that
4:(xy+xz+yz-\-ri) is a square, Now/=0 if z=x±y+2v, where v*-=xy-\-n>
To satisfy the latter take any integer for v and take x and у to be any pair of
ll* Vlja-gaoita, §§ 143-4. Algebra with arith . * . from Sanskrit . « - of BhAscara, transL
by Colebrooke, 1817, 218-^.
i" Мбщ. de M&thtonatique et de Physique, Paris, 5, 1768, 479^84.
u* The Gentleman's Math. Companion, LoDdoa, 3, No. 15, 1812P 348-9.
"•Novi Comm. Acad, Petmp.r 6, 1756-7, 85-114; Comm. Arith., I, 245-259; Opera
Omnb, (I), II, 3№-427. French transl., Sphinx-Oedipe, g, 19!3P 97-109.

Chap. XIX] SUMS OF PRODUCTS AND LlNEAR TERMS SQUARES- 523
integers whose product is n—v1. Then
2) xy+xz+yz+n=(x+y+v)\
the right members being the reduced values of Р*/4, for the respective P's.
To solve an interesting related problem (§§ 35-39), take
and P=x+y±z±a for the four combinations of signs» Then
F=txy+ia(x-\-y)-\-a*+bt
and the expression obtained from the last two by permuting the variables,
are all squares. Now/=0 if z=x+y+a±v> provided x and у make F = iA
The latter is the case if x+a and y+a are two numbers whose product is
(y2—b-|-За*)/4. In particular, if a = l, 6— —I, we see how to find three
numbers x, у, z such that
xy+z, xz+yf уг-\-х7 xy+x+y, хг+х-\-г, уг+y+z,
are all squares» The simplest solution is x = lt t/=4, z = 12. Solutions in
which also the numbers themselves are squares are
9 2 Г> 4 0. 35 64 1 S6
иг? «T> ~nr> fl~> Tr, 9 *
Eulcr12U asked for numbers p, q, ry - ■ ■ such that the product of each by
the sum of the remaining numbers is a square. Hence if S be their sum,
p(S—p), q{S—q), • • • are to be squares. Take p{S—p) =Лр2, etc» Hence
the desired numbers are
A ^
Take/=а/«, etc. For three numbers, let them be
The sum of the last two is 1—4aab0/d. The sum of all three is therefore
unity if а\Ь^02)^4алЬ^} whence a : a = 4h0 : Ь2+0*. Taking а=4Ь/3
and multiplying the initial numbers by d, we get the solution
For four numbers, Eulcr gave the solutions A, % 2, 5), A, 10y 34P 125),
E, 9, 26, 90), E, 32; 6J, 512) and solutions involving two parameters» For
five numbers, he gave 2, 40, 45, 58, 145»
Euler121 gave a special method of treating the last problem. Select any
number, like 3 = 130, which is in several ways a sum of two parts whose
product is a square, viz.>
p= 2, 5, 13, 26, 32, 40, 49, 65,
S-p = 128, 125, П7, 104, 98, 90, 81, 65»
ue Novi Comm, Acad, Pctrop., 17, i772p 24; Comm. Arilh,, I, 45Э-66; Op» Om,p A), III, 188.
i" Opera poBtuma, ly l862p 260 (about 1769).

524 History of the Theory of Numkebs. [Chap, six
Selected values of p give an answer if their sum is 130, as for 2, 5, 26, 32,
65, and 2, 13, 26, 40, 49.
Euler12* found a} h} cy dso that ab—cd, ac—bd, be—ad are squares. Call
the first two expressions x2, y1, and solve for Ъ, с. Take 2x=a-\-d-\-v,
2y = a+d-v. Then
4(a-<0
For v=d-S, a=24, we get 6 = 21, c= 13.
S. Tebay123 found four positive integers aif ■ • ■, a4 such that
aiaz-\-Viui, а^а^-\-а^аг7 Ъсцйз are squares.
A. Gerardin12*1 made zy+zt and xz—yl squares by several methods.
Squakes increased by linear functions made squares.
Let a^Xi+Xi-l-z* Diophantus, II, 35, and Bombelli made xj+<r
a square for г = 1, 2, 3. Diopbantus, II, 36, made each x\ — a a square.
Diophantus, V; 9, made each х)±а a square. Diophantus, III, 1, made
each <r—x\ a square by taking x1=x, x*^2x, <r=5x2, 5 = B/5J+(lI/5J,
x3=2x/5, whence х^17/25. J. Whitley125 took Xi*=xt x2 = nz, Х1=тх,
c—x^atz*, which gives x> Then l-\-a2-n2 and l-\-a2—m2 are to be
squares, which is the case if \n2=a=m.
Diophantus, IV, 17, made xi+^-i-Sa, х\-\-х^ xl-\-Xa} x\-\-xl all squares
by taking x2=4s, x^x-1, 16x?+x^ = Dx+lJ, whence x3=8z+l. Then
D =:
whence у *= 55/52, x = 13y2.
Fermat1* suggested that a more elegant solution is obtained hy setting
3 whence
a ^double equation" with squares as constant terms» He stated that a
similar device will solve the analogous problem in four or a greater number
of unknowns.
J. Anderson1*7 took Xi+x2 = (p~xiJ, xi+Xi=(q~x2J, х1-\-^ = (г-хгУ7
which give xb x2, x^ In 2хь equate the coefficient of r2 to zero, whence
g = l/4. Other writers gave essentially Diophantus' solution»
S. Ward128 took x2 = l-2x:j (l-2x1J+x^=A\ 1-xi+x^B*. Then
Л2-В2=4х?-Зх,. Take A+B = 2xu A-B=2xx-3l21 whence ^=3/4,
Then Щх1+Х1) = Dхз-р!яУ determines x?.
. Acad/Sc. St. JPetersb,, 5, anoo 1812, 1816 A780), 73 (§21); Оошш. Aiith.,
385-91.
w Math. Queat. Educ. Timee, S2,1S90, 117.
»«• L?interm«iaire dee matb., 36,1919, 17-18.
^ L'aJ^ebra opera di R. BomUlli, Bologna, 1579, 4S6.
« Ladiee1 Diary, 1807, 37, Q. 1156; Leyboura'e Matb. Quest. L, D., 4, Ш7, 72-3.
«Oeuvree, I, 301; French traneL, III, 249.
u> The G^ntlemaQ^ Math. Companion, London» 6, No, 26, 1823, 204-7,
»■ J. R, Young's Algebra, Amtx. ed., 1832, 337-8.

Chap, ХЩ SQUARES TlAJS LlNEAH FUNCTIONS MADE SqUAMS. 525
Diophantus, И, 34, made x*—y, y2—z, z*—x squares. In IV, 18, these
and x+y-\-z are made squares.
T\ Strong129 made z*—y, x2—z, y-—x, у*—г all squares. Take
х*-у = (х~ауУ, z*-z=(x-bzJ, y2-x=(y-cx)\
Hence xt y, z are rational functions of a, b, c. Equate the resulting expres-
expression for у*—г to (e-1/6J. We get h rationally in terms of e, a7 c. For
a=l, c=e=2, we get z-5/4, y = Zj2, z=14/9»
ШсаШш found three numbers such that if the square of each be added
to the remaining two the sums are squares. He used the numbers x, 2x} 1.
R. Adrain131 took
x2+y+z=(m~zy, y*+x+z = (n-yJ, z2+x+y=(r-z¥>
and solved the resulting system of three linear equations for xt y7 z.
To make &+x\ 8-\-y2t s+z? squares, where s=x-\-y+z, "A.B.L.35
equated them to (x+iO2, (#-MJ> &-\-kJ and solved algebraically the result-
resulting linear equations. "Epsilon" took $/-|-z = l/4. Then
gives x, and x-\-l-\-y*= □ if bp = <?+2pq—2qy, which ^ves y. W. Wright
took(y—l)r, (x-l)rand(j/—ljrasthenumbers^and r* as their sum, whence
r = v+x+y—3> The conditions become ^—2v-\-2^ \3 = (p—vJ, etc», which
determine v} xt y.
H» J. Anderson1" found n numbers whose sum s exceeds the square of
each by a square. Express s=x2+yi as a sum of two squares x*-\-yn,
x'^+y"*, ■ • -, in л ways (Euler, Algebra, II, § 219) by taking х* = Ыу-Ъ'х,
У' = а'х+Ъ% х"=а"у-Ъ"х, у"=а"х+Ъ'% < -., where
ym*~n* а''= 2р9 Ь" = ^-*2
Take x, xf> x'\ * - - as the required numbers. Their sum 8 is of the form
Ax+By. Thus s^x2+y* if 4Ву-4у*+А*=П. For n=±t С Farquhar
used the numbers w> wxf wyt wz. Set <r=l-\-x+y+z. Then w<r— i^^D
gives a. Then take
which determines г.
J. К. Young1" found three squares x\ and a number a such that x\±a
are all squares. Take x\=m]-\-n\, a = 2-mI-n,-l т^г\—д?, ni=2Tisi> whence
a\-rJ+*J. It remains to make the values 4nSi(/i-sl) of a equal. Take
n=r2=r3=7\ Thus 8<(r*—s*) are to be equal. The values for i=l and 2
are equal if r*=sj+*!$*+*!. Thus 4т*-3а1 is to be a square. Hence take
u' Amer. Jour. Sc. ^d Art» («Ц StUioian), 1, 1S18, 42S-7.
Ul Institutiones analyticae a VinceDtio Riceato, Bononiae, 1,1765, 64»
ш The Math. Corrapondeiit, New York, 2, 1807, 13-14.
J« The Gentleman's Math» Companion, Londcm, 5, No. 25, 18Й2, 125-30.
«■ Math. Diary, New York, 1, 1825, 151-i.
ш Algebra, 1816» S. Ward's Amer. ed.t 1832, 346-7. A like discussion for two squares
had been given by J, Cuniiflfe, New Series of Math. Repository (ed., T. Leybourn),
1, 1806, I, 221-2,

526 History of the Theory of Numbers. [Chap, xix
P+g, i=4fy> For/=2, g=l, s1 = -5or -3, r = 7, *2 = 8. We may
take as S* the second value —3, whence a ^=3360, xt=74, хг = Ш, xj = 58.
A. B. Evans1*6 found л numbers a, such that а?+а^, = П (i = l, •••,
n—l)}al-\-a1=\J. All but the last condition are satisfied if aT=mz-\-2mar-i
(t=2, - >, n), whence
Then aJ-|-ai=Bn-1ffin-Ia1-|-pJ gives ab D. 3. Hart took w«L
OF EACH OF THREE NUMBERS PLUS PRODUCT OF REMAINING TWO
A SQUARE.
L- Euler1* found solutions of x2+yz=p\ tf+zz^f, zz+xy=B. Then
p*-q2 = (x-y){x+y-z). Set p-q = z-y, p+q=x+y-zt whence
Then x*+yz=p* gives ^=4(^c-|-y). The third condition becomes
вау» Dx+4y+e)«. Then(x-8s)(y-8s)^65sf. Hence set я-8s-5ts/ut
y—$8^13u8/t, and to avoid fractions take s=tu. Thus ar=8iw+5^,
y-8tu+lZu2. He stated that the same solution is found if we Btart by
taking z^(yz-s?)jBs), the resulting numbers being «(&-И), t(t—Ss)t
4A4-1").
To give another method, set х=а?+2Ъ, у=¥+2а, z=ab(ab-4). The
first two conditions are satisfied and the third becomes
which is not discussed. But he noted the solutions x, y} z=33,185, 608 and
297, 377, 320. Nesselmann,9*» p. 141, treated this quartic with a = — 1/p. •
J. Lynn1J7 tookl^x-1, 4x as the numbers. Then two of the conditions
are satisfied and the third is Dж)Ч^-1^П-Dх±а)!, say, which deter-
determines x.
S. Ward13* took х=тг,у^пг, m+n —1/4. Then the first two expressions
are squares. The third is a square if l-j-£n—л2=П, say A— cnJt which
gives n.
J. H. DrummondIM took w2, mu?t nw* as the numbers. Then 1-J-wm,
m2+nt m+n* are to be squares. Taking п = \-т, it remains to make
l+mn= П, say A— pm)\ which gives m.
W. Wright140 made «=z*+4yz, jBj*m y=z2+4xy and x+y+z
squares. Take x~y+z. Then 0 and 7 are squares. Take
1» Math. Quest. Educ, Timee, 20t 1874t 8<Ь7.
"• Opera poetuma, 1,1862, 258-9 (about 1782),
i*« J, Cunliffe, New Series of Math. Repository («1., T Leyboum), 2, 1809, I, 172-3, chcee
it equal to {\ry — 4x)' to obtain x rationally in tertna of у, г. We may give any de-
desired value to x + у + z.
i»7 C. Hutton'a Mbodlaaea Mathematics London, 1775, 23oW.
"»J.R Youngs Algebra, Amer. ed., 1832,336.
ы Amer. Math. Monthly, 9, 1902, 232. Misprint of mV for m&.
"•The Gentleman1* Math. Companion, London, 3, No. 15, 1812, 346-7.

Systems of Equations of Degree Two. 527
Then 1а = и*+2иЧ-ь2={7гя-и*)гИ 2=2u2(m+l)/(m*+l)- S. Jones took
$ = {2z-yy and found 2(т?-ап)к, 2(a-\-n)k, 2k?t where k=a?-\-n\
W. Wallace^madea^xy+z2, $^xz-\-y\ y=yz+x*, and а^+р^+у1*2
squares by takingtf = By+zM,0=Bz+g)* whence x=±(y+z). Then7 = r*
if yz=n{rzk4:(y+z)}. Equate the factors to ym/n and znjm. We get y, z
and hence x as rational functions of m, n, r. Omitting the common
denominator, we have х=4:(т2-\-п2)т, 2/=(8win+n2)r, z = (m2—8mn)r-
Then л, ft 7 equal the squares of (m2+%mn+2ri*)r7 Bm2—Smn-\-n2)r,
Dm2-\-mn—4n2)r. The sum (Jvf+mn—n^T of these is a square if г equals
the first factor or the quotient of it by any square.
Miscellaneous Systems op Equations of Degree Two.
Diophantus, III, 2, made **+з,- (i = l, 2, 3) rational squares, where
s=x\-\-X2+x^ In Diophantus, III, 3, s2—xv (i = l, 2, 3) are made squares»
T. Brancker treated the latter problem. A» G6rardin14* gave several
integral solutions of the last two problems.
Kophantus, III, 4, made Xi-s2 (i = l, 2, 3) rational squares.
To finda?!, Хг, • •• such that
where 5 = 2^, ^Comes43 noted that since p2-, e2, q] are squares in arith-
arithmetical progression we may use the known values
Then s = 2xi gives s. For Diophantus' solution, see the first page of Ch. VI.
A. Gerardin and R. Goonnaghtigh144 made я?-х] (г = 1, 2, 3) squares;
also s?-(s—xx); also **—(s—Xi) (г = 17 2, 3, 4), where s^Xii \-z*.
The latter1^ made 8?+х* (г=1, •■-, n) Bquares, also s2—(s—х£), where
Leonardo Pisano14* treated cases of х*-\-хг-\- \-*п-У*, У*-\-х\ =
уЪ ,yli+yl
J. CunUffeHr made tr-\-Xi (i=l, 2, 3) squares, where <t =x\-\-x\+x\>
S. Ryley14* made a=x±-\-yz-\-y\ p=x*+yz+z2, y=V2+yz-\-zz squares.
Take *=a2, p=b*. Then у2-2*=а*-Ъ*. Hence take (a+6>= (y-\-z)s,
{a-h)s = {y-z)r, which ^ve a, Ъ in terms of ^, г. Now y=D if
Then tf—yz—y1 becomes a function of т,8,тщп of degree 4 in nt which will
1И* New Series of Matb. Repository (ed-, T. Leybouni), 3, 18U, I, 21-23»
ш Ал Introduction to Algebra, trans!, out of the High-Dutch by T. Brancker, much altered
and augmeoted by D. P[eUi Londoo, 1668, IQ2-4.
M I.*interai6diaira des math., 22,1915,197-&
ш The G^ndeman'B Math. Companion, London, 4, No. 21, Ш8,752-7.
w L^inieimediaire dee math., 22, 1915, 22СЫ, 244; 23, 191^ 138-141, 155-7, 209-11: 24,
1917,13^14.
i* Nouv. Ann. Malb., D), 1в, 1916, 401-26.
ш Scritti di L. Pieano, 2t 1863, 279^83. Cf. F. Woepcke, Jour, de Math., 20, 1865, 61-62:
A. Genocchi, Annali Be Mat. Fto.. 6,1855,193^205, 357-9.
«* Math, Repoeitoiy (edv Leybourn), London, 3, 1804, 97-106.
"The Gentleman's Math. Companion, London, 1, No. 8, 1*05, 42-4.

528 History of the Theory op Numbers. [Chap, xix
equal the square of
x=n*(f-s*)+rtmBr< -4sV-234) /\r'-s'i)-2sbn2
if m ;п=8*-\-т* :2&-2г*.
To make a=x2-\-y*-\-s, p=x2+zi+st y = y*-\-z2-\-s squares, where
zt s. Ryley14fl took y = lt z=3. Then
if x = {l2-n*)lBn-4:), and 7B«-4)< = D-14nJ if и=-16, whence
x :y :г=61 :9 :27.
J, Cunliffe took х=3г, y—n—z. Then 7 = (n-)-zJ. Make /?—a2, by choice
of л. Then l^«-a4-10aV+153^-n if a = 19*/3. Ortakea = G7i-3^)',
3=^-1, whence n = 2Cr+l). Thenj8=D if r-5/3, whence
r :y :z=4 :32 : 12.
"Limenus" took a=^a2, ^ = b2, 7=^. Then x2-\-dt=-y2+hi-zi-\-a\ Hence
take a number {wi*-\-7n\)(n*-$-n\)(q*-\-<fi) which ig a sum of two squares in
three ways, whence
while у (or z) Is the similar expression with only the second (or first) term
negative. Set v^m/mi, r=nfnXi s=qjq^ Then (x-\-y-\-zJ-\-x2^aT+bi he-
comee/v2—4^(r-\-s)v=f-\-4rs-\-4:J where/=(r2—l)(s*—1). Thus the square
of fv—2r$—2s is known; equate the root tof-\-2rs-\-2-\-Can& take <7= —2
to cancel the terms in s\ s1. Hence 2rs= —1, r—Br^—3r—l)/Br-+r—1).
Take g= — Hi, qi=2n, т = 2п?—Зпщ—п\. Then
The least positive numbers found are 19, 13, 2.
To make ^-|-V4-r2+2£tj-2ff-i-2>?r, ete., equares, W. Wright150 put
i=x-\-yf r)=z-\-z, f=y+j and noted that the problem is reduced to the
preceding one, for which he took y = px, z=3x, and found p so that
^_j_4p4-12^(p-rJ; finally^p-l-lS^nifr^ie. Others equated the first
function (£+,+f)*-4fr to (Е-Ит?J, whence f=2f-2ij, or to B^-f/2J,
whence J^ij+r/^- Then the difference of the other two initial functions
factors.
J. Cunliffe made x2+y*-\-a(x-\-y), x2+z2+b(x-\-z), y2+z*+c(y-\^)
squares by taking я—n>, у—щ z = tv, where rq-\-s*=e\ т2-\-1*=Р, ss+^=^i.
Take m=a(r+8)je\ п=Ъ(г-\-1)/Р,р=с(*+1Iд*. Then the quotients of the
initial functions by e2,P, ф are tfi-\-mv7 v*-\-nv, tfi+pv, which are made squares
(Cunliffe1 of Ch. XVIII).
D. S» Hart152 equated the same initial functions to the squares of x-\-yy
x-\-z, y+z. Then a(x-\-y) = 2xy, etc., determine x, yy г rationally in terms
of ащ Ь, с.
'The Gentleman's Math. Companion, London, 2, No. 9, 1806, 31-35»
• 1Ш., 5r No, 29t 1826, 502-fl.
Itod^ 3, No, 14» 1811, Э00-2. Same by J. Matteeoo, ТЪе Andyet, Dee Matnee, 2, 1876,
46-9.
Math. QptsL Edue, Time», I7t 1872, 37»

Chap. XIX] SYSTEMS OF EQUATIONS OF E>EGKEE TVo. 529
W. Wright1*1 noted that sx—yz=m*t sy—xz=n2, &z—xy=*r*, 8=х-\-у-\-г,
lead to the problem (х+у)*=т*-\-п\ (х-\-гу=т*+г\ (y-\-z)i=n'1-\-r2 at
the beginning of this Chapter.
S. Jones1*4 made a=sz-\-yz, ft=8z-\-yxt y=$y+zx squares, where
s=x+y+z, by taking y=a~x, а=Ь?, y=c?, whence z^(a2-|-b*—c?)jBa),
etc., and0=O.
J. K> Young1" found four number» whose sum is a square and such
that if unity be added to the product of the sum by any one of them there
results a- square. Let the numbers be x±l, x^=y. It suffices to make
4x, 4x*±4xy-\-l squares. Take s=4 and set 65 —16^ = wi2, $5-\-lGy-nzT
Then *л*Н-п*=130, which holds if m-3, n=ll*
W. Wright and others found1* four numbers vy xt у, z whose sum is a
square n* and such that vn*±l^ П, etc» Equate wi?+l, хп2+1, уп2-\-1,
zn*-\-l to the squares of l-)-s, 1+r, l-)-g, l-\-p. By addition, s2-b2s-)-l=7i4
if rZ-\-q*-\-p*-\-2r-\-2q-\-2p = I. The latter is solved for r after taking
g—m — lf p^=lm— 1. Several solvers used the numbers x±l, x±y.
To make x2+y2+St x*+z*-\-Sf y*+z2-\-S squares, where
S=2xy+2xz-\-2yz,
W. Wright noted that the functions factor, being a(b-\-c)t Ь(а-\-с)} с(а-\-Ъ),
where a*=x-\-y9 b—x-\-z, c^y-\-z. Take b=naf c=mat n(m-\-l) =пЦ2,
m(n-\-l) = (n£—p)K We get rn and n. Then m+n = NfDt where N and
D are quadratic in $. Take N = {pi -\-q)* to get £. Then D = П becomes a
quartic in q. C. Holt noted that one condition is satisfied if the numbers
are 5n—m, m—4n, 4n. Baines wrote s=x-\-y-\-z; thus $*—z2, &~y2y
s*—x2 are to be equares, say of («-(-г)/?», (s-)-^)/n, (s-\-x)fr. We get ^ ^^ г.
To satisfy 2s-s, take r-3, n= -37/36, Л1=25/21.
To find three numbers double the sum of any two less the third being
a square, double the sum of any two of their squares less the square of the
third being a square, while the last three squares have the same property,
W. Wright16* used the numbers x, y} x-\-y. Then all but the first three
conditions are satisfied. Takex-\-y=d*} 4x-\-y=Ba—pJ. For the result-
resulting x, a, 4y-\-x= П if р*-\-54:р*у+9у2= D = C#—vJ, which gives y.
To make169 2(v+x+y-\-z) = n = 4a2t a=2(x-\-y+zJ<-2tfi=n, etc.,
note that a=4a2(x-\-y-\-z—v)> Hence take х-\-у-\-г—г=4№, etc. The con-
condition a?z*b*-\-c?-{-d?-\-e2 is satisfied by taking a—e+r and finding e.
Several1" discussed the problem to make a-\-bt b+c, b—c} a-$-2b-\-c-\-d
and a2+bc+bd-\-cd squares, (a+b)(b—c) = b+c, and b*—c?=l.
ш The Geutfem&n'a Math. Companion, Loodon, 3, No. 17, 1814, 462-4.
т1Ш.. 3, No. 18,1815, 317-8.
«•Algebra, 1816. Amer. ed. by S. Ward, 1832, 331.
ш The Gentteman'a Math. Companion, London, 5, No. 26, 1623, 240-2.
"* Ibid., 5, No. 29» 1826, 600-2.
»• Ibid., 5, No, 30,1S27, 675-6.
"• Math> Mieeellany, Flushing, N. Y., 1, 1836,151-5.
35

530 HlSTOBY ОТ THE TjTEORT OF NUMBERS. [Crap, XIX
J. Matteaon found four squares such that fifteen linear or quadratic
functions of the squares or their roots shall be squares.
A* Martin and H, W. Draughon1* found three integers such that the
square of the sum of any two less the square of the third is a square.
A. Gerardhv" treated x*-(y-zy=a, y*-(x-zy=b, г*-(х-уУ=с.
Set y=z+u, x=z+u+w, z=w+h. Then c=hrf a=rs, b=h$, where
r=h+2w, s =
Rational Октнооовсаь Substitutions.
L. Euler164 stated that he had a general solution of the problem to find
16 integers arranged in a square such that the sum of the squares of the
numbers in each row or column or either diagonal are all equal, while the
sum of the products of corresponding numbers in any two rows or columns
is zero. The example given is the following:
68 -29 41 -37
^17 31 79 32
59 28 -23 61
-11 -77 8 49.
Euler115 treated orthogonal substitutions on w = 3, 4, 5 variables, i. e.,
linear substitutions leaving unaltered the sum of the squares of the variables.
He expressed the coefficients in terms of trigonometric functions. For
n=3, he noted the rational solution
2qs-2pr
2qr-2ps р*-?-\-т*-з? 2pq+2rs
2qs-\-2pr 2rs-2pq pz-tf
each entry being divided by р^+дЧ-г2-)-*2- For n-A he gave two similar
rational solutions of which the second is
ap+hq+cr+ds ar—ba-cp+dq -as—br+cq+dp aq-bp+cs—dr
-aq+bp+cs-dr as+br+cq+dp ar-b$-\-cp—dq ap+bq-cr—ds
ar-\-bs—cp-dq —ap+bq—cr+ds aq+bp+cs+dr as—br—cq+dp
—as+br—cq+dp — aq-bp+cs+dr —ap-\-bq-\-cr—d& ar+bx+cp+dq,
in which the sum of the products of corresponding numbers in any two
rows or columns is zero, while the sum of the squares of the numbers in
any row or column is <г = (а*+Ъ*+с*+й*){$?+<1*-\-т*+а?)л For his164 former
problem, we require also that the sum of the squares of the numbers in
* Math. Quest. Educ, Times, 18, 1873, £5-7, Same in his Colfectttm trf Diophantine Prob-
Problems with Solution <ed~, A. Mutin), WanhiiigUm, D, C, 1888, 22-*.
™ Amer. Mat*. Monthly, 1, 1B94, 361-2.
юйрЫпхЮоБре, 8, 1913, 30-1.
» Opera poelum^l, 1862, 576-7T letter to Lagr»ngpy Mar. 20,1770, Quoted by Legendi*,
Theorie dea nombree, 2t 1830, 144; Masses Goman tmrwL, U, 240.
» Novi Сошщ. Acad. Fvtrap~t 15,1770, 75; Comm. ArHh.r 1,427-443.

Сядр. xixj Rational Orthogonal Su випт и noxs. 531
either diagonal shall be <r, viz.,
{4C+bd)(j>r-\-qs)=Ot (ab+cd)(pq-\-r9)-\-(ad+bc)(ps-\-qr)=0.
He gave two special cases, one of which is his181 above solution»
G. R- Perkins1** employed as the numbers of the first row of his square
pr'+q*' — rp'—9q\ psf—^/ 4-rtf ^sp\ <pq'—qp'—T*' + *r',
—pif±qr'-\-rq'^sp't ргЧ^+гр'^»/, ppf -trqqf—г/—ssf y
pp'—qqf z-rr'—s»', —ptf—qp'+fV+wf, psf-\-qr'+Tqf+$p\
— pq'—qpf—rs'—sr\ — pp'+q<f-\-jf — *tfy pr'—qs'+rp'—sq'
those whose sum of squares equals (р"+^+^+^)(р'*Н— ♦)- By writing
in reverse order the functions of the first row and changing the signs of
r, 8 in the first two terms and the signs of p, q in the last two terms, we get
the entries in the second row. We derive the third row from the first,
and fourth from the second, by moving each term one place to the right or
left without crossing the middle vertical column, and changing the signs of
g, s or those of pT r according as the term is moved to the right or left.
Two of the various possible such squares are given» Of the conditions re-
required by Euler,1*4 all are now satisfied except those relating to the two
diagonals. Take g—0. The latter conditions become
By further specializations, he obtained the solution
42+25 -11+4$ 24- q 2-8g
-18+8g -16+ q 24+4$ 38+2g
ц_|_4д 42-2$ - 2-8g 24+ q
16+ q ^1в^8д -38+2$ 21^4$.
C. Avery1** proceeded as had Perkins, without describing the process to
choose the signs, and obtained the solution
48+4$ -44+3$
-47_j_6$ 21+2?
44+3$ 48-4$
-21+2? -47+6$
The case q= 5 yields Eider's1** answer.
V. A» Lebesgue1*8 gave orthogonal substitutions in 3 variables in trigo-
trigonometric form. He11* quoted Euler's1"-5 solution of the problem of 16
integers.
L. Bastien17* took four integers л, &7 yfi such that a0l(yB) is the square
of rjs, where г, в are relatively prime integers. Write
51-2$
64+3$
7-6$
-12-Ho
-7-6?
12+4*
51+2?
64-3$.
«* Math. Miscellany, New Yortc, 2, 1839, 102^5.
**Ibid.t 10b
»' Nouv. Ann. Math,, 9, 1850, 46-51.
™Ibid.t 15, 1856,403-7.
lWSphinx4)edipep 7,1912, 12.

532 History of the Theoby of Numbebs.
Then a solution of Euler's164 problem is
2U 2cty+yu+6t yt-bu
yu-U fa+cty
лх-ffi/ yt+tol
—yu-Bt —2py+yt-8u ax+fiy+2yt Px—<*V>
the sum of the equares of the numbers in any row, column or diagonal being
Fuss* of Ch. V made p*-)-8*, tf+&, r1^-^ squares with pq-\-pr+qr-=&*.
NOT AVUbASLS FOB ВЕРОЯТ.
S. GQnther, Ziete u. Remiltate i neueren Math, ffiat, IbnctnmE, Шад«еп, 187ft, 60-53.
J, Fftvaro, Notixe Btoriobcriticbe sulla ooetruiKme сЫ1е equa&onit Moddnet 1878-
G. de Lougcbampe, Jour, de muth. ^m., 1882,192; 28, I8d4t 5.
Ferreot, tWrf.r <2), 3r 1884t 12lf 155, 1Ю, 193, 217, 241; 1885, 3, 17СЫ.
J. Novuk, Ueber unbent. Gl. 2 Grade* mit 2 Unbek., Pro«r. Budweie, 1890.

CHAPTER XX.
QUADRATIC FORM MADE AN NTH POWER.
Binary Quadratic Form Made a Cube.
Diophantus, VI, 19, to find a right triangle the sum of whose area x
and hypotenuse Л is a square and perimeter is a cube, took 2 and x as the
le^ and h-\-x=251 noting that the square 25 when increased by 2 becomes
the cube of 3. Then ^=^+2» gives s=621/5O.
Jordanus35 of Cb. XII noted that x(x-\-l) is never a cube.
C. G. Bachet1 noted that from 54-2=3* we can find other [rational]
numbers x making a^+2 a cube. Let x=5—N. To make 27-10ЛГ+ЛГ3
the cube of 3—2, equate the second term —27z of its cube to — ION, whence
z=10N/27. We now get N. In VI, 20, we have 17^2*+3* and веек а
cube which increased by 17 gives a square; take N—2 and 3-K as the sides
of the cube and square, and equate the second terms 12N and Ы of the
expansions, whence tf=10, *=20.
Fermat* stated that he could give a rigorous proof that 25 is the only
integral square which is less than a cube by 2.
Fermat' stated elsewhere this result on 25 and the fact that 4 and 121
are the only integral squares which when increased by 4 give a cube.
L. Euler4 proved that s*-)-l^D has no Qxraitive] rational solution
except x—2. To show that, for a and b relatively prime, с^Ъ^-Ь>=[3
only when a=2, 6 = 1, set a-\-b=c. The condition becomes bcg~ П,
g^c?—3bc-\-3№. First, let с be not divisible by 3. Then b, с, д are rela-
relatively prime and hence each is a square. Set g= (bm/n—cJ and solve for
6/c. If m is not divisible by 3, с = ± (m2 - 3n2), b = ± Bmn—3n*). For the
lower sign, с is not a square. Hence c = m*—3n2= \3 = {m—npjq)t>
m/n=(^-i-3g2)/Bp?). Then b/n>=G!(pq), G=р2-3рд+3^ Thus
pqG=n, so that the method of descent applies. Next, for m=3k,
b : c=n%—2kn : n?—3H As before,
Hence (S^+j/)(j)-q)(p-dq)^U. Let p-q = t, p-3q=u. Then
and the method of descent applies. Finally, let c—3d. Then
is of the initial type with the former b, с replaced by dt b. Since b is prime
to 3, the descent applies. It is stated that a like proof shows that x* — 1 ф П.
* Diophanti Alex. Aiith,, 1621, 433-6.
* Oeuvrea, I, 333-4; French tranfi, Ш, 269.
* Oeuvree, П, 345,434, letter*toDfeby, Aug., 1657,andtoCareavi, Aug., 1659. E.Braadnne,
Ptfds dee Oeuvra math, de P. Feimat et de TAiiih. de Diopbanie] Mem. Acftd. Sc.
Toulouee, D), 3, 1856, 122, 164,
mmu Acad. Petrop., 10,1738,145; Comm. Aritb. CollM I, 33-34; Opei» Omola, A), II,
56-58. Proof ^published by E. Waring, Medit. Algebr, ed- 3,1782,374^5,
533

534 History of the Theoky of Numbers, [Chap, xx
Eukr5 applied to х*-М = П his1*4 method of Ch, XXII to make a cubic
or a quirtic a square, finding no solutions except 0, — 1, 2, and stated that
thei?sre no others. Cf. Euler1" of Ch. XXI.
Enler/ to make ax*+cy* a cube, assumed that
of. For Fermat's case zJ+2, we have
(ArtJ3) a—l,c=2, y=rbl, whence дСЗр3—2д*) = ±1, and g divides unity,
Tafcngg = l, we have Зр3—2=±1, whence p*=l, x*=25. A like proof is
given (Art. 192) of Fermat's result that 4 and 121 are the only integral
squares which when increased by 4 give a cube. But (Arts. 195-6) for
2r*-Hhe method leads to no solution, whereas the solution z=4 exists
and tip above assumption is shown to fail.
AIL Legendre7 treated Fennat's problems as had Euler.*
Y,A. Lebesgue8 proved that x5—7=^ is impossible. For, if у is even,
x uoU and х*=8п+1ФB*)э+7; while, if у » odd, x»+l = (tf+2)Q is
imposble since the prime divisors of B = (y-~lJ4-3 are of the form
LMtinger* noted that x2 — yl=z^i? has the general solution
TPepin10 criticized Euler's* proofs, noting that there may exist sets
of fotmalas for x and у other than the set deduced by Euler's assumption.
He piwed FennatV"8 assertions. He studied the solution of x*+cn?*=^
for c=l, 2, 3, 4, 7, n being 1 or an odd prime, and z being odd if с = 7, and
proroi that the following are not cubes: хЧ-1 (х>0); х'+З; 4^+7;
««Н-Ы+л1 if п=108И-Л(* = 23, 35, 59, 71, 95), or n-83, 263, 407, or if
n ia a prime 12i-|-7 with 7<n<1350; ^+2nl if n is a prime 24^+5 or
2il-\-1\ x*+3n2 if n is a prime 6^+5 or its square; s*+5. Also, x5-!-^^^8
oniykx-±46t^+72=^ only for x^±524; д^+И2^^ only for x'=4.
HRrocard11 and others gave various solutions of x5-|-17=y2.
G.C. Gerono1* proved that x5=y2+17 is inxpossible in integers by use of
(хЩ{х—1J+3} =y2+52and the divisors of a sum of two squares.
Kde Jonquieres18 proved that x*+a=y* is impossible in integers if
«=M, k] = l, 3, 7 (mod 8), or a=c*~4', |c[^3, 5, 7 (mod 8), *>1, or
a=H,c=2Bd-hl), and hence if a=-3, -5,7, -9, U, -17,23, -43,
61;«bfora=4, 6, 14, 16ifs+0.
FProth14 stated and E. Lucas14 proved that x*+3—у* is impossible
sin«3=7^4-3s2, while x7—3 = ^^105 only for x = 2, y=\.
, 2, 1770, Ch. 8, Art, 121; French tranel,, 2r1774, pp. 135-^152; Opera Onmi»»(lX
Ir 392, Мбш» Acad. Sc. St. Peterabourg, 11, 1830 A780), 69; Comm, Arith. Coll,
II, 478,
■%Ьгв. II, Ch. 12, Art*. 1S7-196. Opera Omnia, A^, I, 429-434.
тШопе dee nombrea, ed. 3, IIf 1830, Art. 336, p. 12,
%v. Ann. Mmth., BJ, 8,1869, 462-6, 559.
«inhiv ЛШЬ Phys., 49, 1869r 211.
"Jut. de Math, C), lr 187S, 318-^, 345-358, Detaile in Pepin.71
4nv. Corresp. Mmth., 3, 1877, 25, 49; 4,1878, 50. Cf. Eecott," Brocani.»
"Xuv. Ann. Math., B), 16r 1877F 325-6.
»M, B)F 17, 1878, 374-380, 614-5.
>Kmv. Согтетр. MaUb, % 1878,121, 224.

Chap. XX] BINARY QUADRATIC FORM MADE A CuBE. 535
T. Pepin1* applied dc Jonquieres'13 method to obtain the generalization
that я?+а=у* is impossible if a is of the forme3—44>2, where b and с arc odd,
while b has no divisor 4J+3, andc=l, 3y 7 (mod 8) if <* = 1, c=3y 5, 7 (mod
8) if <*>1. Abo if a=HBd+l)*—b2, and Ь is prime to 3 and docs not have
two factors (equal or distinct) of the form 4J+3; for example, a = —17 or
47» Also, if a=8^—26*, where c=4£-|-l and Ъ is an odd number not having
two equal or distinct prime factors of the forms 81+5 or 8J+7; for example,
д=6, -10,118,-58. Also, ifa = 8cJ+26g,c=4A;+3> and b is odd and with-
without two prime factors 8J+3 or 8J+5. Also in several analogous cases.
E. Catalan1* noted that some, but not all, solutions of x2+3y*=z* are
S. Bealis17 gave identities showing solutions of x* +k ^ y*\i fc = &*(86 - 3a2),
Ь*(Ь-За*), Ь(За2+&I, 4л2(аг+1). Given one solution <?+k=&\ another
follows from the identity, obtained by Euler's* process,
\ 4^
Realis1* stated that, if г*—Ъаг—а3 4-/^=0 has integral roots,
«»+[(«+lj'-tf+l)*]*-*1
has integral solutions x=cc—z, y=0—z; for example, if a=a2, 0 =
«=2,^=rfcl; «=32, /3 = ±64. If
/J/4has integral solutions other than :г = а,р^0. Cf. Ch. XXI.™
T. Pepin19 proved there is one and only one square which becomes an
odd [Pepin*] cube on adding 2, 13, 47, 49, 74, 121, 146, 191, 193, 301, 506,
589, 767, 769, 866 or 868. No square >0 added to 1, 3, 5, 27, 50, 171, or
475 becomes an odd cube. The only solutions of a^+ll^y3 are s=4, 58;
the only solution of s?+19 = y3 is y=7. If a is one of the primes 11, 17,
29, 37, 47, 83, 96, 107, 181, 197, 233, 359, 421, 569, 757, 827, there к а
single square which becomes an odd cube on adding Ha2. If a < 1000 and
a is of one of the linear forms 38J+3,13,15, 21,27, 29,31, 33,37 and а Ф29,
89,173,281,331,569, 953, no square increased by 19a2 is an odd cube. Also,
similar theorems.
Pepin" gave sixteen special theorems on x*+g=z*f proved only under
the assumption that x is even and г odd.
Pepin*i proved that x*+n*z4f ti = 5, 6, 10, 12, 14, -У98; 4^+пФ*3
if n-7, 15, 39, 47, 55, 63, 71, 79; zz+44=z* only for s2=81; and gave
several theorems on x^+lly1^? [all provided z is odd, Pepin33].
№ Anoales Soc. Be. BnixeUee, e, 1881-2, 86-100.
* Mem. Soc. Sc. Liege, B), Ю, 1883, No. 1, p, 10.
" Nouv. Ann. Math.r C), 2 ,1883, 289-297.
Ы&., 334-5, Proof of firet by E. Fauquembergue, C), 4, 1885y 379; of secoDd by H.
Brocaid, C), Ю, 1891, p. 7* of Exercicea.
i' Mem, Pont, Accad. NdqvI Lmcei, 8, 1892, 41-72; Extract, Sphmx-Oedipe, Ш8-0, 188-9.
«. Pepin.*
"Comptee R^ndue Paris, 119, I894r 307-9; corrections, 120r 1895t
я Ibid., 120, 1886, 46

536 History of the Theory of Numbers. [Сил*, xx
E. FauquembergueB gave an insufficient proof that 2?-\-2^y* if яф — L
C. Stormer28 solved x*—y*=^ by means of the identity
A. Goulard24 proved that x*—l=z* only for x*=9, since z*—l=8u>* has
no solution except when w=0 or 1 [Xegendre,81 of Ch. I]. T. Pepin (pp.
283-5) reduced the question to us4-x5^2y3 which holds only for u=x
[Legendxe, Thdorie des nombres, ed. 2, 1808, 347].
Б» de Jonquieres26 treated x2—o*=^. For a=3, E. B. Escott24 noted
the solutions у = —2, 0,3, 6, 40 and stated that there are no others <1155.
Concerning Fermat's assertion that 25 is the only square which in-
increased by 2 gives a cube, H. Delannoy27 remarked that EulerV proof is
incomplete since if applied tos*+47—z*it yields z=500 but not the solution
X = 13. P, Tannery18 replied that the proof as given by Legendre7 depends
on the fact that every divisor of xi4-2 is of the form tf+2(?t while not
every divisor of я*+47 is of the form p*+47g". I. Ivanoff (p. 47) explained
the difference by the fact that in the domain 2?(V^2) of the complex
integers depending on V—2 the introduction of ideals is superfluous, but
not fat R( V—47). E. Landau25 supplemented Ivanoff's remark by noting
that a second circumstance is necessary to justify Euler's conclusion that
(x+V^2)(x— V^2)=^* implies that s±V^2 are cubes in .R(V^2)> viz.,
that, in .R(V^2), ±1 are the only units (complex integers dividing unity).
From the superfluity of the introduction of ideals, we can conclude only
that, if a product of two relatively prime complex integers is a cube, each
of the two factors is a product of a cube by a unit. For R(^2O the intro-
introduction of ideals is unnecessary, but (x+V2)(z— <&)^& does not imply
that x± V3.aie cubes of integers <*+0 V2. Cf. Euler1* of Ch. XXI.
A. Boutin» stated that x5—7^ = 1 for x=l, 2, 4, 22, but for no other
values <196. Other writers31 stated that any new solution has at least
1400 digits in y.
E. Fauquembergue51 noted that pxi+vwy+qy2=z? for
T. Pepin" remarked that all the theorems in his1*-*1 papers on insolvable
equations x2+cyJ—£ were subject to the restriction that z is odd. The
enunciation of this restriction is necessary if c~8Z or 8Z+7 since in these
» Mathesie, B), 6, 1896, 191. Criticised by L. Aubiy, Гш(епо&1. dts matb., 18r 1911, 204.
" L'frtermldiiure dee matfa.r 2,1805, 309.
»1Ш.Ь 3,1896,135.
»Ibid.te, 1899, 91-5; 5,1896, 257 (a-3). Cf, Deecartce," Ch. ХШ; Taib,« СЬ, XXI.
»/«rf.r7,1900,135.
«TWA, 5,1896, 221^.
"/Ш., 6,1890, 48.
*Ibid.t 8,1901,145-7.
»ШЛ, 8, 1901,278.
л Ibid., U, 1904, 44 (9,1902,109,183-5).
" iW.,0. 1902,311-2.
* Ann. Soc. 8c Bnuelleflr 27, П, 1902-3, 121-170. £xttact ш Sphmx-Oedipe, 5,
10-13 (of пшпёго арёеЫ), 42-6.

. XX] Binary Quadratic Form Made а Сита:. 537
two case» г can be even without x and у being even. That tha solution of
the equation is effected by different formulas according as г is even or odd
is shown by the caee с=47. Then all relatively prime solutions in which
z is odd are
where / and 0 are relatively prime and one even. All solutions of
x -j-47^3 — Bt*)% where и is odd, are
with similar expressions when 2—4», 8«, IGw, etc. The cases c=35,
c^499 are treated (p. 142, p. 155)
Pepin* noted that 2x*=Zy*-l has the solution z«6I, y=389, but left
undecided the question of an infinitude of solutions. One of two methods
is based on the theorem that all relatively prime solutions of 2z3 = 3j/*—г*
are given by
x=F-3g*, y=/A+3gB or ZfA-lbgB, z=fA+9gB or -5>fA+27gB,
where A =f*+Qg*, B=f*+tf* It remains to find/, g such that 2-±1.
G. de Longchampsw stated that рхг-\-ду'1=£ always has integral solu-
solutions, tin fact, a solution is 2 = at, y = 0f, z^f^pa^-l-gjS*.]
IL Brocard3* listed the known values of a for which x5—y*=a is impos-
impossible and the values for which there is a single solution.
E. B. Escott,*7 A. Cunningham and IL F. Davis* treated x^-Y7=yz.
A. S. Werebnisow* expressed EulerV solution of z*+cy*=& in terms
of «= -2p, fi= -р^-1-Зсд8.
Several* solved x*-\-3y*=4J completely by use of identities.
U. Bini41 gave a solution of а^+Зр*=г* involving two parameters.
An anonymous writer" noted that X7y7—1— 2x* has no solution with
y
A. Cunningham*1 gave a tentative method to solve з?=уг+а. Choose
a modulus m, preferably 10* or 10*, and find the values <m of x for which
x2—a is a quadratic residue of m. By use of various m's we finally get
the possible linear forms of x. Application is made to a= —17, a= — 127.
Several" solved x*+x±l =^
Welsch* applied the theory of binary quadratic forms to justify
Legendre's7 determination by use of V^3 of all solutions of хг
Noqv. Ann. Math.» D)r 3, 1903t 422-8,
Lrintcfmediaire dee math., 9, 1902,115.
TfcdL, lOr 1902,2S4.
L, 12, Ш*5» 43Ч5. Amer, Math. Monthly, 26, 1919r 23941. a.
h. Quest Educ-Timee, B), 8, 1905,53-4- Cf. Cnnningbiiit^
»L'iDtenD6dimiredeeto*tlb, 10,1903,152. a. К В. Eecott, И, 1904,101-3.
»1Ш.Р 14,1907,168; 18,1911^ 279.
* Ibid., M, 190Г, 192.
«Sphinx-Oedipe, 1906-7, 79.
"M*th. Qaeet. Educ. Timee, B), 14, 1908r 106-8.
« L'mtenn6diaire dee math., 15, 1906,244; 16,1909,201; 17,191OF 126; 23,1916,4.
M, 17,19l0r 179-180.

538 Histoky of the Theoey от NoMBEae. [Сил*. XX
E. В. Escott4* noted that, if ^-=2^-1 is solvable, y^24nJ-l or 2л5-1,
U Aubry47 proved that x*+1 +#* = jr1 is impossible. If x is odd, x*+2*
is a sum of two relatively prime squires, so tbftt the factors of y3— 1 are
= 1 (mod4). hii
where z is odd,
Since ?/2+y-fl is odd, its prime factors arc of the form 4f-(-L Thus
y-1 is divisible hy 2?nand heirte by 4, Again, ^+^4-1=3 (mod 4).
L. Aubry and E. Fauquembergue48 proved that 2x2—l—yi has no
solutions other fchanj;<=0, y=—1; x=dzl,* —I; я=±78у y=23.
A Gerardin4*, to mate G^tf+xv+tf a cube, assumed that
and took — Щ+2х+у=0. Then/and »t are expressed m terms of r, y.
To make the result symmetrical, set v=g/3, я=р-г-$/3- Нейсе
for
a result obtained otherwise by A. Desboves.*0
Gerardin" treated aX*+bXY+cY*=hZ\ given one solution a, j8, y.
After substituting X=a+mxt Y—fl-\-my, Z=y+mf% equate the coefficients
of the first powers of m (by choice of /); thus m is determined rationally.
L, Aubry*2 proved that 25 is the only square which increased by 2 gives
a cube [Fermatf]. He5* proved that x4-e=Vis impossible for a=4A3+fi3
if B = \ (mod 4) and A is not divisible by the square of a prime An — \
dividing B, or by 3 if В is not divisible by 3, or by 31 if В is divisible by 3,
Hence it is impossible for a —17.
E, Landau54 proved that х*+2=у* has only a finite number of solutions
by means of Thue's resnit that ct+Scfff+Gap+20*^1 has only a finite
number of solutions (Thue*of Ch* XXIII), and Landau's2* discussion above.
H. Brocard" gave eight sets of solutions of x2—^^17.
L. J. Mordell* investigated у*—*=х* by elementary methods, by the
theory of ideals, and by the arithmetical theory of binary cubic forms.
In particular, he listed the values of к between —100 and 100 for which
he believed there is an infinitude of solutions.
~~« Amer. Math. Monthly, 16, 1909, 96»
« Sptunx-Oedipe, bt 1911, 26-27; stated by R Proth, N<miv. Corrap. Math., 4,1878r 6lr 223.
♦« SphiDx^Oedipe,G, 1911,103-4; 8,1913,170-1 A22-3ЬгE.В.Eeeoit'sрлх»Г Uwl aeulution
y>23 has more than 266 digits).
" Aeeoc. franc, av-ec., 40, 1011, 10-12.
"Nouv. Ann. МаЙьД2)Л8г 1879r 269, formula (8) witha=6 = l.
« Bull, Soc. PhUomathique, {10)r 3, 1911, 222-^.
"Sphinx-Oedipe, 7, 1912F 84«
« L'intermediaire Лея mftth.r 19, 1912, 231-3.
" Ibid., 20, l»13r 154.
u Jbid.t 62-3» Cf» Broeard.i*
v Proc. London Math. Soc, B), 13, 19Hr 60-SO.

Впслжж Quadratic Fobm Mads an «th Power. 539
A- Geranimw «mmawit the kaown results on т*—к=г*\ be noted
the solutions 21—4=2P, 5?—4=^1P, ctmtrary to de Jonquieres'1* assertion
that only one dotation exists. Given one solution xOf ге, G£rardin deduced
(ibid,, 163-5) the second solution [RbVJ
Sei ar*=2fc where p is a prime. Tben г»-^, 2p>", 3p>, 6p> C/=0, 1, 2).
There result twelve integral vahies of к for which the given equation is
solvable, F« fc = Bp—l}*^—2p-M), the solutions include x=2p,
p, }, p^ (pp)p
L. Baetien* listed the values 3, 5, 6, 9, 10, 12, 14, 16, 17, - , 99 of
«^lWforwhkkfl'-^nisknpoi^ble, the values n^l, 2, 8, 13,29, ••-,
81 for which there is a single sohitioa, and the values for which there are
more than one solution.
Cruseol" noted cases wben j*+k=y* has 7, 9, 34 and 41 solutions.
A* Geixrdm {Hb^Ly pv Щ noted cases when it has 21 solutions.
A. Gerardm* proved that all solutions of д^+З^^г8 are given by
T. Hayashi"proved tha*^+l*^ for у +0; ^-l=f=^ for ^Ф0, I,».
А. СшшЬ^мш^8 proved thaV if pis prime, х'—р^г* 10* has the single
sofationz=129, p=383.
L. J- MordelF* noted that no equation x*-\-a=y* has an infinitude of
integral solutions.
For 2зЧ-2*+13=з*, see paper 161 of Ch. L On 2764-1*=^ see
Кютескет2» of Ch, XXI.
ВШАКГ QlTADRATIC FORM MADE AN ПТН
J. L. Lagrange** noted that the *nth power of f=x*+axy-\-by* can be
expressed in the same form F=X*+aXY+bY* by employing the factors
tf-hoy, x+$y of/and taking X+aY to be the expansion of (x-hay). The
resulting values of X, К make F an mth power.
L. Euler» stated tbat he ueed this method for f~x*+ny* in the first
edition of his algebra^
Euk»18 noted that, if N=az+nb*, i\^ is of the form z*+ny7, and asked
for the teast хфО от least y*& for which N*=x*+ny*. Let
«7W., V, 1914, Ifr-U.
21, 1814,129.
Шв 1ДЬ5
- Lottdofl Mstb. Soc. Recife of Ptoe^&v^ N«t. 14,1918.
«AdditMttIZtDi&dtfT8A]erix&tLyo(D}21i774f63i4S44; Eoler1» Opem Omnia, A), 1,1911,
638-M3; Owmt ёб Uwp, VHF 1*4-170. For f**j*-Btft Lagnutge, Mfim» Acad,
t X, 1802; 571-3, letter to Lagnuige, Jan^ 1770; Oeuvrea de
■ Кот» AcU Ami. Fttn^ 9; 1T91 A777)^ 3f Сопшь AittlL, II, 174^182.

540 Hxjstobt of the Theory of Numbers. ССнаг. xx
Then
a+b^n-Vtf (cos ф+i sin ф), A =JV^cos \ф, В = {NAf* sin \ф\ j<fc.
Hence Б is a minimum Ф 0 for a rational value of X approximately equal
t к/ф h к i it
Ф
to тк/ф, where к is *n integer.
Euler* made x*+7 a biquadrate. For ^^(Tp2—g*)/Bpg), it is the
square of (q*+7p2)lBpq). To make the latter a square, tafee q=pz, -whence
we are to make 2zG-\-z?) = G. Since an evident solution is z=l, set
l We get 1С+2%+6^+2у*, which is the square of 4+5y/2 for
A. M. Legendre* treated Lv*+Myz+Nz?=hP, where P is a product of
powers of several variables, in particular, xk.
G, JL Dnichlet* recalled that if I is an odd prime not dividing a and if
S2—a*7=l it is known that <P—a&=l* holds for the numbers <f, e given by
d+e^=(&+t*fc)\ It is proved that d, t are relatively prime. If also
<%—ae2t=klny where db ex are relatively prime, and к is odd and prime to al,
we can find solutions of I2—aw*=fc such that
for a suitable choice of signs. Application is made to show that, if P, Q
are relatively prime, the most general manner of making P1—5Q2 a fifth
power, odd and not divisible by 5, is to set
where M, N are relatively prime, one even and M not divisible by 5. If
P, Q are relatively prime, both odd, and Q is divisible by 5, the most general
way to make P2— Щ2^^ is to set
where фу ф are relatively prime, bofch odd, and ф is prime to 5,
Cauchy's papers on the representation of p* or 4p*, where p is a prime,
by tf+ny1 will be considered under binary quadratic forms. Luce137 of
Ch. XII discussed x*-ny1 = zi.
F. Landry*8 obtained a new kind of continued fraction from
If mjn is a convergent of order и, т2—Ат?=(— 1)V. Hence to solve
x*—Ay*=zmt take as z Airy integer for which A = a*—z.
V. A. Lebesgue70 recalled the fact that, if о is an odd prime and A is
an odd integer dividing P+at but not divisible by д, А*=х*+ау2 holds for
an infinitude of values ju, wben x> у are relatively prime- The least fi is
« А1«еЪпц St. Pelcwbuig, 2F1770, 5 lflO; Fwnch traoeln Lyon, 5, 1774, pp. 1W-^; Opera
Onmia, A), X, 413.
" Theorie dee пшиЬга, 1798, 435-10; ed. 2, 1806, 37^-9; ed 3, 1S30, U, 43-49; German
trand. by Maeer, IIr 43-50.
"Jour, for Math., 3,1828, 354; Werke, IF 21.
" Cinquieme m^moire war 1& tbSorieJck* тютЪкя, ~Рщт, Шу, 1856.
» Jour, de Math^ B), 6,1S61, 2S9-340.

Chap. XX] ВшАКТ QUADRATIC FORM MaDE AN ЛТН POWER, 541
said to be even if A is a quadratic non-residue of a or if A=4n+3, a=4Jt+1.
When p^2v, у is odd» Then A'-x=pz, Ar+x=aqzf where p* and aq*
are relatively prime. Hence 2A' = р*-\-щ? and v is a minimum.
L. Ottinger71 tabulated solutions of x*-y*=zn, n = 2, 3, 4, and gave the
identity
T. Pepin" proved that, if с is positive and such that there is a single
class of positive odd quadratic forms of determinant —с (as for с = 1, 2, 3,
4, 7), the most general manner of solving x*+qf=z1*t where xt у are to be
relatively prime integers and z odd, is to set
where p;$ are any relatively prime integers for which z is odd. Hence we
can justify the method of Euler for c=l or 2. Next (pp. 333-8), let n be
a positive integer such that all the quadratic forma of determinant —n are
distributed into various genera each composed of a single class; then all
relatively prime solutions of xi-\-ny2=z*m+l, with z odd, are obtained from
A) ±х±у^^п = (р+2т1^)*"Ч-\
where p, q are relatively prime. But for х*+пу2=г*т, z odd, we use A)
with the exponent m} and employ the complete solution
q/i+y аГ-Ъ? 2/G
2——' p-~i^' q=T (fc=1<2)
of jp+iuf=z% where for а, Ь are to be taken all the decompositions of n
into two relatively prime factors, except that when к=2, n=8t, the two
factors shall have 2 as their g.c.d. For ах2+су2=г*л, а>1, с>1 (рр,
339-343), when ас ia one of the numbers for which the number of classes of
quadratic forms of determinant — ac equals the number of genera, there is
no solution in integers +0 if m is even; while if m is odd we get all relatively
prime solutions with z odd from
where p, q are relatively prime. Thus 2x*+3 and 2+3^ can not equal
cubes.
Pepin11 proved that x*+a is not a square if a=32Bd+l)fi—5b7, where Ь
is prime to 10 and has no prime factor 20*-|-11. Й d=Q, 6=1, 3, then
a = 27, -13.
M. d'Ocagne78 solved х*—ку*=гп in positive integers by use of
*(«, ft n)-
ft Arefaiv M»th И>у*., 49, 1869r Ш-222.
" Jour, de li&th4 C), 1, 1875, 325. Beiulta for m = 3 cited under PepinJ*
rtCtHdPi99 18841112

542 Нгвтонт of the Theory of Numbers, ссалр. XX
A solution involving an arbitrary positive integer a is
х=йфBа, fc-a5, п)+(к-а*)фBа, k-a?, n—1), у = фBа, к-а2, п),
z = ±(k—a2) for n even; z= — (к—a!), a> -fit, for n odd.
M. Weill" repeated Euler^s* method for ax*+cy2=z*>
T. Pepin76 proved that, if the number of classes of quadratic forme of
determinant — с is relatively prime to n, all relatively prime integral solu-
solutions of х*+су*=гп are given by
For я—3, the solvability depends upon whether or not the triplication of a
quadratic form gives the principal class.
H, S. Vandiver7* found an infinitude ofr but not all, solutions of
О. Candido77 employed Lucas' functions С/я, V*:
Change q to p2—q* Thus &->qy*=zn has the solution x — \V*> y = Un,
pq
A. Cunningham7* noted that у2+у+1 =хл is impossible if n>3,
яж<2а0в. R. W. D, Christie stated that x must be of the form az+a+l
and inferred that 7&#4, 5, n+3 unless a=2.
Cunningham7* noted that the only solution of J(tf*H~l) = P* with
g<1600000 is g^239, p = 13. Christie obtained this solution by making
special assumptions. Cf. Stormer1*7 of Ch. VI; Euler7 of Ch. XIV; Euler*1
and Pepin^ of Ch. XXII.
IL BiniM stated that the method of Desboves1'2 of Ch. ХХП1 [cf.
Lagrange83] does not lead to the determination of the form of the solutions
of х2-\-аху-\-Ъу2=гя for every integer n.
A. S. Werebrusow81 gave polynomials X, Y of degree n in x, у making
AX*+2BX Y+CY*^ (ax2+2bxy+cy*)\
E. B. Escott** noted that solutions of X2—DY2=4Z* are given by
But not all solutions are so obtained.53
O. Degel*4 treated the homogeneous equation obtained from the last
one by replacing X, Yt Z by Xifa (i = 1, 2, 3). The section С by хг=0
" Nouv. Ann. M&th.t C), 4,1885,18».
n Mem. Acead. Font. Nuovi IincaT 8f 1892t 41-72.
"Amer. Math. Monthly, 9,1902,112.
77 Giornafedi Mat., 43t 1905, 93-6, Cf. Candido"
» MMb. Qaeet. Educ. Times, <2), 8,1905, 6^-70.
п1Ш.ь 9,1902,2^24; 14,1902,77.
"L'iotermediAire des matli, H, 1907, 246.
» ib&, 15, 1908, 153; Mat. Sbornik, 22.
"L^intermediairedesmatii^ 15,1902, 153.
»Лй., 17, 1S10, 2,137-8, 229-Э0.
»lbid.t 17, 1910,253-5.

Сиар.хх] fli*i5-i h«^nl Made a Cube oh Higher Power. 543
lies on the cone х*х^~* — 4xl=Qt which every plane x*=px< cute in two lines
having xi=±2^s4. We get rational coordinates of the general point
P on С by taking p to be a square if n is odd. For example, let n=2m*
The general point on the line joining Р=Bрш, 0, /*, 1) and (p, 1, 0, 0) is
Bpm+Sp, 5, ji, l)^(x), which is on the surface if &=4pttmftD—pr) and gives
rational solutions xir The same problem was treated by others.86
F. Ferrari* madefy xt+axy+by2 an nth power. Let/=(x-(-ay)(:r+/fy).
A sufficient condition is x-\-ay=(r-\-a8)*. The latter becomes linear in a
by use of a3—<ux+b=0. Hence we get я, у as polynomials in r, s7 a, h.
O. Candido87 used Lucas' «*, vt satisfying
to show that for p—2\+ар, q^K+apiX+bv?, an infinitude of solutions of
х2+оху+Ъу*=гк is given by 2x=vk— арик> y=fiuk> z*~q. The explicit
formulas are given in the cases A;=2, 3, 4 and for a=0 or 5=0.
F. Ferrari88 used as had Lagrange, the expansion of (ai+utzJa " to
d А'в such that A«+o4j2 = (af+<*i!)*.
find А'в such that (f)
E. Swift8* proved that the number n(n—3)/2 of diagonals of an n-gon
is not a biquadrate.
By Thue211 of Ch. XXVI, x*-h?=ky* (n>2) has only a finite number
of solutions. On l+yJ*sn, see Lebesgue*8 of Ch. VI^ On l+2y2 = 3*, see
Fauquembergue168 of Ch. XXIIL On 1±4*»=D, see papers 7, 8,169 of
Ch. XXVL
OlzjH \~a*xl MADE A CUBE OR ШОНШ1 POWER.
S. Realis" noted that «5
if
J. Neuberg*1 took x^—xh yz = yi, Zt=—zi in the preceding result to get
E. N. Barisien^ noted that any sixth power is the sum of two squares
diminished by a third:
LmtermMiaire deemftth.t 18, 1911, 35.
■»PeriodicodiMat425, 190^10, 59-66. Of. Lagnmge«
M( 27,1912r 265-8, a. Cftndklo."
., 28,1913, 7ЬВ.
сг. Math. Monthly, 23, 1916, 261 2
« Nmiv. Ссямр. Math., 4, 1878, 325.
м МдШш, lt 1881, 74.
•» Le matematicbe рщс ed appUcat*, 2,1902, 35-36.

544 History of the Theory of Numbers, [Chap, xx
An identity (p. 253) shows that 4 times the cube of any even integer is a
Si less a SL
G. de Longchamps" noted that ах2+Ру'!+уг2+№ = и* for
The case b~t=O gives Neuberg's result.
An anonymous writer94 noted the solution x=3, y = 12, z = ll, и=2 of
J, Rose95 noted the solution x=4v*, y — №, z=4t>2(t1— 1), u — 2v} and a
solution with y=2-\-l> Mehmed-Nadir gave the solution
х = Ъ(а2+Ь*)(а*-Ь2)] z, t/ = |((a?d=l)(a2+^=b4b4j; ««tf+b1;
and noted that the same x, u, with У=а(агН-^J, Z = 2ah?(a1+№), satisfy
"V. G. Tariste"* stated that all sets of solutions of x2+y2—zz=u\are
given by seven sets of formulas like u=Aa, x—2b, и*—х*=4:йф; у, z = ot±0.
F, L, Griffin and G. В. М. Zen** discussed x\-\ \-xl =yA.
W» H» L» Janssen van Raay*7 solved x*=xl-\-yi-\-zi.
G» CandidoM found a solution of 2x*=yp by expanding U(a)+fl)).
R, D, Carmichael" gave a four-parameter solution of
M L'mterm^diaire dea math.r lOr 1903r 111-2.
* Ibid,, Hf 1907, 244»
*lbid.t 15r 1908,46.
« 7^19,1912, 38.
*»Amer. Math. MontUyr 17, 19lOr 147-8.
OT Wiekundigc Opfcaven, 12, 1Й15, 209-11,
« Periodico di Mat., 30r 1915r 45-47,
•• Diophantine Aralyaisr New Yorkt 19J5, 46,

CHAPTER XXL
EQUATIONS OF DEGREE THREE.
Impossibility of s^+y2^.
According to Ben Alhocain a defective proof was proposed before 972
by the Arab Alkhodjandi.1
The Arab Beha-Eddin* A547-1622) listed among the problems remaining
unsolved from former times that to divide a cube into two cubes,
Fermat3 stated that it is impossible to decompose a cube into two cubes.
Fermat proposed the problem to find two cubes whose sum is a cube to
Sainte-Croix Sept,, 1636 (Oeuvres de Fermat, II, 65; III, 287), to Frenicb
May(?), 1640 (Oeuvres, II, 195), to the mathematicians of England and
Holland Aug. 15, 1657 (Oeuvres, II, 346; III, 313). Oddly enough,
Frans van Sehooten4 proposed Feb. 17, 1657, the same problem to Fermat,
Fermat5 insisted that the problem is impossible.
Frenicle6 proposed the equivalent problem to find r central hexagons,
with consecutive sides, whose sum is a cube. By a central hexagon of n
sides he meant the number
The sum of Hn, Яя-1, ■ ■ ■, Я*-т-и is thus a cube 23 if and only if
J. Kersey*" stated that J. Wallis proved that no rational cube equals a
sum of two rational cubes, but gave no reference.
L. Euler7 stated Aug. 4,1753 that he had proved the problem impossible,
Euler* gave the following proof, incomplete at one point» We may
assume that x and у are relatively prime and both odd. Set x-\-y-2p7
x~y = 2q. Then we are to prove that 2p(p*+&f) is not a cube. Suppose
that it is a cube» First, let p be not divisible by 3. Then p/4 and РЗ
1F. Woepcke, Atti Aecad. Pont. Nuovi Lincei, 14,1860-1, 30l.
' Eesen* rier Rechenkinwt von Mohammed Beha-eddin ben Alhoeaain айн Amid, arabiech ц.
deutech von G. H, P. Neeeelmani), Berlin, 1843r p. S5. French tttmsl. by A. Marre,
Nouv. Ann. Math., 5,1845,313r Prob. 4; «L 2, Rome, 1884. Cf. A. Genocchir AnnaU di
Sc. Mat. e Ш, 6,1856, 301, 304.
'ObeervatioQ 2 on Diophantue (quoted in full in Ch. XXVI on Fennat'» bst theorem).
Oeuvree de Fermat, Ir 201; IIIr 24L The problem was eeot A637?) by Fermat to
M«eenne to be propoaed to St. Croix; ct P, Tannery, Bull, dee ec* math., B), 7,1883,
8,121-3.
* Correepondance of Huygenfi, No. 378, Oeuvres pompletee de Chr. Huygen«r 2r 1889, 17;
Geuvres de Fermat, 3, p. 658.
* Oeuvree И, 376r 433, letter to Digbyr Apr. 7, 1658, to Carcnvi, Aug. 155ft.
•Solutioduorum probkmatum . . ♦ 1657 [loet]; Oeuvrwde Fermat, UIr 605r 608.
•• The Elemente of Algebra, London, Book Ш, l674r 73.
7 Corresp. Math. Phys. (ed., Fuse), lr 1843, 618, Abo stated ш Novi Comm. Acad. Petrop.
8,1760-1,105; Comm. Arith. ColL, Ir 287r 296; Opera Omnia, A), П, 567, 574.
♦Algebi*, 2,1770r Ol 15t art. 243, pp. 509-16; French tranebr 2r 1774, pp. 343-51; Opei*
Omnia, (l)r I, 4S4-4. Reproduced by A. M. Legendre, Theorie des ootnbree, 1796,
407-8; ed. 3, 1830, H, 7; trand. by Maser, Ilr 9,
30 545

546
History of the Theory of Numbers.
[Силу. XXI
are relatively prime integers, so that each is a cube. Since
he stated without rigorous proof (cf. Ch, 12, Arts» 188-191) that it is the
cube of a number P+3u! of like form and that p+gV—3 is the cube of
t+uj=3. [Cf. papers 6, 10, 27-20, 72 of Ch. XX; also 30r 36 and 183
below.]
Hence p=t{P-ShP)t q=3u(P-u*). But also p/4 shall be a cube. The
same is true of the product 2p of 2t, t-\-3n7 t—3u, which are relatively
prime since p and hence I is not divisible by 3. Thus the last two are cubes,
P and 03, whence 2t=fl+g*. Thus we have two cubes }*, g3, much smaller
than x3, y2, whose sum is a cube 2t. A similar method of descent is used in
the remaining case p = Зт, when the product of the relatively prime numbers
9r/4 and 3r*+g* is a cube. As before, r = du{tz — u2). Since
is a cube and is the product of three relatively prime factors, each factor is
a cube: t-\-u=P7 t—u=^t so that/1—g* is a cube 2u.
J. A. Euler* noted that, if pl+<f+T*=Q is possible, z = p*q, y=<frt
z = T*p satisfy xjy+y/z+zJx=Q or х7г+у*х+г*у=0. In attempting to prove
the latter impossible, he stated that yx is divisible by z3 but admitted in a
note that one can only conclude that the denominator of the irreducible
fraction equal to yjz is a divisor of xy. For v=xy)z, we get x\y -hf/x+yjv=0,
i><x. Continuing, we get solutions in smaller integers.
L. Euler1* noted that p*+q*=r* implies AB(A+B) = 1 for A=pzj(qr),
B=q*j{pr)* Set A=aB. Then B*a(a+l) = lt whereas «(«+1) is not a
cube.
N. Fuss I" noted that a*=b*+c* implies that
oe-4AV = (b3-c3J. Conversely, a?-4<P=D im-
implies a2=p3 -Ьрд* (since the square root of A2—dB*
is of the form p^—dq2), whence р=т*,р+д*=cube.
J. Glenie" constructed oq a given right line
ВС as base a triangle ABC such that AB*+AC*
=BC*. Through the mid point G of ВС draw a
j£ perpendicular GH to it and take
3V5
GH^BC
GF=BC
24
Draw the circle HBC'f let it cut the parallel FA to
ВС at A. Without proof he stated that ABC is the required triangle.
To make AB*+AC*=2BC* or ZBC* (Probe. 2, 3), take
, , or GH~$BC, GF=\BC.
He treated the corresponding three problems on the difference of cubes.
' L. Eider's Open poetuma, I, 1862, 2ЭСЫ (about 1767).
" ТШ., 236-7 («boat 1769).
" №dL, 243 («bout 1778).
"The Antecedent*! Oilimlus . . . and the Constructiona of Some РкоЫетя, London, 1793,
16 pp., p. 13.

Chap. XXI] IMPOSSIBILITY OF l*+jp = &. 547
A. G. Kastner13 checked the construction by use of trigonometric
functions and logarithmic tables.
I. K Hagner» set a = BC, b=GH, c = GF. Then
Having GA*t we see that BA and AC are
Equating this to a3, and writing 4bc+a*-(a+2/)*, which gives c = (a+f)fjb,
we get
By the expression for BA7 we must have h>c, whence /<a@,29- • ■).
The value /-a/4 gives Glenie's solution. Taking /=(£-3/2)a, we see
that the expression for b2 is the square of |3^6*+5#72|а/D&5-0А;+ 6) if
A-24/23, whence b==fc5a/38. If in Euler's11 equation 2p(p*+Zq2) -z3, we
Bet 2p =« rz, we obtain $, whence
and see why the cubic equation is solved by use of a curve of order 2» For
r = 3/2, we get Glenie's case»
C, F. Hauber15 proved Glenie's construction and solved
for x, у and discussed their geometric constructibility, but made no dis-
discussion as to rationality,
J. W. Becker16 gave a construction simpler than Glenie's, as he avoided
irrationals. Take a circle of radius 7Я = 152, lay off HG = 124, №=279,
draw perpendiculars FA and ВС to IR to cut the circle at the vertices
A, Bt С of the required triangle (see above figure). For Prob. 2, take
Ш=6Щ Я<?=198, tfF-550. For Prob. 3, take IR=5, RG=1, RF=±.
In general, let the sum of the cubes of the sides equal e times the cube of
the base a. Denote the sum of the sides by as, the difference by ad. Thus
He asked if s can be chosen to make d rational, stating it to be impossible
derrdnen u. angewandten Math, {«L, Hindenburg), 1, 1795, 352-t>, 481-7.
и1Ш., 2,1798,44^457,
Ш., pp. 458-70,
»Jbid.. 471-80,

548 History of the Theory of Numbers» [Chap, xxi
if e = l. For e=2, take s—2, whence d=Q and the triangle is equilateral.
No general discussion was given»
C. F, Kausler17 gave a complex and inconclusive argument to show that
х*±у* is not a cube. His first theorem is that x—y and х*-\-ху-\-у* are not
both cubes; the proof rests on Euler's8 lemma about р^+Зд2 a cube.
С F. Gauss1* proved by descent that x*-\-y*-{-£ = Q is impossible in
integers, using an imaginary cube root of unity»
P. Barlow19 gave an erroneous proof [^Barlow15 of Ch, XXVI}
A, M. Legendre20 proved that the even one of x7 y, z is divisible by 3
and then by descent that x'+y2 — Bт3пиK is impossible, where и is not
divisible by 2 or 3»
Schopis*1 undertook a proof of the impossibility of
(x+y)z—x* = cube,
in integers. If the equation holds, then y*Q=cube, Q=2Z7 where
Solving for y, we get
V 2(**-l) "
Thus 12zz—З^гя*. The quotient of w*+3 by 12 must be an integer,
whence «?=6гс-ЬЗ, and
He stated that the second member is a cube only when n - 0 or — 1, whence
z — lj and the denominator of у would be zero.
L. Calzolari22 attempted to prove the equation impossible.
L. Kronecker28 noted that the theorem that r3-|-sa = I has no rational
solutions with rs^O is equivalent to the fact that 4a3+27b* = —1 has no
rational solutions other than a=— 1, b — ±1/3. The latter are the only
values of the coefficients of a cubic x*-\-ax-\-b=Q with rational coefficients
and discriminant unity.
G* Lam6M noted that, if x and у are relatively prime, x^H-y3 is the product
of two relatively prime factors 6, q, where 6 is D—z+y or 3D according as D
is not or is divisible by 3, and q is of the form A2+3£2, Then if a sum of
two cubes is a cube, we transpose the single even cube and get х3+^3 = Bг)8,
i7 Nova Act* Acad, Petrop., 13, ad annoe 1795-6 A802), 245-64,
w Werke, II, 1863,387-390r poethummiH MS. Quoted, Kouy. Соггеер. Mathv 4,1878,136.
11 Theory of Numbers, London, 1811,132-140.
*°Mem. Acad. Roy. Sc.de l'Inatitut de France, 6,ann6e 1823,41, §49 (*=Suppl.2 to Theorie
dee Dombres, ed. 2,1808). Theorie des nombrear ed, 3,1830, art. 653r pp. 357-60; Ger-
German tr*Dsl. by Maeer, II, 348.
n Einige Sfiize sub der tinbeetim. Analytikr Progr, Gumbinnen^ 1825. Repeated in Zeitechr»
Math. Naturw. Untemcht, 23,1892, 260-270.
"Tent&tivo per dimoetrare й teorema di Fermat . . ., Ferrajar 1855; Extract by D.
GambEoU, Periodico di Mat,, 16,1901, 155-8.
^ Jour, for Math., 66r 1859r 188; Werke, 1,121.
* Comptce Rendue Pane, 61, 1865, 92l^r 961-5. Extract in Sphitix-Oedipe, 4,1909,

Chap.XXI] Impossibility or a^-b^-z*. 549
whence д and q must be cubes» It is stated that (cf. Euler*)
q = {
In
(A+B)+(A-B) " 1SB
6 = 2A or 1SB, according as z+y js not or is divisible by 3» But a and 36
are relatively prime and not both odd» Hence Ь is a cube only if а=41р,
а-ЗЬ = р, a+3b=f; or Ъ=4№, Ъ—а = &1 b+a=fi in the respective cases.
In either case, j*+«" = Bk)* and i, j, к are smaller than x, y, z. He noted
numerical results like
P. G. Tait* noted that rf+^*» implies
and said that this leads easily to a proof of the impossibility of integral
solutions of the former equation. Every cube is a difference of two squares
of which one is divisible by 9 since
T. Pepin26 proved the impossibility of x*-\-y*=z*.
S. Gunther" showed how the square root occurring in the solution
x, у of ^-h^^a3, х-\-у=*г, сап be replaced by a cube root which is " abso-
absolutely irreducible/1
J. J, Sylvester28 gave a proof of the impossibility.
R. Perrin28 showed how one (hypothetical) set of integral solutions of
а3-ЬЬа+с'-0 leads to a new set of integral solutions»
Schuhmacher30 stated that Euler* erred in affirming that p-bgV^3
must be the cube of t+и^г, since it might be ах(*-Ь<шK, where a* = l.
He argued that the first of Euler's two cases may be dispensed with.
J. Sommer31 proved Kummer's*3 result (Ch. XXVI) that хг+у*=г*
is not solvable in integral numbers of the domain defined by a cube root
of unity.
H. Krey32 made the impossibility proof by use of the theory of quadratic
forms. Set/(x, y)=x2—xy-\-y2. Then 2/ is an improperly primitive form
of determinant —3 and of class number 1. We can represent properly by/
any positive odd number, not divisible by 3, all of whose prime factors p
have —3 as a quadratic residue. If [u, v) is a representation oi m, and
(u', vT) of m\ then
(uu'+w'—uv\ uu'+inf—vu')
* Proc. Boy. Soc. Ediiiburgh, 7,1869-70, Ш tfn full).
* Jour, de Matb,r {2)f 15, 1870, 225-6.
^Sitiun^pber. B6hm, G«s. Wi»,f Pragf 1878,112-9.
"Amer. Jour, Math,, 2r 1879r 393; CoH, Math. Papers, 3r 1909, 350.
«•Bull. Mftth. Soc, Ffciace, 13, l8S4-5r 1W-7. Reprinted, Sphmx-Ocdiper 4, l909r 187-Q.
** Zdtflchrift Math. Naturw. Uutcrricht, 25, 1894r 350.
* VoHeeungen fiber Zahlcntheorie, 1907,184-7.
« Matb. Naturwiffl. Blatter, 6^ 1909, I7&-18O.

550 History of the Theory of Numbers» [Chap.xxi
is a representation of mm'. Taking u'=vf i/ = u, we get mt=J{2uv-utt
2uv -if). First, if x+y is not divisible by 3, it is relatively prime to
f=[x+yJ—3xy, so that it and/are cubes. By the above,
When this is taken as/, the sum u*-\-\? of the arguments is a cube (corre-
(corresponding to x+y). Thus the method of descent applies» The case in
which x+y is a multiple of 3 leads by a like argument to a descent.
P. Bachmann13 amplified the proofs by Euler* and Legendre.20
R. Fueter* proved that if !*+i?*-K5=0 is solvable by numbers 40 of
an imaginary quadratic domain £(Vm), where m<0, m—2 (mod 3), then
the class number of к is divisible by 3, It is solvable in the real domain
k(^-3m) if and only if solvable in A:(Vm). In particular, Kummer's
result that it is not solvable in fr{V—3) is a consequence of the fact that it
is not solvable in rational numbers. To give a direct proof, let a?+fP=a?9
J(V^) where x, у, г are integers distinct from 0, and set a8,
-*- Then
If m and n are integers prime to 3, the domain denned by a cube root of
m3+27n* has its class number a multiple of 3, and 2?=0 is solvable.
W. Burmide35 discussed the solution of z?+y*+z*=Q in quadratic
domains.
R. D. CannichaeP8 gave a series of lemmas leading to a proof of the fact,
stated by Euler,8 that p2+3q2=&* (p, q relatively prime, s odd) implies that
s is of the form *2+3u2, etc.
Further proofs by Holden80; also Korncck,14' Stockhaus231, and Rychlik282
ofCh-XXVL
Two Equal Sums of Two Cubes»
Diophantus, V, 19, mentioned without details the theorem in the Porism*
that the difference of two cubes is always a sum of two cubes (cf. p. 607)*
P. Bungus3* remarked that while a square is often the Rim of two
squares, a cube is first composed of three cubes, citing 63=33+48+5*.
F. Vieta38 required two cubes whose sum equals the difference Вг—1У*
of two given cubes (B>D)> Call B-A the side of the first required cube
and B2A/D2-D the side of the second. Thus {B*+D*)A^WZB and hence
A)
" Niedere Zahlentheorie, 2, 1910. 454-8.
" SitztingBber Aka<L Wise. Heidelberg (Math,), 4t A, 1913, No. 25.
л Ptoc London Math. Soc., B)r 14r 1914, 1.
M Diophantine AuaJyHier 1916, 67-70.
•T Numeroram Myelcria, 1591,1618, 4вЗ? Pare Alter*, 66.
" Zetetrca, 1591, IV, 18-20; Opera Mathematics, ed. by Frana van Schooten, Lugd.
1646, 74-75. A wrong aigo ia B) te corrected on p. 554.

Снаг. xxrj Two Equal Sums of Two Cubes. 551
Using the same sides for B); sides A—D, B*A!B*-B for C), he got
B, *-*
,3, ,-,
C. G. Bachet,3' in his commentary on Diophantus IV, 2 (to solve
x—V=9, х?-у* = Ю> gave Vieta'e results (l)-C). He was able to express
the difference of two given cubes as a sum of two positive cubes only when
the greater of the given cubes exceeds the double of the smaller.
A. Girard**1 noted that, if D*>%B* in A), we first apply C) repeatedly
until we obtain two cubes the smaller of which is less than one-half the
larger, and then use A)»
Fennat40 noted that in the case B*<2D*7 expressly excluded by Bachet,
we can make B*—D* a sum of two positive cubes. Let, for example,
B — 5, 2) = 4. By Vieta's formula C), we get
Of the new cubes, the first exceeds the double of the second. Hence their
difference is a sum of two cubes by A)» Thus 53-43 is the sum of two posi-
positive cubes, " which would doubtless astonish Bachet." Further, if we
employ the three formulas in succession, and repeat the operations in-
indefinitely, we obtain an infinitude of pairs of cubes satisfying the same
conditions; for, from the two cubes whose sum equals the difference of the
given cubes, we can find by B) two new cubes whose difference equals
the sum of our two cubes and hence equals the difference of the two original
cubes; from this new difference of two cubes we pass to a sum of two cubes,
and so on indefinitely. The condition B*<2D* imposed by Bachet on C)
is not necessary; being given the cubes 8 and 1, we can find two new cubes
with the same difference, Bachet would doubtless say that this is impos-
impossible, Nevertheless I have found that41
Further, after what precedes, I solve happily the problem (not known by
Bachet): To separate the sum of two cubes into two new cubes, and indeed
in an infinitude of ways. Thus to find two cubes whose sum is 8+1,
I first seek by B) two cubes 8000/343 and 4913/343 whose difference is 8+1.
As the double of the smaller exceeds the larger, we apply C) and afterwards
A) and obtain the solution. If we wish a second solution, we apply B),
etc."
Fermat4* proposed as a new problem to Brouncker, Wallis and Frenicle:
Given a number composed of two cubes, to divide it into two other cubes.
" DiopbaDti Alex, Arith., 1621, 179-182, 324.
**• L'&rith. de Simon Stevitt . . . annotations par A- Giiard, Leidc, 1625, 635; lea Oeuvtes
Math, de Simon Stevin de Bruges par A. Giraxd, 1634, 159.
"Oetmie, I, 297-^9; French ti*nel.f Ш, 24ft-&
"By (l),8-l="D/3)'+E/3)». ThcD apply B) for £=5/3, D=4/3.
" Осотов, II, 344, 376; letters from Fennat to Digby, Aug. 15,1667; Apr. 7T1668.

552 History or the Theory of Numbers» [Chap, xxi
He would be content if Brounckcr would divide 8-|-1 into two other rational
cubes.
Without indicating his method, Frenicle43 gave the solutions
J. Wallis4* gave 22 additional solutions
u If these do not suffice, I will furnish as many as he wishes; and so easily
that in an hour I would promise a hundred , . . ." Letter XXVI contains
Frenicle's reply; he points out that all of Wallis' solutions were obtained
from the known solutions by simple multiplication or division. "You
should therefore not be astonished that he agrees so readily to furnish a
hundred such combinations in an hour; what is easier than to multiply or
divide small numbers? Indeed, it would be still easier to indicate the
divisions, not making the reductions, unless he wished to disguise more
his artificial solutions." Frenicle added that it would have been easy to
give essentially new solutions and then cited 13 such (Oeuvres de Fermat,
III, 535), Warns (p. 538, letter XXVIII) claimed that Frenicle had
been guilty of the same fault.
Wallis (p. 599, letter XLIV, June 30, 1658) was not more fortunate4*
in regard to Fermat's problem to express 9 as the sum of two positive cubes;
lie expressed 9 as the difference of the cubes of 20/7 and 17/7, and said that
the method to employ to express 9 as the sum of two cubes would be to
find in a table of cubes two whose sum is 9 times a cube! Vieta and Bachet
had found no difficulty in expressing Bl-\-D* as a difference of two cubes,
but had not attacked the more difficult problem x*-{-y*=B3+D*>
J. Prcstct4fl treated the problem to find two cubes whose sum equals the
difference of two given cubes (even when the smaller exceeds one-half the
greater), using first C) and then A). To find two cubes whose difference
is the sum B*-\-D* of two given cubes, solve B), then ^-(-i^x3—y*} and
then 43—f*=zl-{-t?. To find two cubes whose difference is Ba—ZK, solve
A) and then г3—t^ = x*+#3,
L. Euler4* noted that there exist integral solutions of
D)
Eulef1* derived Vieta's formula B) and noted that it does not give all
the solutions. For B=4, D=3, we have 37$/ = 465, 37я = 472, whereas
л Commeraum Epmtolicum de WaJlie, letter Xr Brounckcr to Wallis, Oct. 13r 1657; French
transl. in Oeuvre» de Fermat, 111, 419-420,
** Commercium, letter XVlr Wallis to Digby, Nov., Ifi57. Oetivree de Ferroat, III, 436.
* Cf. Frenicle, letter to Digby, Oeuvree de Fermat, III, 605, 609.
u Nouveaux ekmens des math., Paris, 2f 1689,260-1,
♦» Correep. Math> РЪув. (ed.f Fubs), 1, 1843, 618, Aug. 4,1753.
"Novi Comm. Acad. Petrop.,6f 1756-7, 155; Cotnm. АтШц I, 193; Op. Om,f A), II, 428»
Reproduced without reference by Б. Waring, Meditationes Algebr,, ed* 3, 1782, 325,

Chap. XXI] TwO EQUAL SuMS OF TWO CuBES. 553
there exists the simpler solution 3=6, #—5. To treat D), he set
E) A = P+q, B = p-q, C = r-$, D=r+s.
Thus
F) р^+ЗД-фЧЗг1).
Taking
p = ax+3by7 q=bx—ayt & = 3cy-dx, r=dy-\-cx,
we have
Hence our equation becomes &(<Kc-\-3by) = yCqj — dx), whence
x = — Znbp+Зпсу, у = Ttafi+ndy.
Writing X, fM^3ac±3bczFad+3bd, we get
The abbreviatons $, yt X, ^ were not used by Euler; but their introduction45
enables us to point out the identity which underlies his solution» In
it is the final factor which vanishes, and this in view of the identity
л? - Cfc- «очз(вс+w)«- (-p J+з( ^)!,
which in turn follows from
(a+6V::3)(c?-h^V^3)=arf-3bc-l-Cac-h^V^3.
Euler noted (p. 206) that we may solve similarly ur — Xp, where
* = тр*+щ2, p^mrz+ns2, while /, X are any linear functions of p, qt r, $,
by setting
by setting
p=nfx+gyt q-mfy—gx,
Then
Hence x/y is ratonal.
Euler10 treated D) by settingr without loss of generality,
A = (m-n)p+q\ B
Then (A+B)(A2-AB+B*) = (D-C)(D*+DC-\-C2) becomes, after division
by 2m(p3-^), m2+Zrt?=Zpq. Thus m = Sk, where pq^tf+Ш. But he
had proved in the same paper that every divisor of п?-\-Зк\ ш which n
and к are relatively prime, is of Uke form. Thus
p=a2+3us, $^c*+3cF, m = 3(bc±ad),
while n is acW-Zhd or its negative»
*»L, E, Dickaon, Amer. Math. Monthjyr 18, 19llr 110-111*
"NoviComm. Acad. pBtrop.r St anaces l76(Mr 1763, 105; Comm. Arith.r I, 287; Opera
Omniar (l)f II, 556»

554 Histoby of thk Theory of Numbers. [Chap, xxi
Euler51 deduced Vieta's formula B) and noted that in F) the second
factors have a common divisor of like form /*+3u2. From
G) р2+Зд2 = (/*+3^)(*2+Зи2), ?+W^{k*+Zk*){p+W)}
he concluded that
(8) p=ft+3gu, q^gl-fu, s=kt+Sku, r^ht—hu.
Inserting the values of p, s and G) into F) and deleting the common
factor *2-|-3*г2, we obtain t/u rationally. To avoid fractions, take и equal
to the denominator. Thus
For/, £, hy к arbitrary, formulae E), (8), (9) give the general solution of D).
Special cases are
71
W» Emerson62 repeated Vieta's discussion and treated the problem to
find three cubes whose sum is both a cube and a square. Cf. HillM of Ch.
XXIII.
J» P. Griisonwgave A).
S» Jones" deduced A) and B).
J. R. Young55 passed from D) to F) as had Euler. Set p — m?t s — nJ.
Then F) becomes
J, if r=^
осп
Take rn — 1, n = 2, c=*3d and drop the common denominator 4d. Hence
He also solved D) by taking55 A=m — 1, В = п?—р, С=п+р, JD^
whence 9пгг=СпУ+3(пв—1) = (g—3/ipJ, iray» Hence
np, w=i
Multiplying the resulting values of At * • •, D by 6ngT we get
F. T. Poselger57 treated the transformation of a sum or difference of
two cubes into a difference or sum of two positive cubes»
J. P. M. Binet*5 expressed EulerV8 solution of
A0) s4-y*=g»+u3 ^
11 Algebra, 2> 1770, nrte> 245, 248; ГгевсЬ traaal., 2, 1774, pp. 351r 300» Opera Omnia, (I),
I, 490-7»
" A Treatiae of Algebra, London, 1764, 1808r 3824,
u Enthiillte Zaubereyen und Geheimnisee der Arith.r Berlin» 179$, L2^-8r aod Zus&U at end
of Theil I.
и The Gentleman's DLaiy, or Math. Repository, London. No. 90» 1830» 38-9.
•• Algebra. I8l6r S. Warti'H edition, 1832P 351^2. Reproduced, Math. Mag.» 2r 1895» 164-5»
* Reproduced. Math. Mag,» 2, Ш98, 254.
w AkA*J. Wies» Berlin Math. Abhaadl., 18S2P 27-dl.
"Comptee Hendua Paris, 152, 1841» 248-50. Reprinted, Sphmx-Oedipe» 4, 1909, 29-^30.

Сна*. XXI] Two Equal Sums or Two Cubes. 555
in the explicit form
у = a'p' — p*t z = p<r' — p'2, u = ptz — p<rt
He stated that we may set/' — 1, ^'—0 without loss of generality and hence
express the general solution of A0) in the form
A1) x=k*~ I, y=—k*+mt z=km—\t u= — kl+1,
where k=a2-\-3b2, l=a—3b, m = a-\-3b> We may take a — тгс/3, £= — l/S as
new parameters in place of at b, and get
where <=А/3=а2+/32-№/3. The case «=jS=l gives 33+43+5a = 63.
*V* Bouniakowsky69 treated D).
C. Richaudw noted that in (x+iy—x* = y*-\-z?, y-\-z is of the form
3«2, whence 2x= —1, 2y=s-\-vt 2z = s—v> where
From one solution of the last equation we get the second solution
Hence from one solution a, b=d—l7 c, d of D), by replacing x, y, z by
d—ltctat and hence t, s, v by 2d— 1, c-\-a, с—о, respectively, we get
another solution:
A_a(a+c)-c-2d+l (a+c)(c+d-a-l)+d
c(a+c)«+2rfl (
+1 '
a+c-1 ' a+c-1
since i4 = i(*-»0t ^вз(*+Ю, D = W+Y). Thus the solution 3, 5, 4; 6
leads to 1, 8, 6; 9 and -8, 50, 29; 53.
H, Grassmann*1 reduced A0) to
by setting x=a-\-ct y=a—ct г — b—d, u—b-\-d, and stated that ajb must be
a square, whence a=ma2} b = m^3,
Giving artibrary integral values to л, £, m, and expressing the left member
as a product pq, we get d, с from 0d±ac=pr q.
C. Hermitc81 derived Binet's solution A1) of A0) from a general property
of cubic surfaces» Let 6> be an imaginary cube root of unity» The lines
" MemoiiH Imper. Acad. Sc>r St. Peteraburg, 6» 1865, 142 (Iq Никяап).
M Atti Accad, Pont, Nuovi LiDceir 19, 1865^6, 183-fi,
11 Archiv Matb. Phys<r 49, I869y 49; Werker 2, pt, I, 1904p 242-3. Error indicated by *A,
HurwitZp Jahreflber. d. DeutecheD Math.-Vcreioigung, 27, 1918, 55-56.
99 Nouv. Ann. Math.r B)t 11, J872r Б^; Ocuvree, IHy 115-7.

556 Hibtory or the Theory of Numbers. [Chap, xxi
x —w, y = u*2 and z = v*t у = ыг lie on the surface A0) with w — L Each of
these generators meets the line
x=az+b, y=pz+g
if
ы—Ъ q o?—b q
a ^w2—p* a w-py
whence p-bt q = (l+b+b*)/a, and the z-coordinates of the points of inter-
intersection are respectively
w — Ъ и?—Ъ
2i = ^' *-—•
The third root of (аг+Ьу+(рг+дУ=г>+1 Ы
()
Then also x and у are rational in a, hm To obtain simpler formulas, replace
a by I/a, & by bja. Then
A2) 5z=r(a+2b)-l, sy=r*-a-2b, sz = r"—a+b,
where r=a2-\-ab-\-b\ s=aa— № — L Passing to the homogeneous equation
A0) and changing b to 26, о to a—ht we get A1) with xt yy z, w replaced by
*, -Vt x, -и.
Several61 expressed 8-1-27 and 1-1-8 as sums of two new rational cubes»
G. Korneckw stated that all integral solutions are obtained by taking
positive and negative integers m, tyfm
where T = m*-\-m42-\-tA.
E. Catalan65 noted that D) is satisfied identically by
S. Realis68 proposed a problem which was solved by P. Sondat;67 if a,
% S is one set of solutions of x^-j-^H-t^+t^O, another set is
u=aA-B, v=0A-B7 x=yA+B, y
The new set yields similarly the given set, apart from a common factor.
G. Brunei*8 treated, for n an odd prime, the equation
l^i y* •" yn-\ 0
A3) "
0
« Math. Queet, Educ. Timee, I6r 1872r 95-6; 17, 1872, 84.
"Aufl6euagxl+flJ+z*="U*uiganzenZ.) Progr, Kempen, 1873»
* Nouv. Corresp, Math,r 4,1878, 352-4, 371-3. а. СаЫдп.»»
" Nout. Adq. Math.r B)y I7r 1878r 526; Nouv. Comep. Matb,, 4T 1878r 35ft
" Nouv, Ann. Math.r B)r 18, I87f)y 378.
" Мбш. Soc. Sc. РЬуя. et Nat. de Bordeaux, C) 2, 1886, 129-I4L

. XXI] Two Equal Sums of Two Cubes. 557
the determinant being #?+yJ if n=Zt and y\ if n = 2> Proceeding as had
Hermite8* and considering the intersections of A3) with the general line
in space of n dimensions
it is shown that the coordinates of any point on A3) are expressed rationally
as functions of n—1 parameters ai, - ■ -, an-i:
where aD=0, 6, = -а„_ь b»=e,'-i-«f.-i (i«2, •• •, n-1),
V. Schlegel69 treated a*+a]+al=al by setting
These become, forai+a2=x, ai^a2=2/,
p-\-Q
u-my= -—•.
The last two give и, у; the second of the four becomes
mx+v
mx—v p—q
Equate each member to r. We thus get x and v in terms of pt q, r7 m.
By x =^i2vt
For any m, we can choose p±q to make r rational; then the a< are rational.
A» Martin™ gave Viets's derivation of A) with B=r, JD= — e, and with
C- Moreau71 gave the ten numbers < 100,000 which are sums of two
positive cubes in two ways»
A. S. Werebrusow72 gave the formula
where М*+Мр+№ = №фф, <^-1 [Teilhet™].
K. Schwering73 stated that the general solution of A0) is
A4) x=mct—n2, y= — mjS+7i2, z = na—m?, u
where
A5)
"El Progreeo Mat., 4, 1894, 169-171.
>° Math. Maga*mer 2r 1895, 153^; Amer. Math. Monthly, 9, 19O2r 79,
« L^intermediaire des math,, 5, 1898, 66 [253^ 4r I807r 286].
rt/t«f.y9r 1902y 164-5; П, 1904, 9Gr 289, Math, Soc. Moecow, 25, 1905,417-437.
" Archiv Math. ГЬуя., C), 2, 1902, 2ЙО4.

558 History of the Theory of Numbers. [Chap. XXI
To get Binet's6* solution, set m-\9 n=a*+3b\ a, /3^a=F3b, By A4),
H. Kuhne74 expressed the preceding solution in terras of three inde-
independent parameters by replacing a by Зрг, p by 3qr, m by
n by dr*, whence A5) is satisfied identically. Thus
where s^p^+pq+if, satisfy A0) identically. Hot only do any p, q, г
lead to a solution a, 0, my n of A5), but conversely, by multiplying them by
a common factor, we can make n/3 a square, necessarily r2, and then
D. MirimanofP wTOte A0), with u= 1, in the form
Set y=u(x—(о)-\-г(х~и?)> z—u<*?(x—w)-\-voi(x — w*), and divide by
(х-и)(аг-ы2). We get
Вх=1+Ъ(ы*-1)ш*+г(и-1)и*г, D = 1+3A -ы)шг*+3A-<^)ы^
Hence we get all solutions (except ar=u, w?) by giving all values to u, v.
Real solutions result if and only if u-\-vt <Ju-\-w7 uu+b?v are real, i. e.,
if и and v are conjugate. Writing b, о, —а—Ь for these three sums, we
obtain Hennite's solution A2).
A. Holm78 derived B) by the tangent method. Set z=X+B, y=Y-D
and take Г=ХБ2/^, Then X-0 or ЗВ&ЦВ*-!}*). The latter gives B).
H. Kuhne77 discussed diophantine equations such that the n variables
are expressible rationally in n — 1 parameters. His7* solution of A0) is an
example of the method.
P. F, Teilhet7* remarked that the solution by Werebrusow73 is not the
general one and stated that all solutions of A0) with 4(x—v)=3(z—y) are
obtained by equating the two expressions
2
or by equating the two
у
2 )
where m, n are both even or both odd.
A. Gerardm79 derived B) from
y+D
by setting x—B-\-mh, y^h— Dt and equating to zero the constant term of
the quadratic for A. Thus т = №1&9к = ЪВ*Щ&—]У). Similarly for A).
w Arcbiv Math. Phys.r C)r 4r 1903, 180. Cf. Fujiwara.»
л Nouv, Ann. Math., D)r 3, 1903, 17-21.
n РЛк?. EdinbuiKh Matb. Soc.r 22, 1903-4r 43.
ъ Math. Naturwiee. Bl&tterr ly 1904p lft-20, 29-33, 45-^58, Cf. КйЬпе1", Ch. XXIII.
'* L'intermediaire dea raatb., llr 1904, 31.
T»Sphinx-Oediper 1906^7, 90-93y E2); J'uiterroediaure dee math.» 16, 1909, 85.

Chap, XXI] TWO EQUAL SUMS OF Two CUBES. 559
H. Holden" obtained all integral solutions a, b, c, d of
for such values of p that any factor of a or c, not of the form l*-\-pm*, is
a factor of both. This is true if there is a single properly primitive class
of quadratic forms of determinant — p and if, when there are improperly
primitive classes, the highest power of 2 which divides 12-\--ртгР has an even
exponent. The conditions hold for p = l, d=2, 3, -5, -13, -29, -53,
—61. For p=3, we have the equivalent equation
and hence the complete solution of A0). He proved that there is no integral
solution of the initial equation with a = b and hence none of я?=у*-\-г*щ
J. Jandasek81 gave the identity
K, Petr*2 noted that EuIerV* solution of x*+y*+2?=u* may be written
in the form
x:y:u:—z
~A*E+2BC-BD:-A*E+BC+BD:B*E+2AC-AD:-B*E+AC+AD,
where C, D are arbitrary and ABEP^CP-CD+D2. It is thus not essen-
essentially different from Binet's solution.
Binet's68 solution is claimed*3 to be not general.
R, Norrie** treated D) by taking A=rxi-\-\t B=ra2+P, C=rx3—y,
D=rzo+\. Thus ora+3j8r2+377'=0, where «=^—xf-^-^
We may make 7=0 by choice of x0. Then err3+З/Зг2=0 for r= —Zpjot*
The resulting values of At Bf C, D in terms of x,, x&, xs, X; ^ are of high
degree and much more complicated than the complete solution by Euler*
and Biuet.w
M. Fujiwara85 showed that the formulas by Schwering™ and Kuhne74
can be deduced by simple substitutions from formula A1) of Euler and
Binet.
A. Ge'rardin80 gave the identities
and one similar to the latter»
« Меакпдег Math., 36r 1906-7,189-192.
* C^HopiB, Ptay, 39, 1910, 94-5»
* Ibid., 40^ 1911, 99-102» In tbe Fortechritte report the sign before AD in я is wrong,
«yintennediairedeemeth., 18, 1911, 2в5-6; 19Г I9l2r 116.
u XJnivendty of St. Andrews 500th Annivei«aryr Mem. Vol., Edinbui^h, I911r 5O-I.
■ Tohoku Math. Jour., lr 19Ur 77-8; Archiv Math- Phye.p C), 19r 19I2y 369.
•• L'mterm6diaire dee math., 19, 19127 7. Cf, pp, 116-S for reference». He gave the first
in Аанос. fmn^. av, »c.r 40, 1911, 12.

560 Hi&tory of the Theory of Numbers. [Cbap, xxi
G. Osborn87 gave Young's" identity and
J. W. Nicholson,88 using one solution of тг —n*-\--p*-\-r*, found that
holds if x :у—т^Ь—п^Ь—р^а-\-т2а wnW—rbtf—ptf—ra1.
J. E. A. Steggall89, to solve x* — u* = y3 — f*, took x—u=pt x-\-u=qt
y-v=&, y+v=r* Then F) implies p*+Zq*=fiS, з^-ЬЗг-^Рг whence
A:2=3^ we get p-H= {в-+рC«-Л)]/C0, etc. Hence
where L^PH-3^, is the most general rational solution.
R. D. Carmichael90 obtained a rational solution, involving four para-
parameters, of
^+y3+23-&n/7=wa+r3+w3—Zuvw,
by employing the factor х-\-у-\-г of the left member. Taking z — w=0,
he deduced formulse A1) of Euler and Binet, which he proved to give the
general solution.
T. Hayashi&l noted that C. Shiraishi published in his book of 1826 the
solutions*1* (attributed to Gokai Ampon) of xz-\-yi-\-zi — ui:
u = y+l, z=3a\ x=6a*±3a+l, у = 9а3+ва2+3а or 9a*
Replacing a by «/0 and passing to the homogeneous form, we get
and in like manner
я^ба^-З^-ЬД2, « = 9а3-
Further, S. Baba, Mathematics^ vol. 2, 1830; gave the solution
of A0); S. Kaneko, Mathematics, vob 2, 1845, gave the first solution of
Frenicle," Kawakita, in Algebraic Solutions, vol, 2, compiled from a
b. GA£etter 7, 1913-4,361.
м Amer. Math. Monthly, 22, 1915, 224-5.
" ftoc. EdinbuiKh Math. Soc.p 34, I9l5^p 11-17.
» Diophaatiae AmJyste, New Yorkr 191.% 63-G5.
m ТЛЬоки M*th. Jour*, 10, 19l6r 15-27 (in Japanese).
** For a briefer account, see D. £. Smith *d<1 V. Mikamij A History of Japanese Mathe-
Mathematics, Chicago, 1914,233-5.

Chap. XXI] THREE EQUAL SUMS OF TWO CUBES. 561
manuscript by Baba, solved A0) by setting
x=a+b, у=а—Ъ, z=hcy u = d—hct 2a3+6ab2—^+36^-3^^=0.
Take a = &&\% Then 12ta-dD—c'). Take ca=«, a?—4=0, and multiply
the resulting values of xt у, z, и by 12c/d; we get
M. Weill92 noted that if x^ y^ z^ Ui give two solutions of A0), we can
evidently find 5 rationally so that х\-\-Ьх<^ ---, щ+Ьи*. is a solution»
Given only one solution, we obtain a new solution
if AP+Wt+3C=Q, where
We may choose X, ■ *-, p to make C=0 or Л = 0 and get t rationally.
For three consecutive cubes whose sum is a cube, see papers 245-267,
For minor results on our subject, see Sehier67 of Ch. XXIII.
Three equal bums op two cubes.
Fermat's*0 method of solution was given above.
W. Lenhart93 found four integers the sum of any two of which is a cube.
Three of the conditions are satisfied if я, т*—х, n*—x, r*—x be taken as
the numbers. The remaining conditions require that т*-\-п?—2х, m2-\-r*
~-2xt n*-\-r*~-2x be cubes, say ss, а3, &3. Eliminating xt we have
A) r3_1_^=a3_1_7l3=^+m3^
By his1" table of numbers expressible as a sum of two cubes,
Rejecting the common denominator, we get integers (one of 24 digits and
three of 22 digits) solving the initial problem.
A. B. Evans*4 obtained the last result otherwise» By Euler;" for
Now 13-3613 — 41s—283 can be expressed as a sum of two cubes by the
usual method» The final answer involves numbers of 22 aod 24 digits.
J. MattesonM obtained Lenhart's result by the method of Evans»
H. BrocardM noted that the sum of any two of the numbers 20012£,
-15916±, 19291£, -20020J is a cube. E. B. Escott" noted that 6044,
7780, -1948, -6052 have this property.
E. Fauquembergue98 gave an erroneous solution of A) with 5 parameters.
и Nouv. Ann. Math.r {4)r I7r 1917, 41-46.
я Math» Misoellanyy New York, I, 1836, 155-6.
H Math. Quest. Educ. Timeer 15» l87ly9l-2. НЫ factor 2» should be 2*.
» Collection Dioph. РгоЫетнг pub. by A. Martinr Waahingtoa, D. С, 18Й^ 1^.
14 L^uterm&liaire d«a math., S, 1901, 183-4.
*ЧШ., 9,1902^16.
"/Wrf.,9, 1902r 155; 10г 1903г82 (Sphinx-Oedipe, 1906-7, 80r 125),
37

562 History of the Theory of Numbers. [Chap. XXI
A. S. Werebrusow*9 gave the solution72
in which M*+MN+N2=3^$9 «J=b Helw noted that
B)
holds for
xy-xvyi
and the values derived from the latter by interchanging xly 9,. He101
used this result to get the general solution of B).
Fauquembergue103 remarked that the last formula follows from the
identity
due to A, Desboves103, by taking x*-\-y*=xj-h#i and dividing the result by
the product of (ху—$\У\)* by x*—x\ = y\ — #*.
A. Gerardin104 stated that the least solution of B) in integers >1 is
probably x=5GQ, y=-70} ^=552, yi = 198, ar2 = 525, y2 = 315.
Fauquembergue1Oi noted that if Cauchy's287 formulas are applied to
^19^, which has the solution x = 37 y= —2r 2=1, we get
(i)+(*)m+m(M)+(ftr(**if5)+(iW^),
that 19 -363510a is a sum of two positive integral cubes in various ways
Solution of (
IL Amsler1M noted the solution x = un^\tz=vtli y^un-\-u^h t=vn-\-vn+lj
where un and vn are the nth coefficients of the developments of
A. Gerardin107 noted the identities
G6rardin10S gave several solutions, as
»L>bterm6diaiP& dee m»th.r 9r l902r 164; 11, 1904r 288; Matem. Sbora. (Math. Soc.
Moscow), 25, 190бг 417-37.
icq L'mteimedmire dee math.r 12,1905, 268; 25,1918, 139, for numerical examples in which
2i and fa are integers
1<n Matem. Shorn. (Math. Soc, Moscow), 27r 19O9y 146-169.
iM LTintermediaire dea дааНц 14r l907r 6&.
l« Nouv. Ana. Math., B)r I8y 1879r 407, Special cade of Desbovea.'"
lM L'interm6jliaire des math,, 15, l9G8y 182; Sphinx-Oediper 1906-7, 80,128.
i<*Sphinx-Oedip<\ 1906-7y 126.
i" Nouv, Ann. Math.y D), 7, 1907, 336. Proof by L* Chan»yr D}y I6r I9l6r 282-5; a*me in
Sphinx-Oedipey 9y J.914, 93-4.
1OT SphinxOcdipe, 1910,179.
"ЧШ.> 9r 1914P 143-4^ Nouv. Ann. Math.r D)r 16, 1916, 285^7, where Yt Z should be
iDterchanged.

Chap, XXl^ RELATIONS BETWEEN FlVE OR MORE CUBES. 563
Relations between five or more cubes.
To divide a given cube fc3 into n (n>2) positive cubes, J, Whitleylw
took at k—vt як*1а*~-а, dv, ev, * — as the roots of the required cubes. Then
1)'
S. Ryley took a, v—a, k—a*vjk*r dv, evt * * - as the roots; then
F. Elefanti110 noted that
and that 28s is a sum of 9 cubes, also of 11 cubes; etc. For the second
relation see Bouniakowskyw of Ch. VIIL
Y. Hirano111 noted that
A. Martin1" noted that the sum of the cubes of rmy q-тт, sm,
*, Pn-xQ will equal the cube of sm+qr^fs? by choice of mjq. Also,
S. R6alis113 noted that z\-\ \-z]=z* if
This is not the general solution since 2z,=0.
Д Catalan1» noted that ^-6(x-IL(^-2K+2 gives
Taking x = 7/4 or s = l+6(a/6K, we get a solution of
in positive integers. If we multiply each term by 27(£e—ж*+1Kяг9, combine
the third and fourth terms and replace x3 by x, we get
D, S, Hart115 found cubes whose sum is a cube by taking 13-|- ■ ♦ ■ H-n3 — S
and seeking by trial to make S^ (s+?«K-|-sJ a sum of cubes»
S. Tebay116 noted that, if x=a«i> y=aa2) z = aa3t 2u3=n,
A) x8+^3+^ = 2us
becomes a=nSaf, First, solve aJ+a^nH+s3 by setting
diee» Diary, 1S32, 41-2r Que»t, 1506,
Qr. Jour. Math,r 4r I861y 339.
111 Easy Solution of Math. Problems, 1863. Cf. НауаяЫ, T6hoku Math. Jour.r 10г 191бг 18»
"«Math. Queat. Шад. Timeer21r 1374,1<H,
111 Nouv. Соггеяр. Math., 4r 1878r 350-2.
>Ш>, 352-4, 371-3.
ll*Math, Qaeet, Educ, Times, 23r 1375, 82-3; Math. Magazine, lr 1882^r
ll* Math. Queet. Educ. Times, 38r 1883, 101^3.

564 Histohy of the Theory of Numbers» [Chap, xxi
whence 8* = ЪигР. Take * = 3и3, г = 4и*тп3, whence $=6umn2< Hence solu-
solutions are fli, a2—4u3ffi3±3n3. Next, for a3—p—8, our initial equation
becomes
71 71 71 \ 717*^ у f\ft 'w^ о
Special sets of five cubes whose sum is a cube have been noted»117
A» Martin118 noted that the sum of the cubes of p-\-q, P~Q> r^V> s &
the cube of r-\-p if p = i$*l(T*—q2); that of a+b—c, a-\-c—b, &+c—а, у is
the cube of a-\-b-\-c if ^=24аЬс, whence take а —Зр3, b—З53, c—r3 or take
y = 2a, c=a2l{Sb)'} the sum of the cubes of pa+ntf qa—ntt ra—nt} nt is of
the form stf+Rand is a cube if s^^H-^-J-r3 is a cube and if R=Q, which
determines t. Next, he gave Whitley's109 result.
Finally, given that p\-\- • * ■ -\-pl is a cube, to find n-|-l cubes whose sum
is a cube. If n is odd, take xt pi—xy pi—x,
as the roots of the desired cubes, where
If n is even, take x, PiH-x, pj—a?, Pa+a?, pi^x, * * *, prt-i-fj, prt^x as the
roots, and {t-\-xy as the sum of their cubes, where
Martin.11* found cubes whose sum is a cube &3 by selecting № between
n* and jS = 1*+ • * '+Я1 and seeking by trial to express S—&3 as a sum of
distinct cubes ^ n3* Also by seeking to express p3 — g3 as a sum of distinct
cubes Фд3. He tabulated the values of S for «^342.
R. W. D. Christie™ gave 14 cases like 4J = l-|-l-|-23-|-31+33 of a cube
equal to a sum of five cubes»
Ed. Collignonm noted that there is no positive integral solution of
x9+(z-l)p-\ \-(z-ky = (x-\-l)*-\- Hz+fr)* (p = 3or4).
A. Gerardin112 gave numerical examples of equal sums of three cubes»
A. S. Werebrusow123 noted that A) holds if
x—u-\-v, y—u—v, и = аЪп*, v=bn*, 2— —6mn2, ab^C.
From two sets of solutions a third set is derived.
A. Gerardm124 gave, besides two more complicated identities of like type,
Fa£K+(9*4i^ ~ «Й3+(9«2 - ^
G6rardinm discussed а?-\-Ъ*-\-}1с* = (а+Ьу+}и1*> For a^pm, c—d-\-m,
117 Amer. Math. MoDthiy, 2, 1895, 329-331.
™ Math. Magazine, % I895r 156-l§0»
«• Ibid.t 185-190» Two examples, Martin" of Ch. XXIII.
m Math. Quest. Educ, Time», B), 4, 1903, 7l.
»i SphiDx-Oedipe, 1906-7, 29^3
** 1Ш 12(M
l» Math» Soc. Moscow. 26,1908, 622-4.
*« Aaaoc* fmn?., 28,1909,143-5.
1*ЗЫОФе, 5r 1910,178.

Свар, XXI] RELATIONS BETWEEN FlVE OR MORE CUBES. 565
it becomes
To make the constant term zero, set h = b'7 p=d2; then, for b
By annulling the coefficient of my he obtained
Again,
E> Barbette12* employed the first method of Martin119 to show tbat
б3+83 = 93
Ч 10я = 12
are the <;n1y sets of distinct cubes ^lO3 whose sum is a cube»
R. Norrie8' would find n cubes whose sum is a cube by taking
according as n is even or odd»
A. G6rardin127 noted that the sum of the cubes of x- l,a,s+l, 2f-lt2f,
2/+t is of the form Zt(P-2q) if t = x+2ft q=3fx-l.
K. D. Carmichael128 noted that A) has the special solution
and obtained a set of solutions of а^-\-]^-\-^-\-иг=дР involving five param-
parameters. A special solution of x*+2y3+3zi=t? is x)t = 2n*zFmz, y = m*,
The double of a cube may be a sum of four cubes»129
A. G6rardinlM derived a solution of а^+^+г^/иМгот agiveo solution,
and deduced a solution of
M. Weill131 derived a third solution x=Xi+b(x2—x1), * - from two given
solutions of xP—yb+zt+P+u-'; likewise for ax*-\-by3-\-cz3-\-dl* = Q<
E. Fauquembergue132 treated a^+^+z3-^3 by setting x^ 2a, y = 4h+1,
z=ic-l, 2b-2c+l=fy b+c^g. Then 2а*+ЗРд-\-4д* = и\ which is satis-
i* Lee sommea de р-16тев pu«csancea dietinctcs 6galea h une p-i^me puissance» Liege, 1910,
105-132.
 L/intennediaire dee matb.t 19r 1912r Ш.
u»Amej. Math. Monthly, 20, 1ЩЗ, 304-6.
^L'intennediairedesmath.,21,1914,144, 188-190; 22yl915,60»
»»;w<ir22, Ш5У 130-2 (error for h=2); 23, 1916» 107-110»
л Nouv. Ann. Math., D), I7r 1917, 46r 51-53»
1Л L'ratermddiaire dee math.r 34r 1917,. 40.

566 History of the Theory of Numbers. [Сидр, XXI
fied if a = 6,/ = 1, 0=9, u=15t giving 12a-j-17a-|-193=4-153. This contra-
contradicts the statement by E. Turnere133 that a^+^+^^ni3 is impossible if
n=4 or 5 (mod 9).
A. S. Werebrusow1" gave two equal sums of four cubes.
Sum of three cubes made a square.
V. Bouniakowsky1M used fx(x-\-b)dx to get the identity
Set 2x-b = (x+b)\1, 2х-\-ЗЪ=ц\ Then
Multiply by (8-A1K. Thus
(ЗХK+B-Х
E. Catalan,m by use of the toroid, obtained the identity
which gives an infinitude of, but not all, solutions of
E. Lucas13' deduced from formulas of Cauchy28* the generalization
of Catalan's13* identity-,
A. Desboves1*8 gave a new proof of the last identity»
A» S. Werebrusow1** derived from one solution а, Ъ> ct d the second solution
(а-\-сехУ-\-(Ъ-<ххУ-\-(с-\-хУ = (d+ fa)*,
2db=Z{a?-h*)ct+Zc\ лг = ^-3
We may start from the solution (n?y= (и3)*»
A» G£rardinliW gave the identities
where t=4a*b-\-4b*(<?+(F).
tabulated solutions of
Binary Cubic Form made a Cube.
Fermat solved Ax*+Bx2+Cz+D =& MD^d* by setting
+/(),
or if A =a8 by setting z = ax-\-B/Ca?)f while if both D=d? and A = as there
i» L'eneeignement matb.r 18r 1910r 421.
1W Uintermediaire dee math.r 25r 19l8r 75-6.
M Bull. Ac. Sc. St. P6tetBbourgr PhyB. Math,, llr I853r 72.
^Bull. Acad. Roy, de Belgique, B)r 22r I866r 29; Mflangee Matbr I868r 58; Nouv»
Corresp. Math.^ lr 1874-5, I53y foot-note,
i" Bull. B[bl. Storia Sc. Mat. Fi».r 10r 1877,176.
»* Nouv. Ann. Math.f B), 18r lS79r 409.
"• L'inlerm&toure dee math., 157 l908r 136-7.
i" SphuuuOedipe, 8, 1913, 29.
"l L'tntennediaire dee math,, 23P 1916, 9-10,
w J. de Billy's Inventum novumr Шг И 27-30r Ocuvree dc Fermat, III, 386-8.

Chap. XXI] Binary Cubic Form Made a Cube. 5G7
are three ways of solving» Thus, for xi-\-2x2-\-4x-\-l =2?, z=x+l gives
x = l, 3=3+2/3 gives x^ -19/72, z^l-\~ix gives x= -90/37, and each of
these primitive solutions furnishes new solutions as above. Cases when
the preceding methods fail are noted in § 30; there is no rational solution
хфО of l+3x+3x2+4^ = ^ or of х*-дх*±дх±1^г?; for
x*+2x*+Zx+l=z3t
z = l+x gives x=Q, while z=x+2/3 gives the only primitive solution [von
Schaewen150 noted the additional primitive solutions x- —l,x= —1/2]»
L. Euler,143 after reproducing (§§ 147-151) essentially Format's methods,
treated the new case in which a particular solution x = ht z=kt is known.
Taking x = h-\-yt we get a cubic whose constant term is a cube» Since
4+z2=23 for x^2 or 3 = 11, we may apply the last method, or set
x=B+2y)i(l-y) &nd get (S-\-Syi)(l-y) = u>a от set x = {2+lly)/(l±y).
L. Euler144 proved that py*±p*x*=z* is impossible if p is a prime» For,
z=pA, whenceр*А*^рх* = ^Л Then y = pB, whence р*1Р = рА*^х*. Then
x=pC7 etc., and x, y, z are divisible by an indefinitely large power of p.
W, L. Krafft145 would make хг+пу* the cube of р^+п^+пУ - Znpqr by
setting
which determines x, yy subject to the condition p2r+pq2+nqr2 = 0f whence
To make the radical rational, set q = s2, 8*-4пг* = Р, whence take $*-\-t~2P,
z*—t=-2ng*. Then $*^Р+пд*у which is like the initial equation, but m
smaller numbers»
P. Paoli146 treated а+Ыхг = у* by setting у = Ъх+т, solving the quadratic
in x and making the radical rational Thus l2am-ZmA is to be a square,
which he accomplished by trying values of m< Ща. A like method was
stated to apply to а+&х+Л3 = уэ»
D. M. Sensenig147 treated without novelty axs-\-bx2-\-cx-\-d^y3t when
a or d is a cube»
A. Desboves14* stated that if T=cZ* and F=cZ\ where T and F are
binary forms of the third and fourth degrees in X and Y, are such that
T=0 and F—Q are solvable in integers, one can determine a solution
(X, Y, Z) of one of the equations knowing a solution fx, ?/, г) of an equation
of the same degree by formulas giving X, Y, Z as cubic functions of xt y, zt
in case of T=cZ*9 and, in case of F=cZ^, by functions of degree four in x, у
and of degree eight in z.
и* Algebra, St. Petersburg, 2, 1770, Ch> 10r §§ 147-161; French transL.Lyon^. 1774. pp. 177-
195; Opera Omniap A), Ir 406^14.
lU Opera postuma^ lr l862r 217 (about 1775).
l7Wd2
^ОршсоЬ analytic», Libumi, l7803 128-130.
ы The Analyst, Dcs Moinea, 3, 1876, 104.
"• Comptcs Rcodns Paria, 90, 1880, 1069. Cf. Desboves'" of Ch, XXII,

568 History or the Theory or Numbers, [Chap, xxi
E. Landau, A. Boutin, P. Tannery, and A. S. Werebrusow14* considered
яЧ&с^+й^+Зг/3^! or 2*.
P. von Schacwen16* treated Ax*+Bxhj-\-Cxy*+Dy*~z*. If A =a\B=0,
we have
)( )
which is satisfied if m(z - ax) = ny, n(z2-{ )=m(Cx+Dy)y> Eliminating
z, we get
^ \, E =
We can always make E a square. Next, if A—a*, БфО, we replace
ax+By/Ca?) by xi and у by Scfyi and are led to the first case» Finally,
if neither A nor D is a cube, but x = p, y=q, z^r is a known solution,
set qz=py+s to obtain a cubic in which the coefficient of y* is r3. For
Fennat's example, xi-\-2x2y+Zxy*+y3=z*1 set X-x+y7 x=Y. Then
Many solutions are found: B, y, *) = (!, —I, 1), C, —7, —1), A, —2, 1),
F, —13, 5), etc», whereas Fermat's method gave the primitive solution
J. von Sz. Nagylbl noted that a principle of Foincare's15 of Ch. XXIII
enables us to transform the cubic curve f=a*x?+pxy2-\-qy*—z1=Q without
double points, treated by von Schaewen, by the birational transformation
into the quartic curve iptmA-\-\2a?qm*n—Qapm*n2—3aV—r3m3 = 0y and con-
conversely the last into/^0 by
m=y\ n^y{z—Qx), ±r=3a(z—axy-\-ba\z—ax)x—py\
To pass to the non-homogeneous form, use x/yt zjy, п/т, r/m>
E. Haentzschel,162 starting from a given solution x^h7 y=k, of
derived a second solution by applying the substitution
x = (ht — atA2—2azh—
giving
у
where С г and G3 are the quadratic and cubic со variants of /(x), and choosing
t во that ZCi(h)l-\-C*{h) =0. We may begin with the identity
where D is the discriminant of/, set v= — Ca//, tf2=4s3+D; then
dee math., 8,1601, 147,309; 9, 1902, 111, 283; lOr 1903, 10S; 13, l006r
196-7»
"* Jahreflbericht d. Deutechen Math.-Vereinigungr I8r 1909r 7-14.
»" Ibid., 40Ь2.
wibid., 22, 1913,319-29.

Снаг. XXJ] Binary Cubic Form Made a Squabe. 569
Given a pair of values vt s satisfying yz=4s3-j-D/ we can find new pairs by-
use of the addition theorem for the elliptic function p(u). Only such a
value v is useful for which the cubic equation163 v= — C3// has a rational
root x. The simplest case D = \3 is treated at length and illustrated for
L. Hobser treated1* (z+y)(z*+y2)=4Cz\ J. de Billy166 (p. 41) treated
Candido178 of Ch. XXIII made the product of a linear and a quadratic
factor a cube»
Binary Cubic Form made a Squahe.
J. de BillylW treated many problems/= Q, where/ is a cubic or quartie
in one or more variables with numerical coefficients.
Fermat15* treated 20z3+5x*+40x-\-16 = zt. For z = 4+5x, x=l. To
deduce a second solution, set z^l+y> Then
, -112
for ,-gj-.
From the latter, we get a third solution.
L» Euler^made/^—^+bz+c^+iis3^ П by setting F = (f-\-pxJ, where
2fp = bt whence x = (pi^c)ldt or by setting F={f-\-px-\-qxvJ and choosing
p and q to make the terms in x and x* cancel, whence
But it often happens that neither of these two methods leads to a value Ф =b/
of jr} as for example for^+dz3, and then we resort to triah For 3+£' = Q,
set x = l+y to obtain 4-|- l-j/3. But for 1+x3, x = 2+y gives 9+12y
-\-$уг-\-у* and neither of the two methods leads to a value of x other than
0, 2, — 1; in fact, Ц-а^ = П only when z-0, 2, -1.
Еи1егж of Ch» XXII applied to cubics his method to make a quartie
a square.
W. L. Krafft,™ given ma?+n=b*t made mx*-\-n=z2 by setting x=a-\-yt
z — b+3ma?ylBb)=zi or z=zi+py2 and in the latter case requiring that the
terms уг shall cancel. A* J. Lexell treated the свзеп=к2 by setting x = ay7
whence (b2^№)y3=z2—kif and taking (b±k)y*^z±h7 (b^Fk)y =z^k>
L. Euler169 noted that l+z-z*=n forz = ll/9.
Krafft160 made x^+ny3 a square for relatively prime integers xy y, by
setting
x+y<*y*<n = {p+c?ql&+^r4n*y G=0, 17 2; c? = \)>
«Treated by Haent2achdr Sitcungeber. Berlin Math. Gesell., Юг 19ШГ 20.
ш MoDatehrfte Math. Phye,r 26r T915r 289»
M Diophanti Redivivi, Lvgdvni, l670r Pare Posterior,
ш J. de Billyre Invent иш novtim . * , t Оеиттея de Fermatj III, 385.
™ Algebra, St. Peterobwg, 2r l770r Cb. Bt §S XJ2-127; French trane1,r Lyon, 2,1774, pp. 135-
152; Opera Omnia, A), lp 19llr 388-396. Reproduced, 8phinx^Oediper 1908^, 49-57»
J Opera poetuma, 1,1862, 211-2 (about 1770).

570 History of the Theory of Numbers. [Chap, xxi
Thus x = p*-\-2nqrt y = 2pq+nr*, 0 = 2pr+qiJ which holds ii p = 2a\ r= —Ъ*,
q=2ah. The product of the three factors is the square of
—Znpqr.
J. L. Lagrange161 proved that r3—As?=q2 for
A) r = 4t(P-Au*), s=-u(SP+Au*), q=
He took a cube root a of unity and set
Thus
Т=12+2Аих, U = Ax*-\-2tu, X =
Then the factor r — asVA. of the given cubic function will be of the form
p2 if r = T, s = - U, X - 0. Substituting the value x = - u2[Bt) from X = 0
into the first two conditions, we get
„ Аи* Аи*
In the product P=L3-t-Au3 — ЪАЫх+А^-х* of the expressions p in which a
takes its three values, we insert the above value of x and obtain q. To
avoid fractions multiply r and s by 4^r and q by $&>
Euler163 noted that this product P may be made equal to any power»
Lagrange163 extended the method from аэ = 1 to a*-aa*+b<x—c — 0, with
the roots ai, az, ая. Then
F(xt yt г)= П (z+aM+aU) =x*-\-ax*y+(a*-2b)x4-\-bxy*+(ab-3c)xyz
is such that its product by F(xl} yu Zi) is F(X} Y, Z), where
In particular, the square of F(x, yi z) is F(X, Yt Z), where
X=x2+2cyz-\-acz*, Y = 2xy-2hyz^r{c-ah)z1t
Z = 2xz+y*+2ayz+(a? -b)z\
We may make Z = 0 by choice of x rational in y, z. Hence
X*+aX*Y+bXYs+cY*= V2
lias solutions involving the parameters yy z, with V = F{xy y, z). The same
method leads to solutions of F(X7 Y7 Z) = Vm>
A» M. Legendre164 made Z = 0 by taking y = (u—a)z, 2x=(h—u-)z.
Then replacing и by u/vt we see that X, Yt V are proportional to
V = и*-2а
ш M6m. Acad. R. Sc. BeHin, 23, annte 1767, \769; Oeuvrcs, II, 532.
lM Opera poatuma, 1, I862y 571-3; letter to Lagranger Jan., 1770, Oeuvree, XIV, 216»
■« Addition IX to Enler'e Algebra, 2t 1774, 644-9 [misprint of sign in X, § 92]. Oeuvres de
Lagrange, VIIr 170-fl. Euler's Oper& Omnia, A), I, 643-50.
l" Theorie des nombrce, cd, 3r IIP1830, § 4E5, p. 139. German transl by Maserr 2r IS93r 133.

Chap. XXI] BlNABY Cubic FoHM MaDE A SQUARE- 571
A. Desboves166 gave for a = b = 0 this result with v replaced by v/2.
He1M reduced ax3+Ъу* — cz* to Lagrange's1*1 case by multiplication by a?(?.
H. Brocard1" noted that x*-\-Ba+l)(x—1) = уг has the special solution
R. F. Davis1*8 made Sx3—8cr+16 the square of px2+x—4, obtaining a
quadratic for s with rational roots if 8p*—8p-|-16= П. Hence solutions
like p=*0, ±1, 2 lead to new solutions x>
G. de Bocquigny1** proposed for solution x*—x±l=y2. H. Brocard170
noted that for the upper sign it has solutions x = 0t 1, 3, 5. E. B. Escoti171
noted that for the lower sign it is impossible as shown by use of modulus 3.
JL C. Walker172 reproduced Lagrange'siw work, applying it to a^+a/ = z2.
The least positive integral solution173 of a^ — 662/3= □ has x—25.
L. Aubry17' found restrictions on possible solutions of z*-J-x*-j-2:c-j-l = П.
A. Gerardin175 assumed that x0) yOt z^ is a known solution of
and took x=Xo+mf, у=Уъ+тд, z=za-\-mht There results a quadratic
equation Am*-\-Bm+C=0> He took in turn
L. J» Mordell176 wrote the proposed cubic in the form
B) д2=№-д№2~№\
which is the syzygy connecting the seminvariants a,
h = b1-act д*=ае-Ш+Зс*г дг = асе-\-2Ьс<1-а<Р-Ъ2е-с?,
and g=a*d—V+3bh of the quartic
Given integral solutions of B) in which a is odd and prime to h, we can
find integers a, • • ■, e such that / has the invariants Qi and g$, and b is
prime to a. Conversely, every such quartic yields a solution of B) with
a odd and prime to k. Hence to find all solutions (with у odd and prime
to x) of
C) *-4*-g&f-{itf,
take a representative / of each class of binary quartics with the invariants
Яь 0*; aPply to / a suitable linear substitution (JJ) of determinant unity
to obtain a quartic /' having a' odd and prime to br; then x=h\ y = a',
* Comptee Rendus Paris, 87, I878r 161.
ш Nouv. Ann. Math.^ B)r 18, 1879, 398.
^ Nonv, Correep- Matb.p 3r 1377r 23-24.
ia*Proc. Edinb. Math, Soc<s 13P 1804-5, 179-80.
i" L'intcrm6diaire dea matb.r 9r 1902r 203.
*ibrfO19033
,rr
t» Ibid., 132.
lrt Amer. Matb- Monthly, Юг 1903, 49-50,
** Math. Quest, Educ, Timce, B)p 14, 19O8P 29.
lrt L'intermddiaire dee math., I8r l0llr 276-7.
^Sphinr-Oedipe, 8r 1913P I6l.
& Qoar. Jour. Math.r 45r 19134, 170-186.

572 History of the Theory of Numbers» [Chap. XXI
viz., y=f(Pt Ф, x = U{pi q)t II being the Hessian of/. Thus the complete
solution of C), in relatively prime integers x, yy is given by a finite number
of pairs of quartic forms in two parameters p, q. In particular, five such
pairs of quartics give all solution* of z2 =хг-\-уг in which у is odd and prime
to x.
JL F. Davis177 noted that if x = p is a solution of ах*+Ъх+с2 = □ , two
further solutions are the rational roots of (apx —bJ-*4ac*(x-f-p).
E. Fauquemberguem proved that x2 = (y-\-l)(y2-\-4) has do integral
solutions except (x, y)~B, 0) and A0, 4), since p*$!—l^p4—q* implies
p=q=l.
A. Gerardin179 propped that special cubies be made squares. He and
L. Aubry1** gave a partial solution for 2х3+я2+1 — П.
E. Haentzschel180* made use of Weierstrass1 tP-function to study
where h* and Л>аге rational or conjugate complex numbers. As an ex-
example he treated EulerV*7 problem a^+l = П»
For х?-\-х*+х+1 = П see pp. 54-58 of Vol. I of this History.
For f= □, where / is a certain cubic, see papers 154-6 of Ch. V, 82 of
Ch. XV, and 163 of Ch. XXII.
Numbers the bum or two rational cubes: x*+y3 = Azi.
Fermat40 indicated a process to get an infinitude of solutions from one.
J. Prestet1Si employed Fermat's process to get the solution
X=xBy*+x>), F=-y(ac3+y3), Z=z(x*-y*).
J» L. Lagrange1*1 reduced the problem, by means of his theory of poly-
polynomials which repeat under multiplication, to the solution of А
Setting u= ft, v=fgty and dividing by /*дР9 we get
Set l = llf-Vg. Then k(h?-P)=±A. Set l = kk. Then 4Л/A-^) is h\
so that 2Л2A—^2) is the cube of 2Ajh. But he did not complete the dis-
discussion.
L. Euler1*2 proved that y=xii A = 2.
L. Euier183 proved the impossibility of x*-\-y* = A£ and that the problem
is equivalent to the impossibility of 1-|-2жа= □ in rational numbers, хФО.
To discussxz-\-y* — n&tsetх—а-\-Ъу y = a—btz = 2v. Then a(a?+3№) =4ntP.
** Math. Qaest. Edtic. Tim^e, B), 24, I9l3r 67-8.
i;i L7interiD6duu№ dee math., 21, I9l4r 81-3.
*»lbid.,22t 1915, 104» 12&
JW Ibid., 23, 1916. 132-3.
^Sitxunpber» Berlin Math- GeseILr I6r 1917, 85-92.
ш Nouveaux elemeoe dee M&tb,, Paris, 2r 1689, 260-L Cf. Lucas, Amer. Jour. Matb4 2,
1879, 178; Canchyr"» end.
i» Algebra^, 1770, Art. 247; French tranaL, 2, 1774, pp. 355-60; Opera Onmia, A), 1,491.
"•Opera pcwtumAr 1, 1862, 243-4 (about 1782).

Свар» XXI] NUMBERS THE SlTM OF TWO RATIONAL CUBES. 573
Take
Then о*+36я = (p*+3qz)\ а = 4пг*. Hence take p = <rf*, p+Zq=2&g*,
p—3q^2yh?f <x$y = n, fgk=T.. Substituting the resulting values of p, q
into p—aFj we get afa = Pg*+yh?> If the latter be solvable, the proposed
equation is solvable. He noted (pp. 244-5) that 162-3-232 = (l-3>22)\
whereas 16+23л^фA+2V3K- In general, х*-пу* = (р*-щУ implies
but not the relation with the first factor omitted.
A. M. Legendre184 proved that, for A =2, every set of integral solutions
has x=±y, while for A = 2m, m>\yx— —yt and observed that, for A = d=3
or ±4 (mod 9), z must be divisible by 3. He stated that the equation is
impossible for A=3, 5, 6, whereas for A^=6 it has the solutions166 x-37}
у = 17, 2=21.
On geometrical aspects of the problem, see Glenie,12 Becker.1*
Wm. Lenhart™ gave a table of 11 pages expressing 2581 integers
< 100000 as a sum of the cubes of two positive rational numbers» Formulas
used in the construction of the table were deduced as follows from
4?(+y)Q, Qy+y
First, let х-\-у=аг, x>yt where a is even. For,/= 1,2,3, .-, take x^
y-s —jt 2& = a\ Then
the successive values of 3j* being computed by their differences. For a
oddr tukez^s+j, y=s-(j—1); the new right member is s2+s
Similarly for x-\-y = a'o? or 9а'а3. Next, let Q = m*> Then
whence
with three similar formulas. Eulcr's* solution (Ch. XX) of Q = ?/z3 is
quoted. Finally, let Q*=wV; then
F=a2m*+aa'(x
from which is derived four similar formulas whose right members have
i"Theorie dee nombrcgr Paris, 1793, 409; M£m. Acad. R. Sc. <!e lrIttBtitnfc de Ргавсе» 6r
annee I823r 1^7, 551, p. 47 (=pp. 29-St of SuppJ. 2 to ed. 2r 1808, of ТЬбопе dea
nombree). This Supplement ia reproduced ia Sphinx-Oedii>e, 4r 1909, 97-128; errata,
5r 19lOr 112. Theotie dee nombree, ed, 3, 2,1830, 9.
i» G. Lame, Comptes Rendua Pane, 61r 1865, 924,
IBI Math. MiHoeUany, ИинЪ^^ N. Y., lr 1836, 114-12Sr Suppl. 1-16 (tablea).

574 History op the Theohy or Numbers. [Chap xxi
the factor F* In the continuation (pp. 330-6), it is noted that
if Q—wi'wi3. If also х-Ьу=аэ, we may simplify this formula. To apply to
(A), divide each member by a* and set (e*-h3J*)/ma = »i/; hence
G. L. Dirichlet157 proved by descent the impossibility of
Hence ^±^=2^ is impossible, having been proved by Euler for n = 0,
J. P. Kulik1» tabulated the odd numbers to 12097 (to 18907) which
are differences (sums) of two cubes, and gave the cubes.
J. J. Sylvester188 stated that there are no solutions for A = 2, 3» He190
proposed the question: If p and q are primes of the respective forms
1SJ-H5 and 18J-H1, it is impossible to decompose p, qtt 2p, 4q, 4p*, 2g*
into a sum of two rational cubes.
C. A. Laisant1" proved that а3-Ь* = 10пЧ f-10n* is impossible if
£ = 3,4 or 5.
Moret-Blanc1K stated that а*-Ь*=А'1О* is impossible if h-l, 2 or 8.
T» Pepin19* proved that, if p and q are primes of the respective forms
18J-b5and lSJ-hll, the equation is impossible when А = р,р*, q7 q*t 2pf 2q*,
4p\ Aqy 9p> Qq, $p2, 9q2, bp*y bq, 25p, 25q\ If the sum or difference of two
numbers is a cube, their product is expressible algebraically as the sum of
two cubes. Hence the double of a triangular number is a sum of two
rational cubes. Since a prime 6m-H is of the form А2+ЪВ27 it is a sum
of two rational cubes if one of the three numbers 2Л, 3B±A is a cube, or
if 2B or A±B is the triple of a cube.
Pepin194 proved that Euler'e and Legendre's use of numbers a-hbV^3
IB legitimate and hence showed that the equation is impossible for A = 14,
21, 38, 39, 57, 76, 196, and stated that it is impossible for 31, 93, 95, 190.
E. Lucas196 noted that a solution x7 y, z yields the solution
For A =9, we get 919, —271, 438, and in general all solutions with z
even (not given by Prestet, Euler, Legendre). For Л =7, we get1*5 73,
" Werko, IT, Auhang, 352-3.
11 T&feln der Quadrat- und Kubik-Zahlen aller Zahten bis Hundert Tauacnd . .
1848.
«AnnalidiSc. Mat. e Fie., 7,1856,398; Math. Papers, II, 63.
'" Nouv. Ann. Math., B)r 6, 1867, p. 96.
* Ibid., {2), 8,186», 3l5. J. Joffroy stated that а'-б'-ЬМ» is impossible.
*lbid,tB)t 9t 1870,480.
~Jour.de MatA,, B), 15,1870, 217-236; Extract, Sphinx-Oedipe, 4,1909,27-8. Proof for
pt p*t q, <f by Hurwita^11* p. 220.
Jour, de Math., C), 1, 1875, 3fl3-372.
Bull. Bibl. Storia Sc. MaL, 10,1877,174^6. Nouv. Correep. Math.r 2,1876, 222.
Stated by Lucas, Nouy. Ann. Math., B), 15, 1876, 83.

Chap. XXI] NUMBEttS THE SuM OF TWO RATIONAL CtTBES- 575
— 17, 38, and all solutions with z even. This solution is simpler than
Fermat's" 1265, -1256, 183.
S. Itealis15" noted that, from the solution 1, 2, 1 of *3+^=9z3, Prestet's
formulas give the solution 17, —20, — 7, from which the new formulas
give 3 919, -3 271, 3 438 and hence the solution by Ltlcas.lM For A = 7,
an analogous second set of formulas was given by Realis,
Lucas1* noted that integral solutions exist if and only if A is of the form
ab(a+b)f<?t where a, 6, с are integers. For, if x, yt z are solutions, a=x*t
b=y* give a&(a-bb) = А(хуг)*. The converse is true by the identity
Forx^l, у=2, we get 17*4-37*=6-2V, contrary to Legendre.1*4
Lucas1" proved Sylvester's theorem that the equation is impossible
for A = p, 2pt 4qt tf, 4p*, 2g*, where p and q are primes 181+5, 181+11»
respectively. Combining this result with that of Lucas,m we see that
xy(x+y) —Az3 is impossible in rational numbers (excluding zero and equal
values) if A = p, 2p, 4q, 4p*, f, 2fy 1, 2, 3, 4, 18, 36.
A. Desboves** derived the identity B) by Lucas from Lagrange's1"
theory of polynomials which repeat under multiplication.
J. J. Sylvester301 proved that pq, py, ррг1} qq] are not sums of two rational
cubes if p, pi are primes 18Z-h5 and g, qi primes 18Z-H1. These with
P> Q, P*> <F> ihev products by 9, and 2pt 4qf 4p2, 2q2, give all known types
not resolvable into a sum or difference of two rational cubes. He an-
announced the theorem that if p, ^, ф are primes of the respective forms
18n-bl, + 7, -h 13, while each is not of the form/2-h27^ and hence does
not have 2 as a cubic residue, then no one of the numbers 2p, 4p, 2pJ, 4p2,
2^, 4^ 4^, 2<f>2 is a sum of two rational cubes. If у is a prime 6?i-H not
having 3 as a cubic residue, then neither 3v nor 3^ is a sum of two cubes.
By all of these results, we know whether or not any number ^100 (except
perhaps 66) is a sum of two rational cubes. Proofs of the above theorems
rest on the linear form of the divisors of x*—3s+1. He stated the em-
empirical theorem that every prime 18n±l or else its triple is expressible in
the form202 x*-Sxy2±y*.
A. Desboves203 gave two proofs of Lucas' identity B) and noted that
the replacement of x by Xя and у by ^ yields Lucas' A). He showed that
w Nouvr Ann. Math., B) l7r 187S, 454-7.
i« 1Ш.у 425-6. ч a. Candido1" of СЬ- ХХШ.
"• 1Ш.Г 507-14. This and his1" preceding paper are duplicated in Amer. Jour. Math.. 2.
1879, 182-^.
■w Compte* Rendmj Paris, 87,1878, 15fl.
*« Comptes RenduH Paris, 90, t88O, 289, 1105 (correction); Ашег. Jour. Math.r 2t 1379, 280r
38Э-393. СЫГ Math. Papers, 3, 1909, 430, 437; 312, 347-9.
«* A. M. Sawin, Annata of Math,, 1, 1884-5, 5S-63r noted that x and У are relatively prime
integers if and only if n is an integer
«« Nouv. Ann, Math., B), 18, 1879, 400, 491; C), 5, 1886, 577,

576 Histoby of the Theory op Numbers» [Chap. XXI
z3-^^ At* has integral solutions if A =xy{x+y),xlJ\-y*i 2хв-Ьбз/2, x(y*—x)t
or 2?-у*->Зху(х+2у), and hence if А = Ъ, 7, 9, 12, 15, 17, 19, 20, 22, 26,
28, 30, 37.
E. Catalan*" noted that xy(x-\-y)=z* is impossible in view of the
identity B) and the impossibility of т3+$*=-Р. Lucas'1** paper implies this
result.
E. Lucas** proved certain and stated others of the preceding theorems
by Sylvester201 and Pepin,1M and remarked that, if х*-Зху*+у*^ЗАг* has
solutions, then201
the divisors of the resulting A's being of the form 18n±L In the third
paper he cited cases (A a prime 18л+13, A a square of a prime 1871+7, etc»)
in which x*+y*=A& can be completely solved by the method of tangents
and secants, citing Sylvester's theory of residuation.
T. Pepin** proved (p. 110) Sylvester's201 theorem on 2p, 4p, 2^, etc.,
and remarked (p. 75} that the first three are covered by the method used
by Pepin1M for 2 ■ 7, 2 -19, 4 • 19. He proved (p. 109) the results stated by
Sylvester201 on the 16 types pq7 - • -, 2g*, as well as the theorem (pp. 113-4):
If
are primes, no one of the numbers
18(p, *, Г, Ф\ ф\ Г), 36(P, ф, Г, р\ *\ П
is a sum of two rational cubes.
С. Henry208 proved that any number of the form A=*fi7—9gu and its
double are expressible as sums of two cubes:
Л PC }+L~ /К) J'
if
2А
H. Delannoy20* proved by descent that г3-Ь^*^4г3 is impossible»
The problem ^+^=2(F 405489 hae been treated,210
T. R. Benck211 misquoted LucasJ B), whence his criticism is invalid»
K. Schwering212 put the equation into the form
»o*Nouv. Соггезр. M&th., 5, I879r91.
»» Bull. Soc, Math, FraDcer 8, 1879-80, 173-182; Comples Itendua Parb, 90? tS^O, 855-7;
Nouv, Ann, Math4 B), t9, 1880, 206-11. Related reaulte from these papers are quoted
under Lticaa70 of Ch> XXV.
w» Sylveater, Coinptes Rendua Paris, 90, 1880,347 (Coll. Мя1.1ь Papers, Hi, 432), had aUted
that there exiat soliitions in funetionfl of degrc« 9.
"'Alti Accad. Pont. Nuovi Lined, 34, lSSO-1, 73-131.
"Nouv. Ann. Math., B), 20, 1881,418-20. The right member of his formula C) is A, in
error for 2Л»
"•Jour. math. 61e*roentairee, E)r 1 {аппбе 21), 1897, 58-9.
«•Amer. Math. Monthly, 5, l898r 18l.
»1Ofverdiopha](itiakaekvationenxli+y'l=^J Dies., Upaala, 1901, 15-13.
** Arehiv Math» Phys., C), 2, 1902, 285.

Cua!\ xxi] Numbers the Sum of Two Rational Cubes» 577
and found an infinity of solutions from one by treating
l-h^-{ma:-hnK=(l^mJ)(x-«)(x-i5)(a;-r)
by his method2™ for z3-^^ to obtain (У-НI and 7 as functions of
A. S. Werebrusow213 discussed the form of numbers A expressible as the
sum of two rational cubes. Elsewhere he*14 took
whence Ax is of the form (в, О^-М-И*, and *! = («, b). Then
А^ = (8, t){Mt N), x =
with similar formulas derived by interchanging s and t or M and N. Further
treatment was given for*i^l, Ai=l, 3 or 7.
A. Cunningham214 discussed z2—y* = 17z*, obtaining integral solutions
with z=7. From the solutions-* 18, y**~1, z = 7of х2-Ь^^17га, Prestet's
formula leads to positive integral solutions smaller than those given by
Lucas' A).
R. W. D. Christie™ noted results due to Desboves.208
Christie*1' noted that, if р-^-бо^-За^-Ь3, X3-pY* = l has the
solution
and hence also X=lfxf F= — yjx.
A. Cunningham218 treated z3 -hy3=Cz* for ж, у relatively prime by setting
The g. c. d. of Xt Г is 1 or 3. Let С be prime to 3. Then XY=CtlZ\
Since Z3 is a factor of F and is prime to 3,Я=А2-ьЗВ*. Hence Z*~
But, for у even, K-(s-ii/J-b3(&)J. Hence, it Y=Z\ x-\y =
fcB If r=3Z*,2-to=*±3£i,}y = ±Ai. Foryodd,
There is treated also the case C=0 (mod 3).
T. Hayashi219 concluded from the impossibility of rational solutions of
x2-h^=32a that 4а(а-Ь0)(*+20)/6 is never a cube.
R. D. Carmichael220 noted that, if A =2"*, we may take x, y, z odd and
proved that one of the variables must be zero, except for the trivial solution
x^y=z which occurs if m=l.
*» M&tem. Shorn, (Math. Soc. Moeco*), 23, ]902, 761-3.
^L'lntermeduira d^ math., 9, 1902, 300-3.
ш Math. Quest. Bdac. Times, B)r 2,1902r 3S [4S], 73.
**ltod.f B), 3,1903,109-110.
"ЧЬЯ, B), 13,1903,90. a. Deebovee.»
*'/Wd, 27-30.
n»Nouv. Ann. Matb^ D), 10,1910, 8^6.
111 Diophftntiiie An&tyaiB, 1915, 70-72.

573 History of тле Theory of Numbers. [Chap, xxi
J. G. van der Corput2*1 applied quadratic forms to prove the impossi-
impossibility of z*±y*=pmz* if p is a prime =2 or 5 (mod 9).
B. Delaunay*52 stated thatr if p is an integer not a cube, pz?+y* = l has
no integral solutions if the fundamental unit и of the domain defined by
r= Vp is not of the form Br+C, but has the single solution x=Bt y = C, if
it be of that form. Here и is Dirichlet's ат2+Ьг+с, where at Ъ, с are integers
not of like sign, whose powers, with positive and negative exponents, give
all the units ^-1-^+7, where лт 0, у are integers.
M. Weill221 used the identity
to show that, if one solution of z*-\-y* = Az* is known, a second is
where <*=:rs, 0=y*t and to obtain solutions when A — 3c*-b3c-bL
W. S. Baer*24 proved that n can be represented in the form n = ф{и) +ф{р),
where ф(х)=ах*+ух, with и, v, «, у integers and u>t v>$, if and only if
n is a product of two integers: n=klt where fc>2£, Ыаг'+у^'<к2-Ък£+Ъ?,
V bemg integral and AV — № the triple of a square» Then и and v will be
relatively prime if and only if the g. c. d» of к and Г is 1 or 3, and in the
latter case V is not divisible by 3*. The theorem can be extended to cubics
Ф^АХ^+ЕХъ+СХ+В, where A, ■■, D are integers and В is divisible
by ЗЛ, since X—x — B/(ZA) transforms б(Ф—5) into ф> In particular,
Jet a=\, 7=0, £=0. Then n is representable as a sum of two positive
cubes if and only if n is a product of two positive integers к and I such that
Z<^ and 41—№ is the triple of a square; the cubes will be relatively prime
if and only if the g. c. d. of к and Hs 1 or 3, and in the latter case I is not
not divisible by 3*.
If Л is a positive integer, and p is a prime or unity, г^-Ь^^Лр" has only
a limited number of relatively prime positive solutions, and the remaining
solutions are readily deduced. But u3+i>*=u>2 has an infinitude of positive
solutions of which и and v are relatively prime»
L. Varchon*** proved that z3—#3 = 2*5b is impossible in integers +0;
Moret-BlancV*2 result is a corollary.
M. Rignaux2246 derived A), B) and analogous identities from a common
source.
StJM OR DIFFERENCE OF TWO CUBES A SQUARE.
U Eulerm noted that ^-ht/^D for x = pz\r, y=qz!ry z=r*j(p>+<f).
To obtain integers, set r-n(ps+g3); then
** Nieuw Archief voor Wiekunde, B), llr 1915, 64-8.
** Comptee Rendus Paris, 162, 1915, 150-1.
« Nouv, Ann. Math., D), 17, 1917, 5Ф-9,
**T6hoku Math. Jour., I2t 1917, 181-9.
1114 Nouv* Ann. M&tb<j D), 18, 1913, 356-8.
"*• L'intermMijure dee math., 25. I918f 140-2,
ш Novi Comm. Acad. Petrop., 6, ad аппон 1756-7, 1761, 181; Comm. Arith. Coll., 1,
207^ Opera Onmirr A), II, 454»

Crap. XXl3 SuM OR DIFFERENCE OF TwO CUBES A SQUARE. 579
To obtain relatively prime integers x, y, when p, q are integers, we must
employ fractional values for n. To obviate this, Euler gave a second
method. The factors x-by, x2 — xy+y* have the g.c.d. 1 or 3. In the first
case, he put the second factor equal to the square of p2 — ptf-btf2 and stated
that tkx='pi—2pqt ±y~p2—qt> The upper sign- is excluded since
For the lower sign, x+y~ (р-Ь$)г—3p2==O if
p = 2??m, q—3m2—2mn -Ьп2,
In the second case, хг—хуЛ-у%=Ъ{р*—pq+q2J, (з-Ьу)/3=П. As the
three subcases lead to equivalent results, consider the case
z = 2p2-2pq-q2, y=p*-±pq+<?> (s-by)/3=p*-2pg = □.
The last condition is satisfied if p = 2m*, q=7ri*—n2, whence
Euler2" noted the examples 1+2>-32, 83-78 = 132, 373-hlla=2282,
Several237 found that the difference of 73 and 8* is a square by considering
x3, (л-hiI, and, by use of tables of cubes, found that this pair and 73, 143
give the least solutions.
C. H. Fuchs42* discussed х*-\-1?=аг*> Let x, y, z have no common
factor, a no square factor. If x or у is even, set x-hy=/>, x—# = g. Then
) =4аг2. If p is not a multiple of 3,
Since Э is a divisor of p*-h3g*, it is of that form. Thus 40 ^/-h3^. Also
« = J2-h3?72. By use of ^—3, he got
The case p=ZP is similar. For xy odd, set 2р—хЛ-у> 2q*=x—y* One
of the three cases has p^2p', a odd. Then p=2cd*t рг+Ъ<?=*$и*. Не
again got A).
Л. Hoppe2^ obtained the general solution of х*-Ь$/*—г3 in relatively
prime integers by setting pq=^, p=x+y, q=(x+y)(x—2y)-h3y?, where
p and ^ have the greatest common factor 1 or 3. In the first case all solu-
solutions are given by
where о is odd, and 0=3 or 1 according as 3 is or is not a divisor of
Second, if p} q have the factor 3, the solutions are [T
ш Opera poetuma, 1, 1362, 241.
"Ladke' Diary, 18l2r 35T Quest. 1227; Leybourn'a M. Quest. L. D., 4t 1817, 149.
■■ De Fomaub a^+y»-^ Die. Vrfttbbviaer lS47r 33 pp.
mZeitechrift Math. Phy*., 4, 1859, 301-5.

580 History of the Theory of Numbers. [Chap, xxi
where a is not divisible by 3, while rj = 2 or 1 according as a, b are both
odd or not both odd.
C. Richaud230 solved (o?-hl)J-x3=y2 for x and made the radical rational.
Thus ByJ-l=3r*, whencez = 0,y = l; x = 7,y = 13; s-104, y = 181; etc.
The same solutions were given by Moret-Blanc,2*1 who remarked that
x*+(x+1K=^ only for x = Q, 1 (cf. E. Lucas, Mathesis, 18S7, 200).
W. J. Greenfield232 gave numerical solutions of z3—y*= П.
M. WeilP* noted that (-3^L(^+4K==(l+«*)("*-&)*'
P. F. Teilhet2* gave the solutions 65, 56, 671; 5985, 5896, 647569.
E* Fauquembergue236 reproduced Euler's2** formulas with p^n, q = m.
Replacing p by jS—a, q by -<*, we obtain the formulas of Axel Thue236
for х*-Ь^=г?, who noted that, if 2 is not divisible by 3, then x^-xy+yt-^B1*
Thus, for relatively prime p and q7 px=q(B+z—y), qy^p(B-z+y), since
the product of the second factors is xy. Eliminating J5, we get
In case the numerator and denominator have a common factor, it is 3, and
p-2g=3pi; set gi^g-b2?i; we get
Hence in every case we may set
Now x-yy must be a square, Л2. Hence (g-2p)*-3p* = rfcA*, so that the
lower sign is excluded. From 2pq = (p—q)*— A2, we get
where a, fi are relatively prime. Any common factor of the numerator
and denominator divides 6. If it be 3, we reduce to a like fraction as above.
If it be 2, then fi and hence p and у are even; but we may assume that if
either x or у is even, x is even. Thus in every case we may set
It follows that X'-bF^z* is impossible in integers if z is not divisible by
3. For, if the preceding x or у be a square, a=k2, a5—0* = h2t or /В*2*?,
0*4-80*—h*t respectively; in either case, X\-{-Y\=z\ in smaller integers.
Multiplying x and a by VS, у and 0 by Vfi, we see that Аз?+Ву*~г2
has the integral solutions
"°Alti Ac. Pont» Nuovi LiDoeir 19, 186&-6,185.
*"Nouv. Ann. Math., C), 1, 1882, 364; cf. B), 20, 1881, 515; I'intermediAire dee math.,
9,1902,329; 10,1903, 133.
~ Math. Qiieet- Educ. Timea, 23r 1S75, 85^-6.
«Nouv.Ann. Math.» C), 4, 1885, 184. a. Gerardin.M
"L'interm&tiaiie dee niatb.? 3, 1896, 246.
«■/Ш.,4, 1897, 1ИМ2. Cf. the гешагки, 112-15.
** Ibid,, 5, 189S, 95; Det Kgl. Norske Videiukabe» SeUkabs Skriiter, 1898, No. 7.

. XXI] STJM OR DIFFERENCE OF TwO CUBES A SQUARE. 581
if x=3n,y-2n,z^n. E.Fauquem-
bergue (Ш., 6,1899,131) gave [Euler22*]
K. Schwering2" obtained an infinity of solutions by means of the rela-
relation between Abel's theorem and certain diophantine equations, first
indicated by Jacobi148 of Ch. XXIL Set
By the coefficients of x* and x,
— m
Substitute for m, n their values from ma<+n= (aj-hlI for t = lf 2. Thus
Hence we get ma^n and thus {a\-{-\}h. Take <xt=<xt—a. Then
By eliminating a^ we get the desired solution
The corresponding Abel theorem is here (
A. S. Werebmsow1* gave Euler's4* final solution»
R de Hefeuero240 solved (x—y)t=^, where t=z2+xy+y*. Set d=3 or
1, according as t is or is not divisible by 3. Then x—y^da*, t—dfp. Thus
dp2 has one of the three representations by tf+xy+y'*. It remains to
make d(x-y) = \J. According as d=3 or 1, this reduces to w2—1^=«?2 or
F. Pegorier discussed ^
A» Gerardin*41 noted that one solution of o^-h/S8 = y1 implies a second since
W* H* L. Janseen van Raay243 discussed the solution of
Cashmore gave the first solution due to Hoppe.*2"
See Boiiniakowsky,1* Mordell,17* and Baer^4; also Catalan122a and
Tafehnacher1* of Ch. XXVI.
»»L>interm6diaire dee math,, 5f 1808, 75-6. " "*"
™ Arehiv Math. Pl^e., C), 2f 1903, 285-8.
»' LTintenn6diai№ dee math,, 11, 1904, 153.
"Gionule di M&t.r 47,1909, 3624.
>" Bull de math, flfrn,, 14, 1908-9, 61-52.
M LTinten&6diaii«d« matlb, 18, 1911, 201^2. Cf. WefU.»«
"»Wiefcundige Opgaven, 12> 191fi, «7-71 (Dutch).
™ LJinterm6diaire dee mwtb., 23,1916, 224.

582 History of the Theory of Numbers- [Chap, xxi
Sum or cubes of numbers in arithmetical progression a cube.
К Еи1егий treated the problem to find three consecutive numbers
x — 1, z, s4~l, the sum Зх^-Ьбх of whose cubes is a cube. Since x=4 gives
a solution, set x = 4-\-y. Then 6*+15Оу+ЗбУ-ЬЗ^ is to be the cube of a
number, say 6 4-/2/• The coefficients of у are equal if 108/= 150, and then
1871^= -7452, x=32/1871. Or we may take 3x*-b6*=27zaza, whence
x?A8z3^2)~ 4, and 18z*—2 is to be a square. Since this is the case for
z = l, set z = l-{-v] the cubic in v is the square of 4+27^/4 if v= —15/32.
J. R. Young2*6 required that the sum of the cubes of a—a/x, a, a+ajx
be a cube. Hence, as by Euler, З-Ьб/s2 is to be a cube» To make хг=2п*,
take x=2nq, whence n—2ф. Then 3n3-|-3=24tf*-b3 is to be a cube, which
is true if q — 1.
C. Pagliani247 treated the problem to find 1000 consecutive numbers
the sum of whose cubes is a cube. The sum of the cubes of я-hi, *r', x-\-m
is 8=т(у+1)(у*+2у+т2IЪ for y=2x+m. Let m=Sn3. Then s will be
the cube of n(y-\-£n2) if # — 0 or
Writing v for 2n, we see that this is equivalent to saying that
if 6x^(tJ2-lJ-3(^-hl)» Then x is integral if у is not divisible by 3.
The cases v=2, 4, 10 give
11343+ -
W» Lenhart24S treated the problem of m consecutive cubes whose sum
is a cube» First, let m=2n. The sum of the cubes of s-f-1, ■ * •, s-Ьл, s,
s—ly ••■, s—ft+1 is 0" = Bs+l)(>is*4-rcs-Hj,3). Set и—4«i and divide <r
by BrtiK; we get
if 3s = 8«t— 4ni — 1. To make * an integer >1, take щ prime to 3. Por
Tii — 1, the roots of the 8 cubes are 2, 1, 3, 0, 4, — 1, 5, —2, leading to (h).
For «i = 2 or 5 we get (b), (la). Again, we can equate <r to the cube of
7ir4-sBn2-|-l)/C7i) by choice of s in terms of n. Second, let m=
Then
S = o- 4- (s — nK = тп&*-\--$8т(т2—1).
f^ince S is a cube for s —1/2, set s = l/2-M ^nd take m=wtj. Thus
if ^{mi-2mJ-2)/6f whence s* (mj-lO6. Again, let S^p^V. Then
«• Algebra, 2, 1770, art. 249; French tnuul., 2,1774, p. 365. Opera Omfti», (t), I, 497-8,
*■ Al^bra, 1816- Amer. eA, 1832, 332.
 Annies de math, (ed., GeTRonnc)T 20, l829-30T 382-4,
*» U&th. Mindlany, New York, 2, 1839, 127-132; French Imal, Sphinx-Oedipe, 8, Iftl

Chap. XXI] SUM OF CUBES OF NtTMBEKfl A CUBE. 583
for p = 1 -hr,
4sl~ m*-l I
if r=|D-?ra2)/(m^-l)x whence «=4(w2-
V. A. Lebesgue24* stated that, if x and г are positive integers,
B) xt+(x+ry+(x+2ry+---
is impossible except for n=3, x — 3r. If we write
C) e = 2x-h(«wl)r, a-^-h^-lJr2,
we obtain for the left member of B) the expression ти^/8. Не considered
it a difficult problem to make the latter a cube, and remarked that it was
impossible for n=2 by EulerV theorem.
A. Genocchi*50 treated the last problem nw/8=#\ Set s=rf, 2y—rz.
Then n/(^-hn2-l)^^ Following Format's method,14» set t =
z=n-\-puf and equate the termp of the first degree in u. Hence
Tfie cases n—3, r=107; n=4,r—1; n=5t r—13, give respwctively
E) 1493-h256a+3633=4081, 118-Ь128+13эЧ-14э-20э,
F) 2303 -h243s-h256a -ьге^-ьгвг1=440*.
B. Boncompagni251 proposed for solution the same problem B) and
G) хЗ-Кх-И'-Ь • • • +Сх+(п-1)г]"-Л
V. Bouniakowsky282 noted the particular solution го=2, zo=-^n-
»o—л, of G), and that this solution leads to the second solution
where p and и are given by D), and thus derived E), ete. Starting from
the latter, we obtain new solutions. For n=3, nsajk is the cube of vtV2 if
The general solution of the second equation is known to be
x-hr=d:(pa-6pg2), r=d=Cp*g-V), 1>*=р
Taking the upper signs, we see by the first condition that
Frdm the evident solution p'=q=w = lfwe f*etp=Vi=3, q=l, etc. In B),
he set т=\х and noted that the rational cubic for X has no rational root
when n<8 except for n—3t and stated that (li) is the only solution in
positive cubes.
"• АплаН di Mat., A), 5,18K2,328.
Л£у320.
v. Ann. Math., B)r 3,18M, 176; Zefoehr. MaUu Fhyi, 9,1891, 284.

584 History of the Theory of Ntjmb»ks- [Chap.xxi
A. Genocchi265 treated G), i. e., to make пае a cube. Set
m=tt2—1, «-nV\
Then
Set
From the resulting rational expressions for p, q we get
which is of the same form as the initial equation n$<r = 8t?r Hence one
solution r", s"r v* leads to a second solution r, s, v, etc. But not all solutions
are so obtained. More convenient formulae are obtained by setting
r=g+z, 2v=h+pz, where r=gy 2v=h is one set of solutions.
L. Matthiessen2" noted the particular solutions of G):
n=2p+Zf x=—2p-l, r=2, » = 2p4-3;
n=2p+4, x=- p-1, r=l, v= p+2.
Also that 351120* is a sum of к positive cubes for A^3, 4, 5, 6, 77 8.
A. B. Evans265 noted that the sum of the cubes of the first ?ia integers
is a cube only if n = ly since (ns+l)/2 is not a cube if «>1 £Euler18* on
3
D. S. Hart266 took 2n—1 consecutive integers, x being the middle one.
The sum of their cubes is Bn-l)x84-Bns-3n2+n)x. For 2n-l = p\ the
sum is a cube if e=zaH-^(p6—l)x is a cube» Take x = \+y> 8s = Bp4-p*)*;
we get у and я = (р*—1J/6. For 2n cubes, add the term (х+п)г. The
answer is now x= [(p2—iJ—3}/6.
A. Martin257 noted that the sum of the cubes of x7 x+1, - - -, x+na—l
is a cube if x = (n4—Зп^-г^+^/б,
Hart268 expressed the difference of 1*4 \-n* and (S4-m)J-£* as a
sum of cubes by trial.
S. Realis*9 stated that z\-\ i-^=En+3)^ has a solution with
zu * - •, z* in arithmetical progression, and solutions with г=1,пФ2.
A. Martin2" proved that 13+2*H \-пг is not a cube if n> 1, since
nfn+lj^+p2. For, Bn+iy=Sp*+l is of the form z*4-l*=G, which
holds (EulerlW) only if л; = 0, -1, 2. He listed (p. 188) sete of 20, 25 and
64 consecutive cubes whose sum is a cube, besides known cases.
di Mat., 7, 1865, 151^8; Atti Arcad. PoDt. Nuovi Lincci, lfl, 1865-6, 43-50,
French tranel,, Jour, de Math., B), 11, 1S66, 179; Sphinx-Oedipe, 4t 1909, 73-8, Ac-
Account by M. Cantor, Zeitachr. Math. РЬув., 11, 1866,248-251.
*" Zeitecbr. Math. Phm, 13, 1868, 348-350.
m Math. Quwfc. Educ. Tlmra, 14, 1871, 32-33.
■■ Ibid., 15,1871, 24-6 (Math. Magazine, 1, 1884,173-6).
»' Ibid, p. 26. Same by J- МаНдоп, Collection of Dioph. Problems 1888, Ptobe. 4. 5.
« Math. Quest. Educ. Tin**, 23, 1875, 82-83.
*• Nouv. Ourop. Math., 6, 13S0, 625-6.
Mh M 2, 1885, 159.

Chap.xxij Sum of Cubes of Numbers a Square. 585
E. B. Escott*1 proved that, for 2Шп^5,
has only the following integral solutions: A0 and
К Matthiessen2*2 noted that if fractional values of x, v are allowed in
G), we may set r=L Write u = 2x+n—2, v=pu+n!2. The usual form
of G) becomes a quadratic in u:
Evident solutions are obtained by equating to zero the first or third coeffi-
coefficient. In the second case, integers x are found only for n — 2, n = 4.
F. Hromidko263 noted that x^Z is the only positive integral solution of
(y(y = (x+Z)* [Lebesgue2"}
E, Grigorief2** obtained the special solutions
• +348*=4953,
" L. N. Machaut"** treated B) by setting xjr = и and obtaining a cubic
for и with a real positive root (u=3) only fom = 3, leading to (li),
J. N. Vischers266 proved Lebesgue's249 first result when n = 3.
L. Aubry*7 proved that 3, 4; 5 are the only three consecutive integers
the sum of whose cubes is a cube»
Sum of cubes of numbers in arithmetical progression a square.
To find five integers in A» P. the sum of whose cubes is a square (or
sum of squares is a cube), J. Stevenson268 used nx—2x, nx—x, nx7 nx+x,
nx+2x, the sum of whose cubes 5n*z* -f-ЗОш? will equal т*хг by choice of x
(or sum of squares Ьп*х2+10х2=т*х* by choice of x)> Several solved both
questions simultaneously by using x2, 2xr, 3x2, Ax2, 5s2, whose sum of cubes
is A5VJ and sum of squares is 55s4 = aV, if x=a*/5b; takea^55.
Several** made the sum л2Bп2—1) of the cubes of the first nodd integers
a square by using Euler's*3 solutions (Ch. XII) n = 1,5,29, * - * of2»*-l = O.
A. Genocchi253 discussed the rational solutions of
A) «8+(«+r)l+(«+2rL-' + ^+nr-r)^^
In view of C) of Lebesgue*49 the problem is nso-=8^. Set 2y—n*L Solving
2& for 8, we see that n4A-(n2-l)ri = \J = (nf£-rpI. Hence
ds=2n(n2-l)F or
»l L'interm^liaire dee math., 5, 1898, 254-6; 7T1900,141»
m Zeitech. Math. Natuiv. Uaterricht, 33, 1902, 372-5.
"•/ЫЛ,34, 1903,258.
ш LJinterm6diwre dee math», 9,1902. 319-
**1Ш,, 15, 1908, 163-1.
~ Wiekuudlg TijdBchrift, 5, 1908, 65.
** Sphmx-Oedipe, 6, 19X1, 142-3.
**1be Geotleman'a Diary, or Math. Repoetlory, LoDdon, 18U, 36-7, Quest. 1010.
»* Ladies' Diary, 1832, 36, Quest. 1529.

586 History or the Theory or Numbers. [Chap.xxi
where d=n*— 1+p*. The general solution thus involves the rational
parameters p, L
E. Catalan** stated that, if r«l, integral solutions of A) are
where & = 1 or 2, while а, & are relatively prime integers. For example,
if а=5, 6 = 1, we may take 7 = 313, « = 7850 (in place of 1850 in Table X in
Legendre's Theorie des nombres), whence n = 626, x=3600.
Catalan, in treating the integral solutions of A) for r = l, wrote
a = 2nsf 0 = o>, where e, <r are given by Lebesgue's'49 C) for r^L The
problem is then to make aft the square Wy* of an integer. Since «г is
even, у will then be an integer» But his separation into two cases lacks
generality and his solution is incomplete. His*72 later discussion leads to the
following result: Take any two relatively prime integers p, q, one even, and
express pq/2 as a product of a square u* by a number в without a square
factor; then if
has integral solutions y, t>, we have
M, Cantor273 reported on Catalan's271» * discussion of the preceding equa-
equation a0 = l(fy2, where a and j9 are integers divisible by 4 for which 0dba+l
are squares, and obtained two sets of solutions, in which p and q are rela-
relatively prime integers, one an odd square and the other either half of an even
square or an even square» In the first case, {jP+qL)i1—uL = l yielde integers
% u, and then у2=2рд(уи/4У. In the second case, if (p*+q*)y2—2jht2 = 1
has integral solutions y, w, then у* = рд(уш/2)К In each case, n = py,
C. Richaud274 treated A) for r=l, via., P-k*=y2, where 2*=a?(*-
2^ = (x+n)(^+n—1)» Certain, but not all, solutions arise from i=+;
k, y^2abt a2—b2. Eliminating x, y, we get a quartic equation. For
к = 2ab, it becomes
with an infinitude of solutions m = 2*2, n = l; m»32^+6^ n = 16^+l; etc.
Note that the sum of the numbers x,x+l7 - >,aH-n—1 is a square, (a—bJ.
For a general r, A) becomes ш*=8y2 by Lebesgue's2" C). For275 rw/2 = afr2,
cr/4 = oa:7 y=aabf he eliminated 5 and discussed at length the resulting
"'Bull. Acad. Roy. de Bdpque, B), 22, 1866, 339-40. " ^ ™
tn Atti Aocad. Pont. Nuovi lined, 20, 1866-7,1-4; Nouv, Ало, Math., B), 6, 1867, 63-67;
Melanges Math., 1868, 99-103.
s*Atti Aocad, Pont. Nuovi Iinoei, 20, 1866-7, 77; Nmiv, Ann. Math., B). 6, 1867, 276-8;
Melange Math., 1868, 248-251.
^Zeitechr. Math. Phy*, 12, 1867, 170-2.
«" Atti AccwL Pont» Nuovi Lincei, 20, 1866-7, 9Ы10.
*^1d the alternative case nt/A^cd?, <r/2—«ras y—aab, not treated, there are two misprint*
for 4.

Chap» XXI] SUM OF CUBES OF NUMBERS A SQUARE. 587
equation, for the case a = l, whence the sum of x, x-\-r> - • -, x-\-(n—l)r
is bs. In the most interesting case a=lt r-2, the eliminant becomes
a2—(/4H-l)rc*~ —1 fOTb=nt. It has an infinitude of solutions (a, n) = {P, 1),
D/e+3*2p 4*4H-1), etc» Taking t—lt we have я = 1 and the following result:
While the sum of n consecutive odd numbers 1, 3, - - is always a square n2,
the sum of the cubes of the same n odd numbers will be the square of an
whena2-2r^=-l. Examples arc (a, ») = A, 1), G, 5), D1, 29), B39, 169).
E. Lucas275'1 stated that the sum of the cubes of five consecutive integers
is a square only when the middle number is 2, 3, 98 or 120» The sum of
two consecutive cubes is a square only for the cubes 1 and 8.
G. ft Perkins7276 solution of A) differs only in notation from GenocchiV53-
E» Lucas277 asked when the sum of 7 consecutive cubes is a square»
Several275 found that the sum of the cubes of the first n odd integers is
a square if 2n*—1 = D, n^I, 5, 29?
M. A. Gruber*79 attempted to show that a sum of cubes of n consecutive
integers is a square only for l*+24 \-n*={l+ \-nJ*
A. Cunningham280 desired a sum of successive odd cubes equal to a
square» The sum Sr of the successive odd cubes 1, 33, •--, Br—IK is
2 —1) and is a square if r = 5. Next,
is a square г2 if, upon setting x = 2r*, y = 2p
Bx-\)*^By-iy
Solutions are found by making special assumption?.
W. A. Whitworth2*1 expressed V2 as a continued fraction, took a conver-
convergent N/D, with D odd, and got
Cunningham2*2 asked for a sum of successive cubes
equal to the product of a square by q. Since
we set Tm = %Tu and see that &Мр „ч-ф is a square if (£2—l)/g is fr square»
For each such £, we test Tm = %Tn by a table of triangular numbers (de
Joncourt's, 1772) and find suitable pairs m n. Solutions are found for
M. A. Gruber^3 noted that n=l and n = 5 are the onJy cases in which
?7i* Kecherchca gur 1'arisJyec lDdctenriiriee, Mouline, 1873, Я2. Extract from BulL Soc,
d'EmuUtiofi du D^partemeat de PAflier, \2, 1873, 532.
™ The Analyat, Deb Moincs, 1, 1874, 40.
"' Nouv. Coireep. Muth,, 2t 1876, 95.
*>* Math. Quest, Educ Times, 53, 1890, 55. (X Brocard» of Ch. ХХШ,
^Amer, Nfath. МоПШу, 2, 1895, 197-8.
»f Math. Quest, Educ. Timesr 72, 1900, 45-46 (error); 73, 1900, 132-3.
м 1Ш., 72, 1900, 46.
^/Ш., 75, 1901,87-88.
** Amer. Math. Moathly, 7, 1900, 176,

588 Hibtory of the Theory of Numbers. [Chap, xxi
L, Matthiessen5*4 discussed A) in three ways. One way is to multiply
G), the corresponding equation with the right number &, by z8, where
^2a = y\ Thus, for lF+i^+^+U*^^, take z=5f whence #=1000.
H. Brocard, " E. A. Majol," and F. Ferrari*6 discussed a sum of three
consecutive cubes equal to a sum of two squares.
L. Aubry2* treated (y-ky+y*+(y+k)*=3y(y2+2k*)=u\ First, let
y = 2a2, y2+2k* = 6b2, 4=&ab. Then 2a4 = 3b>~№, which is satisfied if
Second, let y = Ga2, yi+2fc8 = 2b1, u=Qab. Then 18o4^^-Jfc2, which holds if
a = 2pqt b = rp*+sg4, A;=rp4—sq4,
(t, s) — G2, 1) or {9, 8). For p = q=l, the second set gives
which occurs in a manuscript of Lucas'. Or we may set y = 3o* or a2.
Homogeneous cubic equation F(xt yt г)~ О.
A. Cauchy**7 derived a second solution from a given solution q9 b, c.
Let ф(х, уу г), Xj Ф be the first partial derivatives of F(x, y, z) with respect
to *, y, z} respectively. Then ^=0 for
A) x ; у : z — as — ta ; b& — tfi : cs — ty,
where, if и=ф(а, ht c), v=x(a> Ъ> с), %о=ф(а, Ъ, с), the parameters a, P, у
satisfy ms+ujS+itvy^O, while
We may take а, 0Р т^0, w, —v; —wr 0, w; or f, —v, 0. In each case one
of the terms A) is very simple. He showed that we may take such a
simple value and obtain the following solution
F@, wp -p) F(-wf 0,
These become
for the case
D) Fs
If at 6, с and a\ Ъ\ с' are two given sets of solutions of F=0, where F
is any ternary cubic form, Cauchy obtained a third set by expanding
F(as-ta\ bs-tb\ c$-ic') = 0
w Zeitechr. Math. Naturw, Untemcht, 37,1906, 190-3.
ш L'mterni^diaire des math,, 15, 1908, 41-43.
** Sphinx-Oedipe, 8, 1913, 28-9. a. Luc»**- of Ch. XXIII.
«^Exercioee de matbematiques, Pane, 1826, 233-260; Oeuviee de Cauchy, B), 6, 1887, 302.
For a leas effective method, вес Cauchyu* of Ch, Xlll.

Homogeneous Cubic Equation F(x,y,2)=0. 589
and obtaining stL^Q, where L is a linear function of $, t, which is яего for
а-аф(а' V, с')+Ъх(а', V, с')+сф{а\ Ь\ с),
1 = а'ф{а, Ъ, с)+Ъ'х(а, Ъ, с)+с'*(а, Ь, с).
Then the resulting third set of solutions of <F = 0 is
E) x :y :z=as-ta' \hs-thf :cs-k\
By C) for A=B = 1, С=-а*-Ъ\К=0, e = l, we see that
satisfy x'+y^Cfl'+b3)*3 [Frestet1*].
For geometrical interpretations of Cauchy's results, see Lucas.1»6
A. M. Legendre-*8 deduced from one solution of х*-\-ау*=Ъг* the second
solution
Given X, Yt Z, the determination of xt yf z depends on a quartic equation.
J. J. Sylvester2^ stated that D) can be transformed into
AW+B'tf+C'vP+Kuvw (A'B'C'=ABC),
where uvw is a factor of z, provided (i) the ratio of two of the coefficients
А, В, С is a cube, (ii) the " determinant " 27ABC+Ю has no positive
prime factor 6l+1, and (iii) if 2m and 2n are the highest powers of 2 dividing
ABC and K, respectively, then either m is of the form 3A±1 or, if not, ш
exceeds 3n. If a, P, у give one solution of D) and if we set
then x*+y*+ABCz?+Kxyz=O. For the case A=B=*1, С a prime, and
27C+K* positive and not divisible by a prime 6&+1, he290 gave a process
to obtain all integral solutions of D) from one initial solution P= (e, g, i).
The process is to apply to P repetitions of transformation F) and the trans-
transformation, depending also upon P, from one system ?, mt n to the system
\ = Zgm(gl-e7ri)+ZCin(il*-
or to the system obtained by interchanging e and g.
Sylvester291 stated that F=xx+y*+zi+$xyz = ft is not solvable in in-
integers; likewise for 2F-27nxyz when 27n5—8n+4 is a prime; and for
4F=27ru^^when27n2-36n+16isaprime. Set Л/3-27А=ДаД1 where Aj
has no cubic factor. If Ai is even and contains no factor of the form
P+3g2, and if A is a prime, x?+y*+Az? = Mxyz has no integral solution
*M Tbeorie dee nombrrar, ed. 3, 2, 1830, 113-7; МаэегЪ trana]., 2, 1893, 11H
«i London, Edinbui^h, Dulrfin Phil. Mftg,p 31, 1847, 189-191, 293-6 for corrected theorems;
Coll. Math. Papere, 1, 19W, 10Г-13,
'"Phil. Мад.,31, 1847,467-471; CoU, Math. Papera, I, 114-8.
SM Annali di Sc. Mat. e FJs., 7, 1856, 398400; Math. Papers, П,

500 History of the Theory op Numbers» CChap. xxi
except when —M/A is the square of an integer. Likewise if A is of the
form p*w*1, where p is a prime» Also without the assumption that Ax is
even, provided it has no factorP+Zg\ while A~2iv>Jrl; or A/2 is a prime
$г=Ь4, and M/9 ie an integer; or A/4 is a prime ф±2, and M/1S is an
integer; or A is a prime and А, В are of the respective forms $n±2, qn±6,
or gn±4, qn±Zt or qn±Z, qn,
E. Lucas** stated Cauchy's results on the cubic D) as follows: (i) If
й, b, с is one set of integral solutions, another set x, y7 г is given by
а о с
(ii) If в, Ь, с and a\ b\ cp are two distinct sets of solutions, then
x у г
Ъ с = 0, Ааа'х+BWy+СЫг ^ 0
' Ъ' с'
give a third set. But (i) and (ii) do not yield all solutions. Lucas** had
stated as exercises these results without relation to Cauchy. They were
verified by Moref^-Blanc,**4 And restated by A. Gerantou**
Lucas290 stated the generalizations to any homogeneous cubic F(x, y, z)
= 0. 1°. The tangent at a point m\ with rational coordinates xu yt> zlf
and on F==0, cuts the cubic at a rational point m, i. e.,
,.0, ,£+„£+,£,0
' dxi Sy} dzx
determine л-, yf z rationally. The point m is distinct from mi unless the
tangent is parallel to an asymptote or passes through a point of inflexion.
2°. The secant mxm^ through two rational points on the cubic cuts' the
cubic in a rational point (in general distinct from mlf wi*). 3°, The conic
through five rational pointe on a cubic cute it in a sixth rational point.
S. Healis2*7 obtained a second solution (quadratic in at &, y) of
6хуг from one solution a, 0, y,
noted that all integral solutions except z=y=z of
are given by
x = (a-bf+{a-c)\ у=(Ъ
U (x, 0, 7 is one set of solutions of
A
another set is g>en by
-3a(Bp+Cy),
111 Bull. BIbl. Storia Sc. Mat., 10, 1877, 175; Amer. Jour. Math., 2, 1879, 178.
«• Nouv. Ann. Math., B), U, 1875, 526.
™1Ш., B), 20,1881,201.
■»Sphin*-Oedipe, 5, 19Ю, 90.
«•Nouv. Аяп. Math., B), 17. 1878, 507-9; Amer. Jour. Math., 2, 1879, ISO.
™ Nouv» Correep. Matb», 4, 1878, 346-62.
Ш-, 5, 1879,8-11.

Chap. XXTJ HOMOGENEOUS CUBIC EQUATION F(z, y, z) = 0. 591
and values of yt z derived by permuting the triples of letters cyclically.
All solutions of x*+y*+z?=x*y+y*z+z*z are given.
A. Desboves*19 proved that if z, у, z is one set of solutions of
Atf+Btf+Cz* = 0, a second set of solutions is given by
Х=х(Ах*+2Ву*), Y=-yBAx*+By*)t Z=z(Ax*-By*).
For A = 1 this result is due to Legendre.*88
J. J. Sylvester1* called the intersection of the tangent at a point P on a
cubic with the cubic the tangential of P. He proved for A=B=C=1
that C) gives the tangential to D) at the point (a, b, c) and that the point
on the cubic collinear with (а, Ь, с) and («', Ь\ с') has the coordinates
G) bca't-h'c'a?, саЬ'ш-с'а'Ъ\ аЪс'г-а'Ь'с\
A. Desboves101 noted that Cauchy's formula E) becomes, for D),
with similar expressions for y, z. Since а, Ъ, с and а', Ъ', с' satisfy D), we
can express A, £ as linear functions of С, К, Substitute the resulting value
of В into x7 etc» We get G). This result, which is simpler than, but
equivalent to, Cauchy's E), had been found otherwise by Sylvester,800
whose published announcement without proof was limited to the case
А = Б = С=1, and, for K = Q, but At В, С arbitrary, by DesbovesM* and by
P. Sondat.*» From the fact that G) satisfy Ax*+By*+Cz* = Q, we have
the identity
This leads to solutions of the system of equations [cf. Bini438]
or
Desboves304 simplified Cauchy's proofs of B) and E), gave also a direct
proof of B)> and showed that a2 divides ^@, wy —v)f etc., a fact seemingly
overlooked by Cauchy. Hence we may take x=F(Q, w, —v)la2, etc.,
obtaining polynomials of degree 4 for x, y, z. As new results, he proved
that if one solution of ^=0 is given we can reduce its complete solution to
that of a biquadratic equation. He sought an F such that the latter is
Ap+Bi}4 = C$*t where C=A+Bt the only biquadratic hitherto solved com-
completely. The resulting F is
He obtained the solution off(xf y)+c£=Q, with coefficients of special type,
given solutions m, n of the cubic f(x, y) — 0.
A* Holm106 noted that the tangent to a cubic at a rational point, not an
inflexion point, cuts the cubic in a new rational point. In case there is a
*" Nouv. Atxnu Matt» B), 18, 1879, 4OL Same by R. No
"»Ап*г. Joor. Math., 3,188% 61-*; СоЛ. Papcts, 3,1909, 354-7.
"■ Noot. Ann. Mmih., B), 20, 1881, 173-5; C), 5,1888, 563-5.
>■» Ibid., B), IS, 1879, 4ОГ-&
**lbuL, B), 19,1880, 459.
"* UmL, &\ 5,1880, 545^7».
••Proc Sdmborsh M*tii. Soc^ 2% 1ШМ, 4a

592 History of the Theoby of Numbers. [Chap, xxi
rational asymptote, the line parallel to it and through a rational point cuts
it again in a rational point.
A. 8. Werebrusow** obtained solutions of D) with K=0 from one
solution.
B. I^vi307 considered a cubic equation with rational coefficients which
corresponds to a cubic curve С of genus unity (transformable birationally
into a straight line) and determined points on С by use of an elliptic param-
parameter. By a configuration of rational points on С is meant the set of all
rational points deduced from one or more rational points by the operations
of finding the tangential point to a given point and finding the third inter-
intersection with С of the secant joining two points of the set* There are
theorems on the number of points in a finite configuration of such rational
points (ef. Hurwits511). There is a discussion of the cubic
into which any cubic with a rational point can be transformed birationally.
A. Thue808 considered АзР-^В]? = Cz* in which xt у, г are relatively prime
in pairs and z±y^x>0. We can find integers pt q, r, without a common
factor and numerically < V3z, such that px-\-qy-rz. Hence
ax = Ctf-Br*, by^Ar*-Cp*, сг=Ад*-Вр*у
where at b, с are integers. Hence Aaz+Bby=Cc2. From this and the
former linear relation we get the ratios of x, yt z. He introduced further
numbers and deduced many relations with the aim to obtain limits for
a7 Ь7 с, etc.
L. Chanzy509 applied Lucas'29* three methods to the equation
The tangent at {x\7 yi) meets the cubic at the point with the ordinate
The line joining the known points (xh yi)} (s2, yt) meets the cubic in the
point with the ordinate
while xs follows from (x*—si){y2—yi) = (y*)
L. J. Mordell11* considered a ternary cubic form F(x, у, г). Given one
set of solutions, we can find a linear unitary substitution which transforms
^»0 into jSi$24-2jS2$+5»=0j where Sj is a function of degree j of 17, f»
Its discriminant f=S%—8i8t is a binary quartic whose invariants are
'» M&tem. Sborn. (Math. Soc. Moecow), 27,1909, 211 227.
^ЛШIV CbngreatthtOTUULMAt^ Водиц 2,1909,173-7. Supplement to hk four papers,
ЛШ R. AocacL Sc Torino, 41, 1906, 73&-64; 43, 1906» 99-120, 433-434, 67ЭД81.
»>Skitter Vidennfrupwdak. Krietuni» (Mftth.}, 1,1911, No. 4, pp. ]942l; 2,1911, No. 15,
7 pp. Tbe7^^No,20«c<uwdereduiKiefThiK17'of СЬ. ХХШ.
•«Sphinx-Oedipe, 8,1913, 166^7.
ш Quar Jour. Mftth, 45,1913-4,181^.

Ternary Cubic Form Made a Constant. 593
numerical multiples of the invariants S and T of F, If Si=brf^c^ f is a
square for v = c, f- -b Thus (Mordell"» of Ch_ XXII) if we find17* MI
rational solutions of
we can deduce all the rational solutions of F=0. The method is applied
in detail to the canonical cubic я?+у*+2?+0тхуг=*0>
W. H. L. Janssen van Raay*11 solved yjz+z/x+xjy = 3 in integers by
reducing it to a*+J?+c?=3abcr
A. Hurwite111 proved (p, 226) that a curve D) with integral coefficients
has either no rational point or an infinity of rational points if At В, С are
not sero and relatively prime in pairs, while no one of them is divisible by
a square of a prime, and at most one of them is ±1. Next, if A =B = 1,
Сф±1, and С is nob divisible by a square of a prime, the curve has either
1, 2 or an infinity of rational pointe. Finally, if A = B=C=1, X+l,—3,
—5, the curve has 3 or an infinity of rational points. There is a discussion
of cubic curves without a double point (genus 1), the coefficients of whose
equation belong to an algebraic field. A rational point is one whose coor-
coordinates are proportional to three numbers of the field. By use of an elliptic
parameter, there are found all complete sets of a finite number of rational
points, such that the line joining any two (distinct or identical) meets the
curve in a point of the set. The most general cubic curves with exactly
one or exactly four rational points are determined. Cf. LevLw
M.Weill,Jllestarting with one solution а, Ъ,с of Ax*+By*+Cz*=0, wrote
x=a+\Bf у=Ъ+Х6, z=c+Bf and equated to zero the coefficient A\a*
+В\Ъ7+Сс? of 35, and hence found $ rationally, thus obtaining the second
solution C) due to Cauchy. Given two sets of solutions a, Ъ, с and a\ &', c\
he wrote x=a+$a\ ete., found В rationally, and obtained Desboves>30J
special case of Cauchy^s E).
Ternary Cubic Form made a Constant.
J. L. Lagrange1*1 determined cubic forms F(x, yt z) whose product by
F(X, Y, Z) is of that form. Cf. Iibri«> • of Ch. XXV,
GL L. Dirichlet111 employed the roots a, 0, у otf a cubic equation with
integral coefficients and without rational roots. Let F(x, y, z) denote the
product of x+<xy+o?z by the similar functions of p and 7. First, let a
single root a. be real. If T> U, V form a fundamental solution of
F(T, TJtV) = \t and Xf Y, Z form one solution of F(x, y, z) = m, an infinite
set of solutions of the latter is given by the development of
x+ay+c&=(X+aY+o?Z)(T+aU+o?V)\
One solution of any set can be found by a finite number of trials. But if all
three roots are real, it is stated that there exist two fundamental solutions
from which all can be found by multiplication and powering.
*" Wiflkundige Opg&ven, 12, Ш6,306-8 "
"ViertdjsriiBchrift <L Natutfor. GceelL Zurich, 02,1917, 207-29.
*"• Nouv. Aim. Math., D), 17,1917,47-61.
*»Beridrt Akad. Гж BertiD, 1841, 280-5; Wetke, I, 625-52,

594 History op the Твдбовт op Numbeks. [Chap, xxi
G. Eisenstein314 proved that, if pis a prime Zm-\-l, 27(xp~1-\ hs+1)
can be expressed in the form
where у = v-\-wp, z=v-\-tPp*, and щ i>, W are polynomials in x with real
coefficients, while p*+p+l^O, and pi, pi are the primary complex prime
factors of p. The product of two forms Ф is of like form» When Ф = 1
has real integral solutions other than u = l, y=z=0, an infinitude of solu-
solutions can be derived from one, as by PelTs equation.
С Souillart™ and E. Mathieu** proved that the product of two forme
x у z
z x у
is of the same form and stated that a like theorem holds for cyclic deter-
determinants of order n. This was proved for С by J. Petersen.316
E. MeisseP17 wrote (x, y, z) for x*-\-Ay*-\-A*z*—ZAxyz, where A is posi-
positive and not a cube» Let & = ). and x, y, z be integral solutions of
(*,у,ж)-1. Let
By the product of this for the three vahies of в, we get {x, y, z) (a, b, c) = 1.
By the three equations which follow from the above,
a=x*—Ayz, b=Az*~xy, c^jf—xz,
which give a second solution of (x, y, z) = 1. An nth solution follows from
(x-\-$yp+№zf?)n. Solutions of (x, yy z) = l are found for each Л<82.
G. B. Mathews31* proved that if the integer m can be represented by
F{xf yt z) — 2*-|-rtj/3-|-rc*z*—3nxyzt
it can be represented in an infinity of ways. F{x7 y7 z) —1 has integral
solutions and all solutions can be derived from a single fundamental solution
£, iy, f by use of
H. W. Lloyd Tanner319 wrote ф(х, у, г) for the norm of x+y$+z1P,
where е*-\-Зкв-Ъ=0, and called u+v6+w& a unit if ф{и, vy w) = l. He
obtained a correspondence between the units and the proper automorphs
of Ф, i, e., linear transformations of ф into itself, and investigated improper
and associated automorphs.
H. S. Vandiver12* noted that the chxulant (cyclic determinant) of order
л is a product of n linear factors
™ Jour, fur Math., 28, 1844, 280-303.
«• Nouv. Ann. Math., \7? 18Л8,192-4; 19, I860, 32<Ъ2. Ct. Math. Quert. Educ. Tim«^ 63,
1895,35-6.
«♦TidMkrift for Math., lS72r 57.
*" Beilrag zur Petl'sehen Gteichung buherer Gmdet Progr., Kid, 1801,
^Proc. London Math. Soc.t 21,1891, 280-7- On ^-0, see Martlet" of Ch. XXIII.
♦" /ftirf., 27, 1806-6, 187-199.
*• Amer. Math. Monthly, 9,190C,

Chap. XXIJ ВГОРВАЭТВДЕ EQUATION OP DEGRfiLE ТнКЕЕ. 595
where «is a primitive nth root of unity. The product of two circulates of
order n is such a eirculant. Hub is used to prove that
has an infinitude of integral solutions for every pair of integers ny a.
Jt. D. CarmiehaeF1 proved that every prime ФЗ is repre&entable in
one and but one way by f=j?+y*-\-£~3xyz, where x, у, г are ^0. All
positive integers are representabte by / with x, у, z each ^0, with the sole
exception of the integers divisible by 3, but not by 9. A prime 6n+1 can
be represented in one and but one way by / with at least one variable
negative.
A. Cunnmgjham822 considered primes of the preceding form /. Take
then /3=7 = ±lf /=3x±2. Since any prime p>3 is of the last form, we
get positive integers x, у such that / represente p. Next, let A = 1; if В
is prime it is of the forms 6a>+1 =#+3fr
E. Turritee8* noted that the above form / represents the rational
number n when x=nty=n+l}Z>z~n-1/3. If n^l {mod 3),it represents
n b()JZ (
MlSCBLLANBOUS SlNGLE DlOPHAKTTNE EQUATIONS OF DEGREE THREE.
Bhiscara12** noted that the sum of the eubes of yy 2y, 2yt Ay equals the
mm of their squares if 100y* =30£*, whence у - 3/10.
Т. Robinson1** found two cubes x*, Л1 and & square wV in arithmetical
progression^ since 2fV=xj-|-toV detenomee x rationally*
A. J. Lerall*21* noted that, if л cubic equation has rational roots, its
discriminant is a square.
J. Iu Lagrange1*4 empbyed the " tangent method " to determine new
solutions of the cubic equation f{xf y) =0 from one set of solutions p, q.
Set x=p+t, y=q+u, and take
Substituting the resulting expression for и into /{p+i, ?+") =®, we may
delete the factor ? and thus express I, find hence u, as a rational function
of the partial derivatives of A, Cf. Lagrange*2 of Ch. XXII.
To express l*-f-2* as a sum of another square and cube, J. CunlinV"
took 9=ifi+B-*I, v=21x*-6x-l, whence x=253/441. J. Whitley took
9=x*+C-nx)*, whence 2s+n*^ -Й4П-И1, which equals 5+pq—q* if
**» Bull. Amer ^ ,
*° M*db QueeL And Sohitkine, 1,1916,1445.
-• Уаитпукпжпк vkkOl, IS, 1916,417-30.
««VliArgiaiU, §119r Algebra . . . fnm intent <tf Brahmegvpta uid ВЫвсмчц tnuuL
by СоЫжюкв, 1817,200.
ш The G^iUaium;B Dtuy, or M&th. R*podUwT» London, No, 25, 1765; Dayib' ed., 2t 1814,
9&
«"• Euler'e Open poatmtu, lf 1882, 504-6 (»bodt 1770).
*■ Nouv. лЬл. wc»L Berfto, шюбе 1777,1770, 153; Oeuvree, IV, 396.
« Им flwflmin'a Matb Gompwuon, Loodom 2, No. 13, 1810,220-1.

596 History of the Theory op Numbers. [Chap, xxi
+-g1I. The last holds
if 10p=28, $--51/60, whence « = 15/16. Cf. Gerardin.848
W. Lenhart1* discussed S(a:?+^) = S(yJ+^), where t = l, ■-, n.
Assign any values to xp> y< (* = 3, - • ■, n). Then seek numbers (in his*06
table of sums of two cubes) t=x*+zl, t'=Vi+yl, such that
where f depends on the chosen values of x<, yif i^3. For n=2> he found
xi = 5, Xi=6, pi = 7, y* = l. For n>2he tooki=f and found
A2, 5, 1; 11, 8, 2), A4, 13, 11, 8; 17, 12, 5, 3),
B1, 14,10,4,1; 20,17,5,3,2).
B. Peirce {ibid*) took Xi—aa+hi, yi=*a&+bn^i+l and found that the
condition gives
R. Hoppe3*7 considered the rational solutions of л^+^=х—у. Set
y=x(l-u)/(l+u). Then x and ^ are rational in и if □
If « is a solution,
is a second solution, etc. The nth such solution is found.
C. Hermite828 noted the solution x=a{ab—c*), y=a*—№cf г = Ъ(с?—аЬ)}
и=а*с~Ы of
A) &у+№+г*и+и*х=*0я
J, Joffroy^ stated that a2—bJ^7>10w is impossible. A. Morel gave an
erroneous extension to а8—Р + ЦГЧ f-Ю"»
S. Realis**0 gave long cubic functions x, yy z, w of а, р> у for which
obtained as solutions of A):
as well as formulas of the third and fourth degrees*
T. Pepin532 noted that a surface of degree m is osculated at an arbitrary
point of a given surface only when there is a positive integer n satisfying
and proved that 1, 5, 20 are the only integral values <675 of m. E. de
Jonquieres used the discriminant of the quadratic in n to show that
«• Math. Mbcellaoy, New York, 2, 1839, 96-7; Extract, Sphin*~Oediptv Bt 1913, 93-1,
■" Zwteohr. Math. Fbye., 4,1859, 3SQ-61.
«« Hour. Ann. Math., B), 6,1867, 95.
*• Nouv. Ann. Math4 B), 10,1871, 96^6, 288.
»» Nouv. Cotrap. Math., 4, 1878, 34^52,
» Nouy. Ann. Math., B), 18,1879, 301-4.
*" Jour, d© Muth., C), 7,1881, 71-108.
««Atti Acead. Pont. Nuovi linoei, 37,1883-4,183-8.

Chap. XXI] DlOPHANTIfrTE EQUATION OF DEGREE THREE. 597
, f,,,; >
whence m=l,n = l, if m< 1000.
КёаНв» noted that the double of any square, as well as the triple of
the square of any even number >2, equate the excess of a sum of two
squares over a sum of two cubes.
M. WeilP" noted that A) has the solution z = pA, y=hp*—l> z=px,
u^-hpy, where A=ph?+1; also, x=HA*, y=-AB, г=Я*А, и = КИВ>
where H=h'-p, B<=p*h+Sph?—h*+l. The last solution is based on the
identity A*-hIP = (l+hb)B.
H. S. Vandiver «md W. F. King proved the impossibility of
G. Bisconcini316 noted that х*-у* = (х+у¥ has the single solution x = 1,
y=0, in integers; xt+y*=x1-\-yi* has only the solutions x=lt y=0 or 1;
(х—уУ=ху or a'+j/3 bas various solutions.
References3" on cubic equations with integral roots are in place.
A. Cunningham1* noted that one method of solving х*±у*=2?+и7 is
to make x+y and з*~xy+y2 both sums of two squares.
A* Gerardinm satisfied 2х* = 2у*Ъу taking
and equating the coefficients of m2 (thusdetermining ^),so that m is found
rationally. Another method is to take ^=0.
R. Norrie*4 noted that from one set tfif ■ ■ •, aft of solutions not all zero
of a homogeneous cubic equation in Xlf - ■ •, Xn we can in general deduce
further sets by substituting Х^=гг^4-а» (* = 1, ••■, и), tbus deriving
аг'Н-^Н-тг^О. Since у is linear, we can make y=0 by choice say of xn
in terms of xi, ■ • -, x*-\. Then take r= — filet. The method is applied to
^J-!^) = u42^ and to
As to this method see Lagrange*2* and the related method of Cauchy287
and Lucas.**
A. Cmmingham and E. B. Escott*" made xy(x+y)±l a cube, where
l=x—y or 2x+2y; also xy±2(x+y) is made a cube.
Welsch*11 noted that 1, 2, 3 are the only three positive integers whose
sum equals their product. For n integers see papers 150-2 of Ch. XXIII.
A solution**2 of 2sJ—2yJ = S«J is x» y<-i(«S±w,'). This and other
solutions are found by decompositions of wl=x2—y*.
** Nouv. Ann. Math,, C), 2P 1883, Я&яГ " '^~
ш Nouv. Aim. Math., C), 4, 1885, 184-8.
«*Amcr. Math. Monthly, 9,1902, 293-4; 10, 1903, 22, Cf. Euler»; abo Hurwiti^ of Ch.
XXVI,
"♦Periodieo di Mat., 32,1907,12M.
M» L'iDterorfdiairt dee math., 15, 1908, 47-^8, 152, 239; 16, 1909, 208.
«•/WA, 18,1911, 210-3.
m BidL Soc. Phibmathique, A0), 3,1911, 226-233. Cf. paper 285 above.
«■ LJintermediaire dee math., 19,1912, 164-6, 273.
^/W.,60.
«TWA, 20,1913,190, 23W0.

598 History of thb Theokt of Numbers. CChap. xxi
L. Aubry343 noted solutions, involving two parameters, of
xyz- (tf+^-f-^Jtr+^^O.
Special solutions (p. 207) are given for 6^7, 61, 2281, 99905 of
L. Aubry*44 treated tf+z-f y*+y=£+z by setting x+j/=2u, x-y=2v,
z—put whence 2(ii2+3tf2-hl) = p(p*ttI4-i), which is solved as a Pell equa-
equation in u, v.
E. B. Escott"* treated the preceding problem by setting y=x-\-d,
г=х-\-Ъу x = k{b—d), and found eight sets of solutions. Next, for
get y=x+d, z^e—x, a=x+bt d=b+ke. The discriminant of the resulting
equation forx must be a square, 9s2. Thus к—Zn—I. For n=0,
2x=e-by 2y=b-et 2г=2а=Ь+е.
For n= 1, we get the solution (in which p ie a rational parameter)
£ = B1e*+4)/p.
He gave (pp. 126-7) solutions of each of the equations x?±;xy-\-y*=z*t
x*±x*y*-\-y*=zz. L. Aubry (p. 47) reduced x*—xy+y2=z2 to a Pell equa-
equation by setting %(x±y) — vy u.
A. GeYardin** noted tliat, if al-&=f*-g*, then
becomes a quadratic equation for m. By equating to zero one of the three
coefficients, we find new solutions of x2—y* = F*—G*. Cf. CunUffe**; also
Realist «of Ch. XX.
P. Bachmann847 solved &3 — (p\-\-pl-\-pb}k=2piptpx in positive integers.
We may assume that pi = kjci (i = l> 2, 3), к=/к\кЖг, where /—1 or 2.
Multiplying the given equation by /ft|, we get
The factors on the left are equated to n*f and ml respectively, by use of
solutions of x1—h2—na3.
Cashmore'48 stated erroneously that л^-f y*=u*+t? for
z, y=
R. GoormaghtighMg solved ^-f2x+^= □-
T. Hayashi3*0 proved that z2y-\-y*z+z*=0 is impossible in integers Ф0.
ш L'interm&liaire dee matb., 20,19LS, 95.
"Sphini-Oedipe, 8, I9l3r 46-7. a. Lenbart~
**lbid.t 1234»
»- Ibid., 14.
" ArchJv M&tfa. Phye», C), 24,1915, 89-90.
"* L'intermediaire dee math., 23, 1916, 224.
4
Nouv. Ann. Math-, D), 16,1916,

Chap. XXI] EQUATIONS OP DEGREE ТНЕЕВ IN TWO UNKNOWNS. 509
E. Maillet1** discussed y*—y=<?(x*—x), where € is rational. For each
value of с there Is only a finite number of integral solutions.
Solutions***6 have been found for the equation in binomial coefficients
*-+--=»■-■
The sum of the first n odd cubes can1**6 be expressed as a sum of seven
squares Ф0. Special solutions of я? +y*+z*=к(х+у+г) are noted (p. 155).
On l(mi?-\-iuf) =4X(mrt^-nai)t where I, X are linear functions of p> q> r, 8,
see papers 48, 51,55,80,89. On tu*+Pv=Auvi, see Lagrange, p. 572. On
*(«)+*(*) =0, see ВаегЯ*
On x(l-x*) =Ay*, see Tweedie74 of Ch. IV.
Onz+a/x^y*, sec Leibniz" and Terquem70 of Ch. XXII. On equations
of degree three involving products of consecutive numbers, see papers 28,32,
56, 58, 59, and 63 of Ch. XXIII. On xy(x+y) =Az*, see Euler10, Lucas189,
Catalan,» and Hayashi21*; also Lucas*" of Ch. I. ChuquetMof Ch. XII
expressed 20 as a sum of three positive rational cubes; on the general topic,
see papers 404-29; also Ch. XXV, end. On x2+y*+z*^hcyzt see the
papers cited under Hurwitz17' of Ch. ХХЩ.
Systems of equations of degree three in two unknowns.
Diopbantus, IV, 29,30, made xy±(z+y) cubes. Take у =x*—x. Then
the condition with the upper sign is satisfied and that for the lower sign
requires s1—2s*—cube^(§s)*, aay, whence x= 16/7.
Bombelli1" treated the same problem*
Bhascara* noted that the sum and difference of 4y* and by1 are squares
and theu-product 20tf* is a cube, (lQyI, if у=50. The sum of the cubes of
y2 and 2y* is %*, a square, and the sum 5y* of their squares is a cube, )
if y—25. Under Bhftscaraw of Ch. XII is given his solution of x—y =
Л and of ^Ч-^= П, y+z= П.
Euler*8 discussed x+y=U, x2-fy4=pJ. Hence take
ЬР)} у=Ь(За'-Ь*). ТЪеп x+y = (a-b)Qt Q=a*+4ab+b*. Set
a-fr=c*. Then д=бЬ1Н-6Ьс2+с4=(сгН-ЗУ/^ if b/c*=2fir@-/)/C.p-2^
Iben x and у will be positive if h=2g(g-f)> &=$p-2tf. The latter is
satisfied if/=11, 0=1, c = 19, orif/=-3, g=l, c=5, whence 6=8, a=33,
^-29601, y=25624. bV>r three numbers he gave only results:
35+9+5=7*, 35J+9J+52=11I; 67+9+5=9*, 67*+92+52=192.
[But the last sum equals 5'919ф1еЛП
W. Spioer,^64 to find two squares whose sum is a square and difference a
cube, took a = \x*+\x* and Ъ=\х2—\х* as the squares with the sum x2
*«■ Nouv- Ann. Matib, D), 18,1918,2B&-292.
«* Z«tedmfi Matt NAturw. Unterricbi. 50,1919, 95-6,
L16dii d b 26,1919, 778 l(»10
-' L'jfebr» oper» di 1Ы»1 ВошЬеШ, Вокдеи, 1579, 553.
^VQft^mta^H^l^ СоЫягооке,»- 201-2,
ш Opera poetmna, 1,1Ш, 255-в (about 1782).
т Ladta* Dmry, 1766, 33-1, Quest. 535; С. Hotton'a DUiian Miecellany, 3> 1775, 220; Ley-
b' M. Qu«t. LD,2,1817, 251.

600 History of the Theory of Numbers, echat. xxi
and difference x*. Choose the squares
*** A-n»J
J'
with the sum 1 and set a—cz*, b=*dz*t either of which gives x.
J. Leslie*56 made x-\-y and x*+y* squares, by division.
W- Cole386 made x—у, x*—y*, x*—tf all squares by taking x—y=a2,
х+у=п?аг> whence z1—у/*=П if Зж4+1-=П, which holds if m=2. J.
Young took m=2 initially.
J. Saul*7 made x+y=**, х*-\-уг—«* and s'+j/3 a square. By elimination
of у from the first two equations, e4—2s*z+2x7^v2. Let v—z2—rx. Then
x=s*B-2r)/B-r*). Then х2-ху+у* = П if r^&V-f 14r2-12r+4= П,
say (r*-3r+5/2)J, whence r=3/4.
To divide*" a given square a* into two parts such that the difference of
their squares and the difference of their cubes are both squares, an anonym-
anonymous solver called Ъ* the difference of the parts, whence the difference of
their squares is (ab)*- The quotient of the difference of their cubes by b2
is to be a square, whence За4+Ь*=П. Put a*=bzt x—2—z. Then
3x<+l=49_l j-a^is the square of 7-48z/7+12 51z'/49 by choice of г.
J. Whitley*" found two positive fractions such that each plus the square
of the other is я square, while the difference of their squares or their cubes
is a square. Let the fractions be (lzh4i^)/8, whose sum and difference are
squares. Hie difference of their cubes is a square if 3+16V= П—а2.
Letfl=£—zy \a=1—г+2Л Hence z=\ = v. B. Gompertz took x—oz and
y=tz as the fractions, where а = A+?)/2. 1Ъеп х*—уг=П. Take
s+y^jtfz*, y+x*=tf&. We get two values of г which are equal if
<**(q—a)=t(p—f),q+a=s(p-\-t). These give p and q. Then x2—у* = (гг)г
gives zy which equals the earner value of z if ся (а^—/*) (а—si) = П y where
с = 1/(а1-^). Take«=3r«-h Hence x=5/32, у=3/32.
S* Jones made x-\-y=a*t x*-\-y2^ D — (bz—fyJ by choice of x, y*
Then ^HV^D if Ъ*-2&+2Ь*+2Ь+1 = П^{Ъ*-Ь+Ь)\ whence ^--J.
W. Wright took a—I, proceeded similarly, and found у from
Then l-ЗуЧ-З^-П if т4-Ът*+ 14тг-12т+4:=* D = (т2-Зт-2)\
whence m=8/3, y = lbi23.
Lowry*0* eUminated x—a2—у from x*+y* and x2—xy+y* and equated
the resulting expressions to the squares of a^—yr/s and a2—yeftstv)', the
conditions hold if w=l, 4r=3sy «=5s/4. J, Cunliffe took x=Rz—S2,
y=2R8, xt-xy+yt^itiP-RS+S*J, whence R^±8; then the desired
numbers are a*xf(x+y), a*yf(x+y).
шТпша Roy. Soc. Edbbufgh^, 1790,211,
«•Ladke* Diaiy, 1787,36-7, Queet. 853; Leybourn'a M. Qu«^t L, DP| 3, 1817, 155-6.
*> The Gentleman's Diary, or Math. Repoettory, No. 55, 17Й5; Davia' ed., 3, 1SI4, 236.
*"JWi, No. 56, 1796; Davis'ed.,3. 1814,245.
»• Tbe Gentleman's Math. Compankm» London, 2, No. 12. 1809,169-7L
*• Ibid., 3, No. 18, Ш5, 323-i.
•^ New Seriea of Math. Repository (ed., T. Leybourn), 3,1814,1, 169-172.

Chap. XXI] EQUATIONS OF DEGREE ТНЛЕБ IN TwO UNKNOWNS. 601
To find two integers the difference of whose squares is a cuoe and the
difference of whose cubes is a square, J. R. Ambler861 took x*+2 and x*—2
as the numbers, the difference of whose squares is Bz)z. The difference of
the cubes is a square if 3xe+4 = П = Bх*—2)*, x=2. J. Davey used the
numbers x, у and set x2—yi=2?, x+y~n*z, which give x, у in terms of z.
Then x*-y*^U if 3ns-f«* = n = (rn4-z)*, which gives z.
W. Snip*2 made x*+y2 and x2+y* squares by t king x=(m7— n2)vy
y=2mnv. Then s'-f-j/3 = а?ЪЧ? determines v rationally.
J. Anderson868 made x+y a square and x—у, x2+у* cubes by setting
(p±qi)\ Thenx-y=p*-3p*q-Zpq'+q*-(q-pytip=3q. Hence
<, 0=26V, x+y = U if g=ll. Ashcroft used the numbers (x*i^)/2
whose sum is x* and difference is x*. Their sum of squares is Dx*-\-4x*)l&,
which is a cube if 4x44-5', x = 11/2.
S. Ward"8* took y=x+Y, Г-8г», s=Fz. Then (^-НуЧ/У1 equals
2z42«+l, which is the cube of 1+2^/3 if 2=9/4. Then*+jf=44r*=n
ifr=ll.
Several found two integers whose sum is a square and di Terence a
cube, while if each number be doubled Ihe new sum i* a cube and difference
a square. Take x-|-p=4af, x—y=86e.
To make x-yy tf-y2, tf-y2 rational squares [Cole3*], J. Whitley365
used the numbers 3 = 2z*-b2t^ y^2z*-2it; then shall
which is true if г=уог z=2v. H. Godfiay look х=тг+пг, у=2тп; then
дл_1_^_1_^=(то«+ mn-|-5nV2)? if n-= -4^/7.
Several8*1 solved x+у=П, x2-f^ = p, ^4-^=x2-|-^
Several8*' found two numbers the difference of whose squares is a cube
and difference of cubes a square.
EL W. CurjeF8 found two numbers T, у whose sum and difference are
squares, sum of squares a cube, and *um of cubes a square. By the first
and last conditions, x*—xy+ti*= П, which holds if х=гBтпп—пг),
y=z(m*—nr). Then y±xare squares if m=9t ra—4, z~ D, whence x=56zt
у=65*. Then хЧу*=7361*2* Thus take г=7361\
P. R Teilhet8" stated that all pairs of numbers whose sum and sum of
squares are squares are (A*—B*)M*N and 2ABAPN, where A and В are
relatively prime and not both even, N~A*—B*+2AB, and where M*N is
an integer. He asked when also the sum of their cubes is a square, as for
345,184.
Dimiy, 1*10,38-9, Quest Л 291; ЬеуЫи1гптв M. Queet-L. D., 4P 1817, 221-3,
m The Gentleman's Moth. Companion, London, 4, No. 20,1817, 659-60.
*■/bid., 4, Na 21,1818, 719-21.
m- Young's Algebra, Amer. ed., 1832, 342-3»
"Lutes' IHmiy, 1821, 32-5, Qucrt-1362.
ш ТЬ« L*4/b and Gentleman's Duty, London, 1849, 49-60, Quest 1779.
~ Math. Visitor, 1,1880, lOO-l, 136.
"Amer. Math. Monthly, 1» 1894, 95^6, 325.
m Math. Quest. Educ. Times, 62,1895, 51-2.
"•L'mterm^diairc dee math., 10,1Й03,12#. a. papers 139-40 of Ch. XVI.

602 Нюютт of the Theohy of Nuhbebs. ГСяар. xxi
A. S. Werebrusow*" found an infinitude of solutions of the last question.
Teilhet*71 gave a more general treatment of the problem.
A. G^rardin1" treated the system *4-fy* =al+W, x+hy=a+kb, and
found many solutions, such as
z, a~ (9m*-1H*^18*40-3jP; у, b=(9mt
In rintermediftire dee m&ih£matitaeiu are dietueeed tbe problems:
*, p^S, 196; 23,1916, 68-9; 24,1917, S5-&
^, 23,1916,141-2,
22, 1915» 53, 232; 24, 1917, Э9.
26, 1919, 145.
Systems of equations of dsghee three in three unknowns.
Diophantus, IV, 6 found that s'+z* is a square and з/'+г1 a cube for
y-lQ!7fx=Zy,z=iy. In IV, 7,8, he found that x3-}-^ ш a cube and ^-j-^
a square for s=5, у =*5, z=10and for x—40, y—20, 2—80.
J. de Billy propos d the problem to find three numbers such that if their
product is subtracted from any one of the numbers or from the difference
of any two or from the product of the second by the first or third or from
the square of the second, there results always a square. He expressed his
belief that 3/8, 1, 5/& is the only solution.
Fermat173 replied that ftf the numbers are denoted by Ay 1, 1—A~\ the
problem reduces to the double equal ty
which has an infinitude of solutions. In addition to de Billy's solution
A = 3/S, Fennat gave A = 10416/51865.
Malezieux*74 proposed the problem to mid three rational numbers in
A. P. such that one obtains a square by adding to their product either the
difference of the s uares of any two of them or the sum of the three differ-
differences of the three numbers.
E. Fauquembergue1™ gave the solution 1/31, 25/689, 1/19.
J. Ozanam** asked for three numbers in G, P. such that one obtains
squares by adding to their product the square of each number, and such
that if these fractional squares are reduced to their simplest forms the sums
by twos of the square roots of the numerators are three cubes in G. P.
" J. Hob7* solved the first part, saying the entire problem is impos-
impossible.
"» L'interm^diairc dea math., 10,1003, 319-20.
™1Ш., П, 1ИН, 167-70.
'"Sphiox-Oedipe, 5, IftlO, 1-12.
'» Oeuvrt», П, 437, letter to de Billy, Aug. 26, 1659.
171 Unedited letter to de Bflly, Sept. 6,1675. Cf. P. Tannery, I'iaterm&toure <Ы math., 3t
1896, 37. Element» de Geometrie de M Ле Due de BourgogneT par de Mal&eux, 1722.
14 L'intenn&U&ne dee math., 6, 1899, 115-6.
'* Unedited letter to de ВШу, June 25, 1676. <X P. Tannery, I'mtoinediatre dee matb,
3, 1896, 57; С Henry, Bull. BibL Storia Sc. Mat. e Efe., 12,1879, 517.
'" L'intermeduke dee math., 4 1897, 253.

, XXI] EQUATION» Of De£HIS ТННХШ Ж ТДКЕВ UNKNOWNS. 603
К. Fauquembergjue*7* called the numbers xjyt x, xy. Then
A)
are made squares by removing the factors x*7 and making the product
(*+tf*)(s+l)(s-H/y*) a square by Fermat's method. Setting y=a/0,
we get x=Nli4<*ifF), where
Then A) are the squares of
These fractions are said to be arithmetically irreducible. The sums by
twos of the numerators are 2Na*7 2Nc?fPy 2Npf which are in G. P., but are
not made cubes as required.
L. Euler** desired three rational cumbers whose sum, product and sum
of products by twos are all squares. Denote the numbers by nx, ny, пг.
Then
Then i?xyz=n requires n = m*xy(x+y)(xy-i?). By the sum of products
by twos,
Set xy—u*—u2, x=tv. Then the preceding condition becomes
Set 5 = f—г and multiply numerator and denominator by Р-\-\—&. Thus
national values of fare found from r=l, 3/2, 3> 9* The simplest numbers
derived from r=3/2, *=6G/1&, are 705600/rf, 106/4157, 361/557, where
d=23t5449. The corresponding integral solution* are 7О56СКИ, 109172c?,
15OO677<£ Euler*' had expreeeed his betief that these give the least integers.
Ё. Fauquembergue*" used a simpler method and obtained
whose product is a square, sum is (cP+l?L, and sum of products by twos is
) Fora=2, Ъ~19 we get 80, 225, 320.
.T 5, 1896, 86-7,
»» Novi С«аш. AcwL Fetrep^. S, П6О-1, 64; Comm, Ari4h^ I, 339; Op, Ош, A), II, 519.
**» Слтеяр. Metk Pby*. C^-r Рш»)р lt 1343s B31, Aug. 23,1755.
m L'interm&fimze dee za&th., 6,180^ 96-90.

604 History of the Theory of Numbers. CCbap. xxi
To find181* three integers whose sum and sums by twos are cubes, take
х+у±г=ф±п)\ х+у=Ъ>,
Then
if
J. CunlinV* noted that z-\-y—z=a*t х+г—у=;Ъ*, y+-z—x=<* imply
x+y±z=<?-\-hl-\~<?7 which has been made a cube by many writers.
Several882" found three squares in A, P. the sum of whose square roots
is a cube by using tbe know expressions vBpqzb^Tpl)r fG^-hg1) for the
roots, and equating their sum vkt where fc=pt-h4pg-hgI, to «*£■.
J. Anderson*81 m&dexyz±l, xy±l9 xz-fl and yz-hi all squares by equat-
equating the last two to (pz— IJ and (qz —1)\ whence %=p4—2pt y=tfz—2q.
Then the first two will be the squares of l-h2pgz and pqz—p—q if
2=4-h2(p-h$)/(pg) and p=g+l, respectively.
To find three numbers whose sum is a cube and the sum of any two a
square, J. Foster1*4 took 2-fy=mV, x±z=mfbiyy-\~z=m*<?, x+y+z=<Pm*t
whence от = (а*+Ь2Н-с2)Д2й8), Many solvers used the numbers 2(s*-hy*
-z*)f 2(х*+гг—у*)> 2{&+y*-x*)t whose sums by twos are squares. To
satisfy 2(x2+yt+f)=Stfi, take ^=-4p6-nI=(cn-2p3J, which determines
n, and take z=2mnl(m*-hi), whence n1—z*=D^s*.
W. Lenhart1* found that the sinn, and sum of any two of, 1982015,
2759617 and 44286264 are cubes; they are the excesses of 366* over the
cubes in
S> Bills1* obtained the same result from
The root z involves the square root of бис3—3^—18^+9 which is equated to
(>v(x+2a)(x—ay. Tor the resulting value of x, 6t?(ar-h2a) = П if
3 = П
whence i?=3/Dai). Takea=^J. JSeveral writers1*7 solved the same problem.
To find three integers in arithmetical progression whose common
difference is a cube, the sum of any що less the third is a square, and the
aum of tbe roots of the resulting squares is a square, S. Bills"8 took з*—у3,
x*i x*+y* as the numbers. To make s*db2jf squares, take x=uyt whence
u*±2y are known to be squares if w^Jf^-hg1), у=2р#*(р*—д*). It remains
k)(?^f2D^(^^
(phpg^)^pgq,yp9(ppg^)^
thue finding t. Other solvers used the numbers x^xy+y*, tf+y2 in A. P.
"" New Sous of Math. Repository («L, T. Leybouro)r 2, 1809,1, 31-33.
*» The Gentleman»* Matk Сотрашов, London, 3, No. 14,1311,282^3.
»- New Seru* <rf Maih. Bepoeitory (ed4 T. beyboorn), 3, 1814, I, 111-5.
ш Tbe Gentleman's Math, Companion, London, 5, No. Щ 1823, 238-9.
« идЫ Diary, 1826, 35^6, Quert. 1434.
** Math. MMcellany, New York, 1, 1836,123.
** Math. Qoeet. Educ Time, 12, 186ft, 30,
» Matb^^tor, 3, 1807, Wr-8.
« Math, Qo«t. Educ. Тшм», 12, I860, 91-2.

Chap. XXIJ EQUATIONS OF DEGREE THREE IN THREE UNKNOWNS. 605
and took x =* 2mnf у=тг—п2* Then shall m*+4mtt+nJ= П, say
which gives m/n. Take p=3/2. Then xy is a cube if «=300.
J. Matteson1* solved the last problem and that with the final condition
replaced by the following: The sum of the roots of the squares is an eighth
power, the squares being a seventh, a fifth and a fourth power, and the
arithmetical mean of the required numbers a square.
To find three positive integers whose sum, eum of squares, and sum of
cubes, are squares, A* B. Evans** took ax, ay, az, where a=x+tf+z, and
set а(з*+уЧ^) =аХх-у±г)*, whence y*±y(x+z)-3xz=0. The radical
in у is rational if э?+1Ш±г*=П = (гт/п-хУ, which holds if x=m2—n\
2=2тпп+14*Л In the resulting expression for 2x*, set m=p—8n and
equate to the square of p*— 16pn—83n*. Thus p/n=1332/83. Then
аз*= 412095790665, etc. Several solvers used 15mz, I5my, Ът(х+у), whose
sum of cubes is divisible by their sum. Thus a linear and two quadratic
functions are to be squares, which is true if т=сР/{23(х+у)}.
D. S. Hart8" divided unity into three positive parts whose sum of squares
and sum of cubes are squared by taking xfs, y/s, zjs as the parts, where
*=x+y+*. The conditions are satisfied if s= □ , 2x*= U, 2x*= □, which
is the preceding problem. He895 found three numbers whose sum and sum
of squares are cubes, and sum of cubes a square. Let ax9, bx\ cz* be the
numbers. Their sum of cubes will equal (x1J if я=2<г\ То make 2a
and Za1 cubes, equate their product to (a+6—c)z; the roots of the resulting
quadratic for a are rational if &<+26*c-№c1+6^-7(ri= D. Set &-2c+d.
The new quartic is a square if d=35c/9 or 116c/315.
To find three integers whose sum, product and sum of squares are all
squares, S. Tebay5" used the numbers xyf x(x+y), y(z+y), while A. p.
Evans used xa*r ya2, xya1, with x-y-^1.
D, S, Hart1* found three numbers, say ах, Ъх, сх, such that if the sum of
their cubes be added to or subtracted from the square of each, the sums
and remainders are squares. Set d=at+b*+cl.
i^
give x=(^-6l)/d=(bt-A2)/(?, while cV=bd^=ikV, Pi2 give
By the numerators of x, &^2a*-p, f=4№—h1, Р=2е*—F. The first is
satisfied if a ^P*-bQ*, f=2PQ~P2+Q*. As in Diophantus V, 8, take three
right triangles of equal area, with the hypotenuses 49+9, 49+25, 49+64.
For P=7, Q=Z, we get a=58, /-2. Similarly, P-7, Q=5 give 6=74,
Ь-4в; P^8, Q=7 give c= 113, J=97. Hence we get ax, eto.
Problems solved in the American Mathematical Monthly: ТЪгее num-
numbers the sum of whose cubes is a square and sum of squares a cube A, 1894,
363), Three integers the sum of any two of which is a cube (p. 208, p. 279).
щт Collection at Diopbaotme Probteme, Washington (e<L, Mxftuk), 1888, pp. 5-7.
~ MMh QottL Ednc. Time, 17, 1872, 30-1.
■«Лн1.г21, U74»ll»-L
"Ли, 23,1875,31.
** ЪЫЬ. Vito 218
Viator 2,1882,17-18.

606 History of the Тшвовт of Numbers. £Сблр. xxi
Three integers whose sum is a cube and sum of any two less the third a
cube B, 1895, 86-7). Three positive integers the product of the first by
the sum of the other two a square and sum of their cubes a square™ (p. 196).
Four positive integers each less double the cube of their sum a cube G,
1900,4^-50). Three positive integers whose sum, sum of squares, and sum
of cubes, are all squares (9, 1902, 145-6), or all cubes B4, 1917, 240).
R. F. Davis and others3* made Я*+У*+2*, У*+Х*+Я« Z*+X*+Y*
all squares- Take Х=2A-у)> Г=2A+у), Я=2A-у*). Then the first
two equal the squares of 2BTjr+jf). The third is a square if y= ±4/3.
A. Martin*7 solved А*+Б*+С>= □, Л'+Д'+С^Я1. As the solution»
of the latter he employed the produete of a by the values given by Young.1*
Take n'+2=(n-rI. Then ZA2=^H becomes a quartic for q whose solu-
solution, found as usual, is a very long expression for q. Take r=3, whence
n<=7/6. Then я=а/0, where «=81420385, 0=11290752. Take u=6^.
Then А, Ву С are integers each of 17 digite:
A=11868013975030087, S-16269106368215226, С=88837226814909894.
M. Rignaux3" noted a solution of the last problem involving parameters
m, n, g such tbat m*+2ttl=U; in his numerical example, A and С are
negative, while А, В, С contain only 6 or 7 digits.
P. Tannery and H. Brocard*» noted tbat 3, 4, 5 yield by multiplication
54+72+90=6*, 54'+72'+90' = Ш*.
E. B. Escolt4* gave numbers without common factor:
3+4-6=1*, 3'+*-6*= -5s;
36+37-46=3', 36*+37*-46*= -3*.
H. Brocard401 gave 9+15-16=2», 9>+151-16x=2» and 24+2-18=2»,
24^+2»- 18*=2(r\
A. Gerardm408 noted that, if х+у+г, 2ж* and Zx* are all cubes, x, у, г
are in neither geometrical nor arithmetical progression. He and others4*1
noted special sets of integral solutions of s+v+z=c* s*+jf+sl=bs; also
values making e1—z—y, f—y—z, s*—x—z all squares or all cubes, where
s=x+y+z (ibi&,23,1916, 5-6); 8*-х,8*-уу8*-г all squares (pp. 157-9);
xyz±x*, xyz+y*, xyz±£ all squares B4, 1917, 37-8); e1—x—y, f—y—z,
s*-z-z all cubes B2, 1915, 220).
** Abo, Math. Quest. Educ. Ътж, 24,1913,63HI.
™ Matb. Qoeet, Edne. Time», 64,1ЗД6, 26.
»t Math. Мдвмше, 2,1898,254-5.
tn L^termedisire dee maUL, 34, 1917! 79-80. Щ corrected a misprint ш a citation of
Martm'e 9оЬ«коц conori^r quoted in 7t 1900,162.
«• Linton^dimini (he math., в, 1899, 190.
ЛЛ 7,1900, Ш.
14-
, 1914,384».
*, 23,1016, 172; 23,1916,93,

Chap. XXIJ CUBE OF 8хШ PbUS ANY One A CUBE. 607
TO FIND П NUMBERS THE CUBE OF WHOSE SUM INCREASED (OR DIMINISHED)
ВТ ANT ONE OF THEM GIVES A CUBE.
Diophantus, V, 18 £191 required three numbers z< such that, if s de-
denotes tbeir sum, jr'-f x€ jV—яЛ are cubes. Set xf = {a}-l)*3 [х> - A — a})*3].
Since Zx<=s, we have Baf-3)s* = l [=—11 For the first problem,
*3 DtkM,=2-ro, aj^2; then
if m^2/15; thus *=5/I8. For the second problem, 3-2а< = П, ,
Diophantus took the square to be 2J, whence ZaJ^f =^162/216. Hence
we have to express 162 as the sum of three cubes. Now 162^ 125+64—27.
By the theorem in the " Porisms," the difference of two cubes is always a
sum of two cubes. Having thus the three cubes [not given by Diophantus]
and 21«* = 1, whence s=2/3, we obtain the numbers я,-. <X Bachet:*4
Diophantus, V, 20, required three numbers x> of sum s, such that X{—s1
are cubes. Set s*=(<tf+l)A Then 2а*-ЬЗ is to be a square I/A Let
«i=m,af=3-m9a»-L Then 9m2-27w-h31 = n = Cwi—7J, say, whence
да ^6/5, s=5/17.
C. G. Bachet401 believed that Diophantus had found by accident the
square 2\ which 3 exceeds by a number expressible as a sum of three cubes
<lf and stated that he could not solve the problem if 2| be replaced by 2$.
He completed the computation omitted by Diophantus. [By Vieta's*
formula AI 64-37 is the sum of the cubes of 40/91 and 303/91. Thus
162/216 is the sum of the cubes 125/216, 20V(913-27), 1018/(913 8). Sub-
Subtracting them from unity and multiplying the remainders by eI=^B/3K,
we obtain the answers 91/27*, ete., which Bachet expressed as fractions with
a common denominator, but with the common factor 27 in all terms.41*
The reduced denominator is 549353259=91**27*.
A. Girard** noted that we may employ Bachet's value 2# since
3—2f=162/9" is a sum of three cubes. Or we may employ 2M which 3
exceeds by the sum 440/1000 of the cubes 216/1000, 216/1000 and 8/1000;
the resulting solution of Diophantus V, 19 is 49/256, 49/256, 62/256 [since
1(8/8I »(l/8)s]
]
Fermat406 would not admit that Diophantus was led to 2J by chance
and remarked that it is not difficult to rediscover hie method. " Take
я—1 as the side of the required square between 2 and 3. Then 3—(x— II
is to be the sum of three cubes. Take as sides of two of the cubes linear
functions of x such that, if the sum of their cubes be subtracted from
2-{-2х—з^г the result contains only two terms in x of consecutive degrees.
This тал be done in an infinitude of ways. Take I—z/3 and 1-^x as the
sides of two of the cubes; then the result mentioned is
Equating this to — cV, we have ^=117/B7^—26). We are to choose
"< Diophanti Меж. ЛгШь, 1621, 324.
*"Oeuvn*,III, 268^9.

608 History of the Theory of Numbers. [Chaf. xxi
с so that 1—x/3>0. Since the third cube is negative, we apply40* the
Porism. Here Bachet was again embarrassed; he confessed that he could
express the difference of two given cubes as a sum of two cubes only when
the greater of the given cubes exceeded the double of the smaller."
Ludolph van Ceulen407 A540-1610), at the end of his Dutch work on the
circle, proposed 100 problems the 68th and 69th of which are to find three
and four numbers such that if each be subtracted from the cube of their
sum the remainders are cubes» For three numbers his solution, communi-
communicated in letters, is the one published by van Schooten,408 his successor at
the University of Leyden. After learning van Ceulen's method, N. Hubert!
obtained answers for four numbers, quoted by van Schooten:
S67160/C, 787400/C, 13527640/7), 14087528/7)
(С=46574бЗ, Д-125751501);
12172736/*, 11296152/fc, 9112168/A, 4724776/Jfc (*=644S1201);
and the further answer for three numbers:
15817S15000/G, 9568152000/G, S925120000/G (G- 86526834967).
Frans van Schooten** first found three cubes such that on subtracting
them from 43, the sum of the remainders is a square. Let the roots of
the cubes be N—1, 4—N, 2. Subtracting their cubes from 64, we get
ббЛГ-ЛГ2, 48#-12ЛГ2+ЛГ3 and 56. Equate their sum
to A1+TV)'; hence #=23/10. Thus the above remainders equal
a = 61803/1000, b=59087/1000, 56.
Now let the three desired numbers be ал3, Ъп*у 56л3 and their sum ±щ
whence «=20/133. Hence the answer is
494424/7), 472696/Д 448000/7) G)^2352637).
Their sum is 80/133, whose cube diminished by the three numbers gives
as remainders the cubes of 26/133, 34/133, 40/133.
J. H. Rahn and J. Pell** treated Diophantus V, 18, 19, 20 at length.
Pell's solution differs little from van Schooten's, except in using A1+mNJ
in place of A1+iVJ, and is given in Wallis' Algebra, Engl. ed., 1685, p. 219,
with van Schooten's answer.
J. Kersey4'0 employed, for Diophantus V, 19, ai=*53/144, a2=27/144,
a3 = 16/144, whence 3-2a'*=B47/144}2, « = 144/247; thus the desired num-
** To secure Diophantua' value (x — 1J*™2$-,, we must take л ™5/2 or —l/2r whence
27cs=3&4/5 or —8*26, во that с is irrational* Hence Ferm&t's process is not general,
although it Jead* to a solution by setting С-5Д whence 2 "=13/11, and the eidee of
the cubes are 20/33, 72/33, —96/33, as n°t*d by Heath, u Ltiophantue," od. 2, 214.
"T Van den Circkel, 1596r 1615. Latin trans!, by W. Sneltiua, 1619. <X Bull. BibL Sterna
Sc Mat. e Ful, 1, 1868r 141-156.
<Oi Exercitatioovm Math., 1«57, Liber V, Sect. ]Д 434-6. Reproduced by С Hut ton. The
Diariaa МшоеШшу, London^ lt 1775,138-9.
'" Rahn*e AJgebra, Zurich, 1659' An Introduction to Algebra, tr&nsl. out of the High-Dutch
by T\ Brftacker, mach altered and augmented by D. P[eDl London, 1668, 105-131.
Cf. Walltf' Algebra, Ch. 59.
«•Tbfi Elemente of Algebra, London, Book Ш, 1в74г Ш-4Г104-5.

Chap. XXI} CUBE OF SUM PLUS Any One A CuBE. 609
bers are the ratios of 2837107, 2066301, 2981888 to 15069223. Or we may
take ai = 103/(9 23), 0,-12/(9 23), ^ = 1/9, whence s=9 23/1053, giving
^ = 7777016/43243551, etc. Or, ^=41/64, а, = 39/64, а,=3/63,
a =8 • 16/185, Xl = 1545784/6331625.
Or, ai=67/88, a*=87/88, a*=22/88, s = 176/221, ^ = 3045672/10793861.
For four numbers, take alf - -, «4 to be the ratios of 4684, 4836, 3485,
3315 to 1360. Then ZF4-a!)=^, ^16027/1360; the desired numbers
are Xi~A6-aV, s=2Xi = l/L His solution of V, 18 is the same as in
Diophantus.
The answer of van Schooten*18 was given without details in the Ladies7
Diary, 1717, Question 51. J. Hampson41* gave without details the smaller
answer to Diophantus, V, 19: 13851/7), 19467/Д 18954/7), where 7)=85184.
He"* also stated two answers to Diophantus V, 18: ratios of 23625, 1538
and 18577 to 157464; ratios of 18954, 4184 and 271 to 132651.
J. Landen1" took zy, zx, zv as the numbers in Diophantus V, 20, and
p*z as their sum, and z$, zr, zq as the roots of the cubes, finding the answer
341/Z>, 854/Д 250/7), where Z>=4913; no details were given.
The " Repository solution um is a repetition of that by Diophantus as
completed by Bachet,** leading to 162707336/d, 134953209/d, 68574961/rf,
where d^549353259. It is also noted that 37 is the sum of the cubes of
18/7 and 19/7, whence s =2/3 and a new answer is 68256/fc, 67229/fc, 31213/A:,
where к =250047.
For Diophantus V, 18, J. Bennett411 took nx as one number and s as
the sum of the three. Let а?+пх = ($±хУ, whence x=£V4n — 3s*-3$/2.
Taking n/*v~21, 31, 57, we get ns-63«*, 124s3, 342a3, which will be the
desired numbers if their sum щ s, i. e., if s=l/23. J. Ryley417 used the
numbers z, y, a—z-y; then a*±x=a*tf, a*-\-y=d*n% give x, y. Let
a?±a-z-y=mta?. Then а^=1, /-я^+и'+^-З. Take n=2-r,
s = l-hr, f={2vm—3rI, which gives r in terms of mf v.
T. Leyboum41* noted that Diophantus V, 18 is satisfied by taking
(a8—u6)^, (V—t*6)*1, (с1—иь)& as the numbers, if u2v is their sum. The
latter requires F=ot-h6'-hc1—Зиб = П. Take a-p+q, Ъ=т-р, c=s.
Then F=3(g-hr)^-h3(^2-r2)p-h^-hr8-heI-3u6- Take 3(fl+r)=n?; then
F = (m—np)z determines p rationally. By trial, he found that F=B3)* if
a=4, b=5f c=7, «^1. For Diophantus V, 20, he419 took ($V
)»1 as the numbers and v?v as their sum. Then
^ The Element» of Algebm, Loodon, Book III, 1674,101.
m Ladie*' Diary, 1747, 27, Queet. 27S.
"M, 1748, 27, Quest» 283,
*" iAdiea' Diary, 1749, 26, Qoeet. 304; С Hutton1» Diarkn MisoelUny, 2, 1775, 270; Ley-
bourn'e Math. Quest, proposed ш L*die6r Diary, 2,1817, 7-9.
mThe Dianan R«po«tory; or, Math. Register . . _ Collection of Mftth. Quart, from
Ladies' Diary, by л Society of Mathematician^ London, 1774, 81-2,
^Ladiee1 iMary, 1805, 43^4, Quert- U32; Leybourn'e M. Quest. L. D-, 4, 1817, 46-7.
417 The Diary Companion, Supplement to Lacliee' IMary, Londoo, 1805, 46-7.
^•Leybourn'e ШАЬ Quest, proposed in Ladiee' Diary, lr 1817, 405-7,
«♦ Ibid.t 2,1817, 7-9.

010 History op the Theory of Numbers» [Chap, xxi
By trial, G=37* if a = 5, 6=8, c=9, « = 1. Setting b=2qi-al we see that
О becomes a quadratic in a, and G = (m—3gaJ determines a rationally.
M. Noble440 gave a note on the history of Diophantus V, 19, citing
papers reported on above. He noted that one solution leads to an infinitude.
For, if 3-2aJ=a2, then 3-2(ai+0**)s = («+,frJ, provided
We may take ZA+-2aJ^$t x=(-f~ZB)/C. He also gave the following
solution. Let xt y, z be the desired numbers and s their sum» Then
x=s*-a*, у = $*-&, ^=я3-сэ. Thus
A) а-З^-а^-Р-А
Take в=и\ a=(p±q)v, b = (u2~q)vf c=(u2—p)v. Then A) requires that
Set E=(u2±pmlny. We get p and hence
Equating F to the square of tPk+qr/e, we get q. Then evidently
Wm. Lenhart found n numbers xx- such that if each be added to the
cube of their Bum s the sum shall be a cube <*J. Thus s-h^is1 = 2a!. Take
s^l/r. Then г2+п = 2(г<ыУ. But in another papei^ Lenhart" of Ch.
XXV, he showed how to express a number (here r2^) as a sum of cubes.
Again, to find n numbers a, such that if each be subtracted from the cube
of their sum s the remainder shall be a cube #J, we have ns? — s = 20J. Take
s^r/t. Then r(nr2-i2) = S(^03- И r=*, the problem is to find n cubes,
each <1, whose sum is n— 1. It was discussed in the paper cited. Here
let t>r7 t being prime to т. The following tentative process was used.
From nr2 subtract in turn the terms of a decreasing series of squares prime
to r and beginning with the first square <w* and ending with the square
just >r*\ Multiply each remainder by r and seek (as in the paper cited) a
separation of the product into cubes ftfoK. For n^4, take r = 12, *=19;
2580=а+Ь, a=1241=9a-h8a> b = 1339 = 23+ll\ Using his table of sums
of two cubes, he found various answers for n =3 and one for n~5.
S. Bills,4* to find x^fl-aV, 9=Xx>, would solve n-a\ a^F
by taking arbitrary values for к, O4, - • •, a« and using the theorem that any
number is a sum of three rational squares. Similarly,423 to find л:^= (aj—l)e»,
we have 2aJ—« = l/e*; set K — l/s and assign arbitrary values to air * • •, an
and solve a\±al±d=K*f where d=a%± h<£—n. Put
then ^-9г?=27/4+<?=У£- Taking K+%v=f, K-Zv~g, we get К and ti
*■ Leyboum'a Math. Queet, proposed m Ladiea' Diaryr lt 1817, 52-62.
« Matb. Miscellany, New York, lp 1836, 283-7»
e Math» Quest Educ, Timee, 22, 1875, 71.
**Ibid.,24t 1876,52-3.

. xxi] Cube of Sum Puts Ant One a Cube» 611
A. B. Evans*24 found three positive numbers whose sum is unity such
that each plus unity is a cube. Take a\&—1 as the numbers. Then
*4. Set p=&-\-aSf 4r*=aJ-haJ—s*. Eliminating the a*, we get
Then x = {4r*-si>)!D7A±2rs11). For r=9, e-5, the condition а?
is satisfied if ax = 1404/133, at = 1637/133. Cf. papers 426, 428.
D. S. Hart4* found N (tf^5) numbers such that if each be subtracted
from the cube of their sum s the remainder is a cube. For 3 numbers
x, y, z, let s1— х=тг, s1—у=п8, 8*-г=р*л Then
which is satisfied if
These give answers involving the least numbers found to date. For N-A,
7w8-hnl+p»+g3=4s8-^ which is satisfied if « = 5/0, ?n=3/27, n=5/27,
p-6/27, g~ 13/27; the desired numbers are the ratios of 3348, 3250, 3159,
1178 to 27». For N~5, take s = i, and m, , r to be the ratios of 1, 3, 4,
5, 8 to 18-
It. Davis<M divided unity into three parts such that each increased by
unity is a cube. He and D. S. Hart (p. 133) treated Diophantus V, 20.
S. Tebay,**7 to make а?—х> a cube where a=2xif and hence ntf—a a
sum of n cubes, would express n—or* as a sum of n cubes, the roots of
three of which are m—99 m—i, s-[-t—m. Let H-n-[-m% be the negative of
the sum of the remaining cubes. Then <*""*=*#+3d Bm—я—t). Equate
the last product to 9rV£, thus determining U Then
Hence a and t are found rationally in terms of r, m, H. He428 expressed 2
as a sum of three rational cubes. But 3s3—s=2 if 8=1. Hence, as by
Hart,415 we have three numbers whose sum is unity and such that unity
exceeds each by a cube. He tested eleven sums of three cubes by the
method of Hart,439 but found no answer in quite so small numbers as Hart's,
his smallest answer being 13/49, 17/64, 351/D9-64), with the sum 9/14.
A. Holm42* treated Diophantus V, 19 by starting with Diophantus'
formula
To find positive solutions of 3 —2а*=П, take ai=5/6, a*=-|—х,а*= — \
Then
> 0 7 1 ^ _, / 3 t V ■, 36r-h7
ш Math» Quest. Educ. Timei, 25, I876r 31, Cf. Strong11 and LenhArt* of Cb. XXV.
<*Ibid.,2»t 1876,06-8.
m Math. Vkitor, 1, 1880,107-
«T M»th. Quest. Educ. Times, 38,1883, Sb2.
**Ibid.t 101-3.
ea Math. Quest. Educ. Tunee, B), 9, 1006, «8.

612 History of the Theory or Numbers. [Chap, xxi
To make the cubes positive take J<z<f. This is the case if r =
whence x ^5/9. Thus the ratios of 351,832,833 to 3136 answer Diophantus'
problem.
A. G&ardiQj R. Goornmghtigh and others diacueeed in l'intennediaire dee akatheraftticien*
the following problems io which t ifl the sum of the unknowne:
f-xondf-y cubea, 22,1915,222; 23,1916,142-4,210-1.
**-*,**-^*'-zaU cubes or all biquadratee, 22, 1915,245; 23, 1916,4-5.
&-x}#-yt «"-ж, «■-I all cubes or all equaree, 23,1916, 28-9, 52-3.
t*—zu " -} x*—Xi all cubes or all squares, 100-1.
«■—jjp * ■ •, e1— x» all cubes or all squares, n odd ^5, 24, 1917, 114-5.
Systems of equations or degree three in foub or more unknowns.
Alkarkhi430 (beginning of eleventh century) solved х*—у*—г*, х*-\~1?=1г
by setting x—2yt z = myy t=ny, whence y—4—т*=п*—4, m*+n*=&; take
mz=4/25, fts=196/25. He treated various similar problems.
J. Ozanam431 asked for four numbers such that one obtains a square by
adding to the product of the first three the product of any two of the four.
W. Wright1*2 found four numbers the product of any three added to
unity being a square» Substitute the value of г from xyz-{*l
into vyz±l and vzz+1. The results are squares if
which determines p, and G = p%—2vzp-\~ux?y= П. The latter leads to a
quartic ш q which is equated to the square of q2—2urgH-2i£ry—vxyi-{*2vxzy
—2tPz*9 thus determining q. Then vxy+l=n2 determines x. J. Baines
to6kwzy±l=a*, гксг-f 1=^ wyz-\-l =c*y which determine w, x, у in terms
of z. Take (bl-l)(c2-l) = l=^ Then аяг+1 = (а*-1)*+1 = D1/9)* if
a=7/3.
J. Anderson45* found n numbers whose sum is a square such that the
square of each exceeds the cube of their sum by a square. Let the numbers
be &Xi, where 2г*=1. Then shall x\~$* = □ =(sp^—x$*, say, whence
х*=я(р*+1)/BрО- W* Watson used the numbers xtf with the sum e1.
Then x\~ 1 — D — (Xi—т<У ^ves x* The condition on the sum gives 8.
Several4** found four numbers x, x-{*yf x-[*2y, x±Zy in arithmetical
progression whose sum s of squares is a square and sum p of the product
of the extremes and the product of the means is a cube. Take y = vx.
Then « = □ if 4+12v-hl4u*^ (rv—2)\ which gives v. Take r=4. Then
v=14, p=47&cft which is a cube if x—478.
S. Ward4*6 found four numbers о, Ъу с, х such tbat the product of any
three added to unity shall be a square. Set m=a&, n=ac, p=bcf and let
mx+l~(l— rxf, whence x = Br-hm)/r2. Then shall
FakbH, French tranaL by F. Woepcke, Paris, 1853,1Э4.
« Letter to de ВЩу, May 9,1676; Bull. Bibl. Storia 8c. Mat., 12,1879, 517.
<* The GcntlemAD'a Math. CompftnioD, London, 5, No. 24,1821, 47-8.
ш Ibid., 5, No. 27,18JHr 266-S-
«* The Math. Diary, New York, lr 1825, 55^6.
ш Amer. edition of J. Ii. Young'e Algebra, 1832r 343-5.

Chap. XXI] SYSTEMS OF CtJBlC EQUATIONS IN n>3 UNKNOWNS. 613
Thus А2«-В2={<2г±т)(п-р). Take A+S=2r+m, A-B^n-p, which
give A. Hence r*+%гп+тп=А* gives
Taking any values of a, b, с which satisfy obc~f 1 = П, we get an answer.
For example, a=%, b=2, c=3give Young's answer x = 16016/25.
On four integers the sum of any two of which is a cube, see Lenhart," etc.
A- Genocchi43* noted that early arithmeticians knew that x=3, y=4,
s=5, * = 6 satisfy xy=2s, хг+у<1=г*у x3±y*±&=tf, and proved that thia
is the only integral solution. If the third condition is replaced by
£*-by*+**—*% where n>l and t is an unknown integer, he proved that
either 6 = 1, a=3, n^2, m=l, *=3 or m=3, t=l, or Ь = 1, а=2, rc=3,
fn=t = lf or 6 = 1, a^2, 7i=2, m=2, *=3огто = 1, t=6.
P. W. Flood437 noted that the six cubes
of which the sum of the first three is 1/8 and the sum of the last three is 1/8,
are such that on adding any one to the square of the sum of the remaining
five we obtain a square.
V. Binr"* considered zyz=uvtv with 2х?=2и*от 2х* = 2иг, and Sjt = Sw,
Zx'^Zt*1, the second pair being equivalent to Xx^lx', xyz=*xfy'z\
L. E. Dicksonw showed how to obtain all sets of integral ^solutions of
the last pair of equations, as well as of the pair440
which express the condition that two rectangular pamllelopipeds shall have
integral edg&, equal volumes and equal surfaces. Cf. papers 16-18 of
Ch. XVII.
A. Gerardln*1 noted that <P—z*, <P—y*, d*—z\ d*—P are all squares,
where d=x±y—z-t, if z = 65, y=488, z=481, t = 7.
Gerardin442 gave three sets of solutions of sa-fjf+e^p+u*-Ья3, xyz=iuv,
including the solution
p* w we» " ¥t 91 p1
The same pair of equations and 2a; = S< have the solution
x t у и , z v
- = - = рд-Л ^--er-p», ---Рг
mAl\i Accad. Pont. Nuovi LtDcd, I9r 1865-6, 49; Amjftli di Mat-, 7, 1865, 157; French
fa»nel.r Jour, de M*th., B), 11, 1866,185-7.
«■' Math. Queet. Bduc. Tim», 70, 1899, 52.
ш L'interm6diaire dee mfttk, 16, 1909, 41-3, 112. Cf. Deebovee«; aleo
8,1913,140, and СЪ. XXIV.
«• Meeenger Math., 3^, 1909-10, 86-7.
«•ШЛ, aod Amer. MMh. Monthly, 16,1909,107-114.
^ L*mlcrmeduu№ dee math., 23,1916, 76,
"Nodt. Amu MMh , D), 15, 1915, 564-6,

CHAPTER XXII.
EQUATIONS OF DEGREE FOUR»
gull OK DIFFERENCE OF TWO BIQUADRATICS NEVER A SQUARE; AREA OF A
RATIONAL RIGHT TRIANGLE NEYEB Л SQUARE.
Leonardo Fisano1 recognised the fact, but gave an incomplete proof,
that no square is a congruent number (i. e<, tf+y1 and хг—у* are not both
squares), while the latter is the area of a rational right triangle. Four
centuries later, Fermat* stated and proved the result thus implied by
Leonardo: no right triangle with rational sides equals a square with a rational
«wfe. The occasion was the twentieth of Bachet's problems inserted at the
end of Book VI of Diophantus: to find a right triangle whose area is a
gjven number A. The necessary and sufficient condition given by Backet
was that BАJ+К4=П for a suitable K. For, this condition implies
that 2A/K and К are legs of a right triangle of area A; while, conversely, if
К and H are legs of a right triangle of area At they are proportional to K*
and 2Ay which are therefore legs of a right triangle. He quoted two condi-
conditions given by F. Vieta, Zetetica, 1591, IV, 16, of which the first is that the
area increased by some biquadrate should be a biquadrate, and expressed
doubt as to the necessity of the conditions
Fermat's proof is of especial interest as it illustrates in detail his method
of infinite descent and as it presents the only instance of a detailed proof
left by him. In the left column is given ft translation of Fermat's account
and in the right column proofs1 of the statements.
" If the area of a right triangle If the sides have a common
were a square, there would be two factor, the area has a square factor
biquadrates whose difference is a which may be removed. Since we
square, and hence two squares whose may therefore assume that the sides
sum and difference are squares, x, yt z are relatively prime, we may
Thus there would be a square equal apply the rule of Diophantus and
to the sum of a square and the set x=2mn> y^m2—nowhere mand
double of a square, such that the n are relatively prime integers not
sum of the two component squares both odd. Then mn(m*—n2) shall
1 Tre Scritti, 1854, «8; Scritti, 2> 18Я2, 272. See Leonardo17 of Ch, XVI.
'Format's nungina.! notes in his copy of Bachet'e edition of DiophantuV Arithmetic*;
Oeuvree de Fermat, Paris, 1, 1891, 340- 3, 1096, 271,
» Cf. H. G. Zeutben, Geeehidite der Math, in XVI and ХУП Jahrhuudert, 1903, 163. In
. АееЪЬогаШпоГ Feroi^'flpitMrfbyA. M. b'g^
ed. 2, 1808,340-3, иве is made erf the theory of quadiat» forme to Hhowtbat |=r*+2**;
while P. Baebmann, Niedere Zahlentbeorie, % 1910, 451-4, employed the uniqueness of
factorisation of the integral algebraic numbers a+fcV^2. Both completed the final
«tep in the proof by comparing the Areas of the initial and new triangles. H. Dutordoir,
AnnaUfl de la Socilte Sc. de ВгихеПея, 17, ДОВД, I, 49r announced in eight lines that
he could fill in an elementary manner the gppe left in this proof by Ferzn&t. For the
elaboration naed in the text, see L. E. Dfck*mr BuD. Amer. Math, Soc», B), 17, 1911,
615

616
History or the Theory of Numbers.
[Chap. XXII
is a square. But if a square is the
sum of a square and the double of
a square, its root is likewise the
sum of a square and the double of
a square, which I can easily prove.
It follows that this root is the sum
of the two legs of a right triangle,
one of the squares forming the base
and the double of the other square
the height. This right triangle will
therefore be formed from two squares
whose sum and difference are squares.
But4 both of these squares can be
shown to be smaller than the squares
of which it was assumed that the
sum and difference are squares.
Similarly, we would have smaller
and smaller integers satisfying the
same conditions. But this is im-
impossible, since there is not an infini-
infinitude of positive integers smaller
than a given one. The margin is
too narrow for the complete demon-
demonstration and all its developments."
be a square, whence m=<&, n = &,
ai—V^=n> where a and Ъ are rela-
relatively prime, one even and the other
odd. Thus аЧЬ5 and а'-Ь5 are
relatively prime. Hence а'-нЬ5—£*,
a*—Ь*=т7*, £ and 4 being odd in-
integers. Abo ^=^-1-2^. Set
Then e and / are integers and
e/^b*/2. A common factor of e and
/would divide & n, Ь5 and a*. Hence
e and / are relatively prime. We
may take e odd (changing if neces-
necessary the sign of 17). Thus e=i*t
/=±2s*, 2rs = b, where r and 8 are
integers. Hence £ = e-\-f= r2-!- 2s1,
tj=i*-2^ Also o2=V+ i>3=r«-h4s*.
The right triangle with the legs r2
and 2a* has the area r2**. It is there-
therefore formed (in the sense of Diophan-
Diophantus, as above) from two squares
nti^al and nv=b\y its sides being
?£ h
m\-n\
i*. By т}пг=^ we get
, a factor of h^2r*. Hence*
aj and bx are each less than Ъ and
hence less than a.
Fermat's6 observations on Diophantus II, 8 and V, 32 includes the
statements that the sum of two biqiiadrates is never a biquadrate or a
square»
Fermat had proposed, Sept., 1636, to Sainte-Croix that be find a right
triangle whose area is a square (Oeirms, II, 65; III, 287); to Frenicle,
May (?), 1640, (Oeuvres, II, 195); to Wallis, Apr. 7, 1658 {Oeuvres, II,
376). Fermat stated that the problem is impossible in a letter to Pascal,
Sept. 25, 1654 (Oeuvres, II, 313). The attempted7 proof by J. Wallis,
June 30, 1658 (Oeuvres de Fermat, III, 599) goes no further than a proof
of the rule of Diophantus for the sides of a right triangle. Fermat referred
in a letter to Carcavi, Aug., 1659 (Oeuvres, II, 431-6, see 436) to proofs by
the " descente indefinie " which he bad sent to Carcavi and Frenicle con-
«Afl translated by Heath, Dioph&ntue of Alex.red. 2, 1910, lOr, by flj+bj^aj+b} — *-
293-5, Tannery (Oeuvreede Fernut,IH, 272)gavethe
incorrect reading: But the sum of these two equatee can
be shown to Ы smaller than that of the first two of which
it was assumed that the sum and difference are squares.
• Oeuvras, ly 291F 327; Шг 241r 264.
'Qiticired by Frenicle, Oeuvree dc FermAt, Щ] 606, 609.

Chap.XXII] Area Right Triangle + D, х* + у* Ф П. 617
eerning negative theorems, and cited in the same letter the theorem under
discussion.
Frenicle de Вевву* (fl765) gave a proof, published posthumously, the
principle of which is doubtless due to Fermat hi view of the letters just
cited. It suffices to prove it for a primitive right triangle. Denote the
sides Ъу 2mn, tn*±n2. If the area is a square, the odd leg wi2—n* is a square
P and the even leg 2mn the double of a square. Thus we have a second
primitive triangle whoee hypotenuse is m, odd leg I and even leg n. Since
mn is a square and m is relatively prime to nt m and n are both squares.
Denote the sides of the second triangle by 2ef7 е?±р, where e and /are rela-
relatively prime. Since n — 2ef is a square, one of the numbers e,/ is an odd
square and the other the double of a square. Let е=г*, /=2s*. Also
e2+/* ;= го is a square a2. Thus a, e, f are the sides of a third primitive right
triangle whose area is the square rt*. Its sides are less than the corre-
corresponding sides of the second triangle:
a<a2=mf f<2ef=n, e<{e+})(e-f)=L
The sides of the second are less thau the corresponding sides of the first:
m<mf-\-n2, n<2mn, l<P=m2—n2> Hence from the first primitive triangle
with a square area we have derived another primitive triangle (the third5)
with » square area and with smaller sides»
G. Wertheim1* reproduced the last proof in slightly modified form.
Frenicle proved in like manner (p. 175) that no right triangle has each
leg a square and hence the area of a right triangle is never the double of a
square. He concluded (p-178) that no square is the sum of two biquadrates
and that x4 — 4z*=y* is impossible in integers»
Fermat had proposed to St. Croix Sept., 1636 that he find two bi-
quadrfttes whose sum Is a biquadrate (Oeuvres, II, 65; III, 287), to Frenicle,
May (?), 1640 (II, 195).
G. W. Leibniz15 proved, in a manuscript dated Dec. 29, 1678, that the
area of a primitive right triangle with integral sides is not a square- The
sides are хг±:у2> 2xyy one being even. Then if x1—y* and xy are both squares,
x and у are both squares; also x+y and x—y (since a common factor 2
would make s2—y2 even, contrary to the above)» But yy x—y} x} z+y are
not all squares. For, if so, the last three give squares in arithmetical
progression whose common difference is a square, " which is absurd." Fur-
thei,\i x2^y2—xy1 then (y-\-x)y~ □, (x—y)x= D,and each of the four fac-
factors would be a square, just disproved. He noted several corollaries» In
view of the triangle formed from x and 1, (x—l)x(x-\-l) is not a square»
The difference of two biquadrates is not a square. For, if i/1—w4= □, the
dm Tnaogto Rectangles en Nombres, Paris, 1676, 101-6; Mem. Acad. Be. P&rie,
5, 1666-1699; 6d. Paris, 5t 1729r 174; Eecuil de pJusieurs traitez de mathemfttique de
PAcad. Eoy. Sc. Paris, 1676.
• Identical with Format's aeeond triangle.
i» ZeitBchrift Math, Яц*ч 44, 1899, Hiat, Lit. Abt-, 4-7,
" Math. Schriften <ed., C. 2. GeAanit), 7r 1863, 120-5, Ь a fragment, dated July, 1679,
Leftcis merely stated that the problem la impossible; see L. Couturat, Opuscule» et
iragmenta ineditie de Leibniz, Paris, 1903, 578.

618 History of the Theory of Numbers. [Cbap. xxii
area of the triangle formed from v* and to2 would be a square. Again,
*Ы-Ф* □ by (х*-уг)ху± U.
J. Ozanam" stated that x*±y* Ф*4. For, a4—b* is the area of the right
triangle whose sides are the ratios of 2a*b\ а4—Ъ\ a?-[-b* to ab, and is not a
square " as proved by Messieurs de l'Acad. Roy. Sc. and also by R. P.
de Billy."
L. Euler1* proved that аА+¥^\3 if ob+0. For, if («*)*+(&*)«=□,
where a and b are relatively prime, then a' = p*—g2, b*=2pq, where p and q
are relatively prime, one even and the other odd. By p2—q*= □, p is odd,
whence q is even. By pBq) = b*7 p and 2q are squares. By tf^a*-\-q2,
we get р = т*-\-п2, q*=2mn, m and n relatively prime. Since 2g=Q,
win— □ and m—z4, л=*#2. Thus x*H-^ is a square p, and я, у are less than
а, Ъ. By a similar proofy a4—fr* + D unless 6=0 or b=^a.
E. Waring14 and A. M. Legendre3* reproduced literally these proofs by
Euler.
C. F. Kausler18 treated х*+у*~г* by use of the lemma that &±уг are
not both squares. Equating x=2PQ, y^P1—Q* (from &+уг = П> х, у
relatively prime) to x> у—р^Ч-З1, p2-^ or to (p4-£*)/2, (p2—g^/2, in either
orderr where p and 5 are relatively prime, and odd in the latter case, we
are led to a contradiction» Now xl=z*~yA requires z?+y*, z^—y^^m^n*, 1
or т*п\ ws, - • - A9 cases); 7 cases are excluded by the lemma, others by
z'+jf >z*-y' or (z*-j/f)">22+jj4. Finally, if г'+у'«m?, ^-i/^wm^ tben
2й2^7п(т2Ч-л4)у while m — 2 is easily excluded. Thus £a prime factor of]
m is a factor of z and hence of y_
P. Barlow17 noted that, if the area xyj2 of a right triangle (z, у, г) were
a square 10*, then г*±:4м?^(х±уJ, whereas it was proved by descent
(p. 109) that z?+y* and x*-y* are not both squares. Also (p. 119),
J. Homer18 noted that if xjy^yjx — Q, where x, у are relatively prime,
then x=m2, y=n2, m4±ni= П, contrary to a known result.
Schopis1* proved x*-\-y*=z2 impossible, using the impossibility of
^-^=2**. Next (pp. 6-10), x*+tf'=2*' is impossible; likewise (p. 11)
A. M. Legendre80 stated that the above8 proof that the area of a right
triangle is not a square shows that а^-^фй if аф&, &Ф0. [But in
« Journal dee Sg»vansy 1680, p. 85.
«Comm. Acad. Petrop., 10, 1747 (I738)r 125^4; Comm. Arith.r I, 24-34; Орел ,
A), II, 38, Same proofs m E\ilerre А1кеЬг*, 2, Ch. 13, arte. 202-8, SL Peterabutg,
1770, p. 418; Freoch Uanet., Lyoit, 2, J774, pp. 242-54; Opera onrnia, A), 1,1911, 437;
Shi4)di 19089 59Ы
ppy у
" Meditationee А1«еЬЫсаер Cambridge, ed, 3,1782, 371-2,
»Th6orie dee nombre«t Pane, 1798, 404; ed. 2t 18OS, Э43; ed, 3, 1830, XI, 5; German
traneL by Maecr, 2,1893, 5.
i* Nova Acta Acad, Petftjp., 13r ad ашюе 1796-в A827), Mem., 237-44.
"Theory of Numbers, 1811, 121 <tf.H4).
к The Geotteman'B DiaryT or the Math. Repository, London, No. 80, 1820, 37.
i» Einige S&Ue aus der unbeetimmtett Analytikr Progr. GumbimicD, 1825.
» Theorie dee nombres, ed. 3y 2, 1830, % 325, p. 4, Cor. (Маяег, П, p. 4).

Chap, XXIIJ AREA RlGHT TniANGIJB Ф □ , S4 + у4 ф П. 619
that proof it was known that a and b are not both odd, a criticism due to
A. Genocchi21].
J. A. Grunert* reproduced Euler's proof that a*-f ЬА фс2.
О. Terquem* proved by descent that xi±.yi=zz is impossible.
J. Bertrand*" proved that х*-\-у* фг2 somewhat as had Euler.
P. Volpicelli" proved that no congruent number is a square. For, if
pfrf)-*, then A'-fr^+wV-eH-W, (^-4pYJ-^-Ba^)S
whereas a difference of two biquadrates is not a square.
V. A. Lebesgue* proved the impossibility of x4-j-j/4=^2 by descent. It
suffices to treat (Tpy+y*=z*, where p, y, z are all odd, and у, г are rela-
relatively prime. The factors г±у2 of (TpL have no common factor other
than 2. Hence
The lower sign is inadmissible. Hence 14~ y2=2u^Lui. Thus
T» PepinM proved the impossibility of xi-y4=z* in integers +0.
W. L. A. Tafelmacher*7 proved the impossibility of xi-\-yA = zi.
D* Gambioli48 proved that z*—y*=z* is impossible in integers Ф0.
Т. R. Bendz29 proved by descent from x*-|-4^=z2 that the area of a
right triangle is not a square.
L. Kronecker^ amplified Euler's1* proof.
G. B. M. Zerr31 employed unproved assumptions in an attempt to prove
that the area of no right triangle is a square.
A» Bang** noted that relatively prime solutions of х*-г* = у* imply
s4-*-2yJ, x±z=2yi,
Thus y*-y\=4yl, so that
Hence u\-v?=2«u\\ so that
u\+i£=2v\\ ul+ul=2v\\
and Ut = ViV2v%tb. By the third and fourth, «?-Н«5 = 2^2+2и^2. Then by
the second,
» Annali di Sc Mat. e Fie,, 6, 1855, 316, foot-note. Hie like critic™ of the proof by Тег-
quem1' im not valid.
« КЮ^'в Math. Wort«rbuchr 5, 1831, 1143.
" Nouy, Ann. Math., 5, 1846, 71-i.
«•Traits elem. dralg«)re, 185lp 224r-7.
" Atti Aocad. PoDt. Nuovi lined, 6r 1852-3r 89-90.
* Exencioes d'aftalyee numer., 1869, S3-4; In trod, a La throne des nombree, 1862, 71-3.
M Atti Accad. Pont. Nwovi Liaoei, 36r 1882-3, &И36,
" Analee de la Univeraidjid de CbiJe, S4,1893, 307-320,
» Pcriodico di Mat., 16, 1901,149-150.
** Ofverdiophantiska ekvationen a;'+i/»=2*p Di». Upsala, 1901, 5-9»
" Vortenmgen uber Zablentlieone, 1,1901, 35-8.
•i Amer. Math. Monthly, 9, 1902, 202.
n Nyt Tidvkrift for Matematikr 16, Bp 1905r 35-36.

620 History of the Theory or Numbers. [Csak xxn
like the proposed equation, but with t£<Vx- Rychlik2" of Ch. XXVI
gave a proof.
J. Sommer3* reproduced Euler's13 proof of the impossibility of
xA+yA=*z2 in integer and Hilbert's'6* proof (Ch. XXVI) of its impossibility
in complex integers а-\-Ы.
A. Bottari** proved a^+^—z2 impossible by use of an unnecessarily
complicated set of solutions of xl-\-yi=zt.
F. Nutzhorn35 gave a complicated proof of the impossibility of X* -h^=^
R. D. Carmichael*4 gave a new proof that neither of the equations
mA—4nA = =h t* is possible in integers each Ф 0. Hence the system
p2 — 2о2=кт?} jyl-\-2q!= =bfen* is impossible in integers each Ф0. Thus the
area of a right triangle is not the double of a square. Hence т*-\-п* = аг
is impossible in integers each Ф0.
Solution of 2xa— у*^П; right triangle whose hypotenuse and sum
OF LEGS ARE SQUARES; X2-\-y2 — B4f X-\-y = A2. AbSOt Xi — 2yi — Of
Fermata7 proposed to St. Martin and Frenicle, May 31,1643, the problem
to find a rational right triangle whose hypotenuse and the sum of whose
legs are squares. Fermat*8 affirmed that the smallest such triangle with
rational sides is that with the sides**
A) 4 687 298 610 289, 4 565 486 027 761, 1 061 652 293 520.
Fermat's40 method consists in forming the right triangle from z+1, x\ its
sides are 2x2+2x+l, 2x-H, 2x2-h2x. The first and the sum 2x^-\-4x-\-l of
the last two shall be squares. By the usual method of Diophantus, we get
s*=—12/7. The triangle is therefore formed from —5/7, —12/7. Employ-
Employing 5, 12 instead, we get" A69, —119, 120)» When a negative result is
obtained it is in accord with a general procedure of Fermat to repeat the
operation and to form the triangle from x+5,12. Its sides are (s-|-5)*±12J
and 24(x-h5). Hence x2-HOs+169 and z*+Mx+l are to be squares, say
a2 and b*/169. Then bz-a*= 168s3-h5736a:. Taking
b-« = 14x, b+a = 12x4-2868/7,
we get a= —x-h 1434/7. Comparing its square with the earlier a2t we get
1343 1525 2048075
X" 7^2938 " 20566 "
w Vorleaungeu uber Zahlentheorie, 1907,176-193. French tranal. by A. L6vy, 1911,184-199.
* Periodic» di Mat., 23,1908,109.
* Nyt Tideakrift for Mat., 23, By 1912, 33-38.
* Ашет, M*th, МооШу, 20,1913, 213-21.
ir Osuvn», II, 259-63.
" OeirvresT I, 336; Ш, 270T obaerv&tioti on Bachet'e o^mment oq Diophantua VIr 24. Also,
Oeuvres, II, 261 B59r 263), letter to Mereenne, Aug. 1, 1643.
" Cited by Frenicle, Meto.Aead.Sc.T 5,166fr-99; 6d. Paris, 1729, 56-71. Since hie numerical
search was fruitlewT be doubtless leam«d of Fermat'e solution from Menenne.
«Inventum Novumy I, 25, 45t HIy32- Cteuvree, III, 340,353, 388.
41 Whence the hypotenuse and leg difference of A69, 119^ 120} ftre equArea,

Chaf.xxii] 2х*-уА=П; System х+у^А2, хъ+уг=В4. 621
The ratio of x-f-5 to 12 is that of 2150905 to 246792. The triangle formed
from these is A). He noted that the problem is equivalent to that to find
two numbers whose sum is a square and sum of squares is а Ыquadrate.
Fennat42 noted that in the right triangle A56, 1517, 1525) the square
of the difference of the legs exceeds the double of the square of the least leg
by a square. Without giving details he added that this triangle serves to
find a right triangle whose hypotenuse is a square and whose least side
differs from the other two by Bquares.
Frenicle41 gave details on the last problem. An analysis followed by
numerical trials led him to the triangle, formed from 6 = 156 and « = 1517,
having the sides
2ob=473304, a1 -b*=2276953, a2+b?=2325625 = 1525*.
The least side differs from the other two by the squares of 1343 and 1361.
Ab remarked by A. Genocchi" these results imply that 2xA-yi=^D has the
solution x >= 1525, у = 1343 [Lagrange,64 Euler* (third memoir), Lebesguew J
E. Torricelli** proposed the problem to find a right triangle with integral
sides whose hypotenuse, sum of legs and sum of hypotenuse and larger leg
are all squares. E. Lucas* stated that this problem was proposed by
Fermat and that its solution depends on x*—2yA=z2. In fact, Fermat46*
proposed the problem to Torrioelli. An attempt to trace its origin has
been made by E. Turriere.1* a. *M. Cipolla.«<
J. Ozanam47 treated the problem of Fermat37 by the method essentially
the same as employed by L. Euler.*8 If the legs are x> yf then x+y is to
be a square and x*-h!/* a biquadrate. In this form the problem was proposed
by Leibniz. Euler made x*+y* a square (pz+q2J by taking x=^p2—qz,
Then p*-hff* is a square for p=r2—z2, q = 2rs, whence
It remains to make
x +y
a square. It will be the square of r2—2r$+s2 if r=3s/2. Taking r = 3,
s=2, we obtain a negative value —119 for x. Setting т = 3a/2+*, we get
16(^4^ +7815^^1160^116^,
which is the square of ^-|-148s^4^ if s/t=84/1343. Taking s = 84, we get
т= 1469 and z, у as in A).
« Oeuvre, II, 265-6, letter to Carcavir 1644.
49 Method* pour trouver la flotation dee probldmea par lee exclu&ioD*p Ouvrnsea de math.r
Pwrie, 1603; Mem. Acad. R. Sc. Parie, 5r 1666-99 [1676} 6d, 1729, 81-5^
"Atti R. Aftcad. Sc. Torino, 11, 1876, 811-29»
«G. Lori^rintermWiairedefimath.^, 1917, 97-& a. 25,1918, 83,
" Bull. Bibl. Storia Sc. Mat. Fie., 10,1877, 289.
«* Letter from Meraenne to Torrieelli, Dec. 25F1643, Bull. Bibl. Storia Sc. Mat. F(s,, Sr 1875,
411; Oettnrea de Fermat. 4,1912,82-3 (cf. p. 88).
"L'eneeignemeat math-, 20,1919,245-268.
^ АШ Accad, GioeniA ec. nit. Catania, E)r 11,1919, No, 11.
47 Nouy. dement algebre, АтвЬЫлт, 2,1749, 480-1,
" Algebtu, 2r 1770F art. 240, pp. 509-5; French tranebr 2r 1774, p. ЗЗв: Opera Omuia, A).
1,483-4.

622 Histohy op the Theory of Numbers. [Chap, xxji
Euler" noted that x*-2y* = (p*-2tf)* for if=2pg, x*-p4-2fl*. The
latter holds if zbp^-2*1, q^2rs. Then 2pq=y*= zb^r2-^. Set
r=P7 «=iA For the upper sign, &—2w*=D, whereas t and и arc smaller
than x, y. Hence take the lower sign. Thus a solution of 2w*—1*= □
yields a solution of ^—2^ = 0, For t=u = l, we get x=Z, y=2. Then
lor £=3, ix=2, we get s = 113, y^84. Again, « = 13, *=1 gives s=57123,
I/= 6214. Lebesgue" (end) noted that this solution is incomplete.
Euler" treated х+у = П, х*+у*=(г*+1У by taking s-^-feH-l,
у=4^-42. Then ж-hy is the product of the two factors ^Ч-B=ь2лЙ)г-1,
which he equated to (г-\-р±д *&)\ By the rational and the irrational parts,
we get
Thus g=13 gives p = 18 or -Ш/7, q= -13 gives p=21 or 113/6.
Euler" reduced B) to G) by setting v-=2xi+y<> whence ^+8Bir)*=-^
Conversely, let g4+8p4=r2; then 8p« = (r+g*)(r—g1), so that g and r are
odd. First, let r-h^-2a, r-g*=4& where a is odd. Then р*=<*0, and
л, /3 are relatively prime; whence a=^, jS=t*, p^«^. By subtraction, and
cancellation of 2, g*=s*—2J4. Second, let r-g*=2<*, 74-^=40, where a is
odd. Proceeding as before, we get g*=2t<—г1. While in the second case
only we obtained B), the reduction can always be made since /l8*/t
implies 2х*—у*=2? for
In quoting this solution, Lebesgue,* p. 74, gave/^р incorrectly for fg2 in *•
Euler« noted that ar-Mz-B2, z*+y*=A4 imply (x-y)*=2A*-B*. The
latter is the square of tf-i-2^-^ if А*=?+12, Bl={^tfy-2^. Taking
v = 2abcdt we have A = a?bt-hAP if $=<№-<?&, and B=aV-2W? if
The two values of f-И are equal if
bc±r d 6c=Fr
rf c'-b1 a
Hence 2A*—B* is a rational square if 2b4—c*is. Taking b=c = lf we have
a=3, d=2, £=5, 17 = 12, A = 13, B^l, 2 13*-l = 239l; since B<A, x
and у are not both positive. Taking b = 13, c=l, we have r—23&f
ajd= -3/2 or 113/84. For a=3, d= -2, then {=1517,17- -156, A = 1525,
B= -1343, which" do not yield positive xy y. For a=113, d=84, then
A=2165017, B= — 2372159, and we obtain very large solutions. [In fact,
Fermat's(l). SinceT-y = ^-h2f7y-$5,a:-hy-B2,wehavez=2f?j,y=$5-^
Thus x, у are the legs of the right triangle formed from {, 17. Here
1=2150905, ty=246792, as in Farmat's solution.]
« Algebra, П, art 211; French trwiel., pp, 260-3; Opera Ошша, A), I, 444-5»
и Opera poetuma, lr 1862, 491 (about 1774).
u Opera poetuma, 1, 1862, 221-2 (about 1780).
« Opuec. anal., 1, 1783 A773)r 329; Comm. Arith II, 47.
« The method of £ulerr Algebra, 2, art. 140^ to nuke 2ac*—1 a square does not give all eob-
tione sinoe 1526/1343 is omitted {remarked by Lebeegue»). E. Fauquemberpie,
1'mtermediairt dee nwth^ 5, 1888, 94, claimed to prove that 2 = 1, x-13 we the only
integral Bolutioiu.

Сяар.ххп] 2х*-у*=П; System x+y=A\ х*+у*=В\ 623
J. L. lAgrange54 discussed Fermat's" problem at length» From
P+q=y*} р*-\-д*=х*, he derived, after setting z — p—q,
B) 2tf-y*=z\
The problem reduces to the solution of B) since we have
C) P = h(vz+*)> * = *(*-*)■
Lagrange was evidently not acquainted with Euler's62 paper of 1773 in
which he derived B) and obtained four sets of solutions A—x, B=y;
indeed, JLagrange omitted the set 1525, 1343, in his citation of Euler.
Given any integers xy у for which 2x*—y*~ D, Lagrange gave a method to
obtain smaller integral solutions; then by reversing the process and starting
with x^y= 1, he concluded that all pairs of larger solutions can be found
in the order of their magnitude» While Euler's simpler procedure appears
to give all the solutions ш this manner, he did not prove that this is the case»
We may assume that x and у are relatively prime. A simple argument
shows that x, y, z are all odd. By B),
Denote these (even) factors by 2mp, 2mq,where p and q are relatively prime.
Then pq must be a square. Hence, replacing p, q by p\ q2,
Etiminatmg z from the first two, by means of the third, we get
x*-y*=mp(p-q), x*+y*=mq(p+q).
Thus m^l or 2, since m is a divisor of 2x* and 2^2. If m=2, set
q—p—p\ Whether m = 1 or 2, we obtain equations of the form
D) x*-y* = v{p-q), **+У*=я(р+д).
Thus pis odd. Set (x-\-y)!p^2mjn, where n is odd and prime to m. Then
x+y = 2iw, p=ns, where a is an integer» By Di), x—y^2nt, p-q =
where / is an integer prime to s. Thus
E) x = ms+nt7 y=ms—nl> p = ns, q=n8—4mL
Tben the product of D2) by (в*-8*я)/л* gives
и
Since m and n are relatively prime we therefore have
F) m = («- Щ/l, n = (s2 - &2) (I (J an integer).
If ?n=0, thene/*-±l, n2=\ixL=y2=l. Hence if B) has a set of relatively
prime solutions xt у not both of numerical value unity, then by E) the
greater of xf у exceeds the greater of the corresponding solutions st t of
G) *+8*-«\
and s, I are relatively prime and not both of absolute value unity. Con-
Conversely, from relatively prime solutions sf t, we obtain by F) and E) rela-
relatively prime solutions xt у of B).
" Nouv. Mem» Acad* Sc. Berlin, annfe 1777 [17791 140; Oeuvrea, 4, 1869, 377-93.

624 History of the Theory of Numbers. £Chaf. xxn
Let s, t be relatively prime solutions of G), Then s is odd and
7
where « and p are relatively prime. Thus /j divides P. Also 8*=p(o>—p).
Hence ji = 1, о> = 2$*, p^r*, or«=^7 p = 2rA, whence
^^ф-т4; or u
Conversely, if 2^—7^ = 8* or tf4—2^ = 8* and we set t=*qrt we have solutions
$7 t of G). If s and £ are relatively prime and numerically distinct from
unity, the same is true of q and r, while the greater of $, t exceeds the greater
of q, r. The first of these two equations is of type B).
Applying to the second, if—2r* = s*, a discussion entirely similar to that
just used, Lagrange obtained
s=8n4-p\ ^вп'+р4; or s-n«-8p\ q*=n4+SpA.
The former becomes the latter if we interchange n with p and change the
sign of 8. The solution of $*—2r4 = s3 is therefore reduced to that of
qi^nA+Sp*l of type G), by setting r^=2pnt s=n4-Sp\ Further, q and r
exceed n and p.
The method leads to all solutions not only of B) but also of G) and of
tf4—2r*=D. Starting with the evident solutions s^t^l, t*=±3 of G),
we deduce the solutions r=>2s*=2, q-u = ±:Zt k^l, of tf4—2rA=kz; and,
by F), E), solutions of B): w=0, «=— 1, J=^7, z=y=z = l, or ?n=—6,
n = -7r 1=1, ^=13, ^/-17 2 = 239. For r-2, g=3r 8 = 7, we deduce the
solutions s=7,t = qr = §, «=113 of G); from 13, 1,239, we get the solutions
8 = 239, t~13, и=57123 of G). Starting again with one of the latter sets,
we obtain new sets of solutions of B) and tf4—2r4= □. In this manner,
the sets of solutions of B) in order of magnitude are (x, y} z) = (l, 1, 1),
A3, 1, 239), A525, 1343, 2750257), B165017, 2372159, 1560590745759),
-■. The corresponding sets C) are p, q = l, 0; 120, -119; 2276953,
-473304; and the last two numbers A). Lagrange therefore proved
Fermat's assertion that A) gives the sides of the least right triangle whose
hypotenuse and sum of legs are squares» But Lagrange evidently merely
transcribed the statement by Fennafc, without making a numerical veri-
verification, as the value 15---9 of z given by Lagrange (pp. 142, 150, 151;
Oeuvres, 380, 393-4) m erroneous [Genocchi44} the correct value being the
difference 350 - -1 of the last two numbers A).
Three of Euler's" posthumous papers of 1780 relate to Format's*7
problem. In the first paper we find a slight modification of liis4* discussion.
Taking 8=2, r=3+t>, we get
if v=1343/42. Thus p = 1385-1553, 5 = 168-1469, yielding Fermat's solu-
solution A),
Euler, in the second paper, employed his*8 notations, and obtained
АВ*Ъ8-8\В=2г8. Taking A ^FH-2^ B=2tu,
»Mfei.Aaul,Sc.St-Pfterabouigl9r 1319-20,3? 10» 1821-22r3; 11,1830rl; Comm.Arith.,
11,397,403,421.

Сялр.ххщ 2х*-у<=П] Sybtem x+y=A2, х*+уг=В*. 625
we have А1—2В2 —(Р—2и2)*. Noting that a solution involving fractions
may be replaced by an integral solution, he took *=1, whence r=tu.
Equating the two expressions for A, we get
For u=l, S=*3/2. The latter leads to the second value u= — 13, which in
turn gives t= -113/84. Then «=301993/1343, etc. Euler stated that
it is easy to see that the pairs of adjacent values of щ t give all sets of
rational solutions. From the formulas for the sum of the roots of a quad-
quadratic equation, we see that
if u71} u\ I* are consecutive terms of the series.
Euler, in the third paper, set AlB = {l-\-x)j{l-x)\n2AA-Bi=U. Thus
In accord with his141 general method, he set l+6s+JCx**X(pf+8g1), z = \pq*
Cf. Euler,141 end.
V. A* Lebeegue" gave a method simpler than Lagrange's (whose article
he had apparently not seen) to obtain from given solutions of B) a smaller
set of solutions. Since p*-\-<f=x*y we may set p=2mn, q=mi—n\
х*=т*-\-п\ where n is even since p-\-q is a square y2. By the third relation,
m=r*—8*f п=2г8, x=r*-\-&, where one of the integers r, s is even and the
other odd. Changing the eigu of у if necessary, we may assume that, of
the factors H-j-^s—8*dty of 8rV (in view of p-\-^=yt)f the one with the
upper sign is divisible by 2 but not by 4. For r odd we may therefore set
r1 -\-2n-f-\-y=2-r* ^-1-2^-^-^=47^,
и t
where u, t are odd and relatively prime. Multiplying the sum by $t*2, we get
(8) r*(P-ul) -2гЫ+#{2ч*-\-у£) = 0.
For a odd, the right members are obtained by interchanging r, *, and the
new sum is derived from (8) by replacing r/s by — sfr, and t by — L By (8)
()
Since ut and 2u2—P are relatively prime, each is a square or the negative
of a square. But tandu are odd, and ?—2u7is of the form Sk—1 and not
a square. Hence, taking и and t positive, we may set и =Д $=p*, 2/*—p4 ^ A*.
Then
J A
If г, у do not nave a common square factor, r, s are relatively prime and
<rr=fA, <r3**QBy where tr is prime to / and g. Then y=i4ju—2&utt and
Jour, de MaUl} 18, 1853, 73-86. Repiinted, Sphim-Owfipe, 6, I9l l, 133-8*

626 History of the Theory of Numbers. [Chap, xxh
where C=PA2+2fgAB-Q2B2=tfA2+2pB*. Now / divides ry g divides s,
while r and s are < Vx. Hence each set of integral solutions of B) with
x2 + l leads to a set of smaller solutions. For/—13, 0 = 1, fc-±239, we
get A=3 113, B= -2-113 or 3 84; for the first, сг = Ш, г = 39, a =-2,
z = 1525, y=-1343; for the second, <r=3, r=13-113, s = 84, z = 2165017,
j/=-2372159.
Lebesgue noted that it*±2J"y4=z? has integral solutions only when
m=4ndb3 and then may be made to depend upon B); likewise, 2mx4 — yi=z*
only when m—4n+l. But я*±у* — 2m22 is impossible in integers. All of
these cases except cti±Syi=zi and &r4—y4 — z2 had been treated by Euler,
Algebra 2, Ch. 13, whose*9 solution of х* — 2у*=г* is incomplete (Art, 211).
E, Lucas57 gave a complete solution of x4 — 2yi = ^bz2 and ^4-8^*=^,
based on the complete solution of и2-\-г?=у*. He4* obtained the usual
results concerning Fennat's17 problem»
T. Pepin** treated 2x^-1 = П by his™ final method. He" remarked
that Lebesgue** merely stated, but did not prove, that his formulas lead to all
solutions of B) under a given limit, Pepin obtained the same solutions
by a simpler method proved complete. If x7 yt z are relatively prime by
pairs,
where p, q are relatively prime and q may be taken even. Then
±у*=(р*-д*-\-2МУ-$рЪ\ ±z=p< ,
the lower sign being excluded by use of modulus 8. Thus
r, s being relatively prime. By the last, р=Ад, q=hk7 r^XA, $=j*k, where
X; ^, kf к are integers relatively prime by pairs, к alone being even. From
P2 — q2-\-2pq^r2-\-2s2 (the lower sign having been excluded by modulus 4),
kiBfi2-\-hi)-2\fihk-\-\\h2-fj2)^Ol whence
к ^ ^
X^ 2ft*+h7 ' v
Thus д, h form a solution of B), while \K-2№= D. The above is valid if
#>1, whence g=HO. Thus any solution except x=y=z=l leads to a
solution xf=fit y' = h) z'— V2^4 —A*, in smaller numbers, and given by
where 2^—ft4 = ^ АД = (/1Л±£)/{2^Н-^2), from whose numerator and de-
denominator common factors are to be suppressed» We can therefore com-
compute the successive sets of solutions of B) starting with x = y—z — \.
"Recherches виг Panalyee indeterminSe, Moulin», 1873, 254*2. Extract from Bull, Society
dr£mulation Dept. de 1гАШег. l2r I873r 467-72. Same in Bull» Bib). Storia Sc. Mat- Fk-
10, 1877, 23&-45.
" AtU Accad, Pont. Nuovi linoei. 30, 1876-7 220-2.
" 1Ш., Збг 18Й2-3Г 37-40.

Сил*. XXII] ax4-\-hy4=cz1. 627
S. Realis" noted that if <**-2fP=y1, then хА-2у*=г* for
a+1130).
Fora-7 = 1, 0=0, x :y :z=№ : 84 r 7967. For^-3, 0-2, 7^7,
s=57123, y=6214, «=3262580153.
A. Gerardin" treated the last problem, assuming that also a second
solution A4—2B4=C* is known. Set
Then
Equating to zero the coefficient of u\ we get S and u. Taking A = 3,
В = - 2, С = - 7, we obtain R€alis> solution. Starting with 34 -2 • 2* = 73, set
and annul the coefficient of wi2; we get g and w in terms of x, у and hence
a solution of the sixth degree. Modifying the last method, we again get
Realis' solution.
A. Cunningham48 noted that the solution of B) by Lebesgue* and Lucas**
appears to be complete and to indicate that the only integral solutions of
xi-2yi= -1 are A, 1) and B39, 13). But Euler's63 solution of B) yields
only half the solutions.
L. C. Walker83 quoted Fermat's last two integers A), whose sum is a
square and sum of squares is a biquadrate.
OX* -f ЬуА MADE A SQUARE OR MULTIPLE OF Л SQUARE.
The cases з*±у\ 2x*—y4, х*—2у*, х*-\-8у* have been treated above.
For it*—h*y*> see Congruent Numbers in Ch. XVI, especially papers 43, 54.
G. W. Leibniz* treated before 1678 the problem to find an integer x such
that x-\-alx=y1, where a is a given integer and у is to be rational. Set
а=Ъс, x=bz, where с and z are relatively prime integers. Set y = v/w, a
fraction in its lowest terms. Then bz*+c=*v7u^, so that z is divisible by иЛ
Similarly, since cv?jz is an integer tF—bzv?, w^ is divisible by г. Hence
z=v? and Ъм*-\-с=1?. Since с is the product of t>dbip*^5, it exceeds each
of the factors and hence their difference, whence c?>4bwA. The resulting
tentative process to solve x-\-ajx—y* is to express a as a product be of two
integers, choose an integer w such that 4Ъм*<с? and test the value х=Ъ\х?
(or what is equivalent, see if but-\-c is a square).
w Nouv. Coirap. Math., 6r l880r
n Sphinx-Oedipe, 6, 1911, 87-8.
« Math. Quest. Edw, Times, B)r H, 1908» 76-8.
"Amer. Math, Monthly. 11, 1904, 39.
44 Math. Schriften (ed., C. I. Gorhardt), 7,1863,

628 History of the Theory of Numbers. [Chap, xxii
L. Euler" proved that 2а?±2Ь* is not a square if a +6 by means of the
fact that х*^у* is not a square. likewise (от 4x4±yitxi—4yit dbDx4^2y*).
fCf. Frenicle,9 ВвшЬ," Carmichaei.MJ He proved that neither ma*±m*b4
nor its double is a square. Alsow that c4+2&4 Ф □ if Ь Ф0.
Euler17 treated a-\-ex* = П, supposing kuown one solution: a-\-ehA=k\
Setx*=h+y. Then
will be the square of k-\-2eK1yfk-\-elit(ki-\-2a)y*lici if
By use of the substitution x=h(l-\-y)!(l—y), a-\-ex* becomes a quartic
having both the constant term and the coefficient of y4 squares, and hence
is more readily made a square.
J* L, Lagrange8* proved that if й*-\-а#=и* a second set of solutions of
*=z* is given by
To deduce this result, Lagrange made assumptions which he recognized
were not necessary ones. Assume that z=m*-\-an2. Then the given
equation is satisfied if y*~2mnt x7=m*—an2. The latter holds if m—p*~{-aq\
n=2pq, x—pP—atf. The resulting expression for y* is a square if р—8г,
q—f*t pi+O5*=«l. From the second solution, one deduces similarly a
third, etc. But not all sets are necessarily obtained in this way. He
remarked that the simplest and most general method for such equations
is perhape that by factors in his Addition IX to Euler's Algebra (Lagrange1"
of Ch. XXI).
A. E. Kramer" treated px*—y4—z7, where p is an odd prime, and x, у
are relatively prime. Let р=пг-\-тг. Then
bf+mz^iy*—mx1) = (nz1-\-z)(inx7— z).
He took m=r*. First, let one of у, г be odd and the other even, so that x is
even* Set y^+rW^ab, nx1-\-z=act where Ъ, с are relatively prime. Then
the long equation gives y*— r2z*=dct nz*—z=db. Then a, bt c, d are odd
and a, d are relatively prime» Since a=na~r5^, /3^r*a~{-nd have no
common factor except possibly p, while ba=cp, we have a=sct P=sbf
where a = rkl or dbp. Let e be theg. с d* of d and y-\-rx; h that of a/eand
y— rz. Since y1—rtx*=dal8, we get y-\-rx^ef, y—rx=ghf d=eg, a/s=fli7
where /, g are relatively prime, as also e, k. Substituting the values of yt tx>
«Comm. Acad.Petrop., 10,1747 A738)r 125; Comm. AriiJi.rI,28; OperaOmnU, A), ПГ47.
Algebra, St. Fetorebwq^ 2f l770r arts. 209-10; French trand^ Lyon, 2. 1774, 254-203.
Opera Omnia, A), lt 442^3.
m The proof in Ьи Algebra щ the ehorter» The latter was reproduced by A. M. Legendre,
Theorie dee nombree, 1708, p. 406: Maeer, Ut 7; E. Waring, Medit. Algebr» ed. 3,1782.
373-L
♦T Algebra, St. Petersburg, 2, 1770, Arte. 138-9; French tranal., 2, l774r pp. 162-7; Opera
Omnia, A), I, 40&-2.
«No«v. Mem. Aoad. Sc. Berlin, annee 1777, 1779, 151; Oeuvree, IV, 395. Reproduced
by К Waring, Meditation» Algebr&ic&e, ed. 3,1782, 371.
« De quibugdam aequationibue indeter. quarti gradus, Dim.t Berlin, 1839.

Chap. XXliJ at*+hyt = asl. E29
d, a, given by the last four equations, into у2+ггхг^сф/я, we get
Denote the quantity hi brackets by A. Evidently s is not negative. Ac-
According as a is unity or the prune p, we get 2A*=f*-\-pg* or j^-hp/*. Con-
Conversely, any solution of one of the hitter equations leads to a solution of the
proposed equation with р=п*-\-гл, since /, g} A determine x, yt *.
Next, let у, г be both even or both odd» The only modification needed
hi the above case is to divide ^ir5^2, nx?±zt у±тх by 2, and use d/2^eg.
The result is B*=tf*-\-4tfipf8, where
For 8^1, we nave !P=/'+4p04J which implies
Hence the initial equation is reduced to a similar one рЪ*— с**=^#, where
с, & are relatively prime. It thus remains to consider c4—pbt^d*. First,
let one of ct d be even and the other odd. Then &±d=-p&, t?^d=k\
b=eh, whence №-\-ре*~2<?. Next, let с and d be both odd or both even.
Then (c*zbd)/2=4p^ or pit, (ct^=d)f2^ui or 4t*4. Then с2=1г*-|_4ру* or
c?=4tt*-h?w*> which is reduced to the former type by multiplication by 4»
0, Terquem70 proved that neither я*+2^ nor a^rk^ nor z*—&y* is a
square if у Ф0, and that zdzlfz is not a square.
♦ J. Bertrand7* treated oz4V= Q-
С G. Sucksdorff71 treated 2^=t2V^2V forx, y, z odd, positive and
relatively prime. It suffices to treat eight cases having n=p—0, m=4p-\-0t
1, 2y 3; four with the minus sign having w*=^=0, n=4/i+0, 1, 2, 3; four
having m=n=0tp=2fi+Qt 1. First, г^+у^г2. The factors
must be a4, ^t where a&=y. By subtraction, 21^hla;t=a<—^, Hence
ElimiBating a, fi9 we get и*-\-Ь*=Р, of the given type. A like method of
descent applies to 2*+1xi-\-yi=zif whence
(lower sign excluded since the sum of two odd squares is not divisible by 8);
thus $p4=<*i-y\ <^±V=2y4t o?=Fy=46\ whence e?=y4+2P. For
reference is made to Euler's87 treatment of e+ex4^ D, where —y* ]& taken
as a; various solutions result. The impossibility of 24*+*я4+у*=г* follows
70 Nouv. Ann. Math.. 6r lS46r 75-78. ~~
n Traite Шт. d'algrf»re, Paris, I860, 244.
"Dfequintio au et qusfcenu* aequatio ^^±2^—2**1 eohjtione gaude&t in iotefri*. . ► .
Hlfv^ 1851,16 pp.

630 Histokt of тик Theory of Numbers, [Chap, xxii
as for the first equation. Next, я4—24*+1y4=s* implies
whence xz*=<x*-\-8p\ х±о? = 2у*> xTas=4547 ±<*=у*—2#. For the upper
sign we have an equation like the proposed. For the lower sign, there are
solutions, as a=7 = 5 = l. The impossibility of xi-24it+*yi=z* (t=Q or 2)
follows from x*±z=2a\ х2Ъ=2и-'р1, ж*=а«+2***0*. The impossibility of
z*+yi=22*+1zi, x+y, follows from (s*-y*J~2*4V-4zy.
Lebesgue's8* results concerning the equations in the last paper have
been quoted. Cf, Schopis" on яЧ-^+гг2.
E. Lucas73 listed and treated the solvable equations
A) ах*+Ьу*=сг\
in which 2 and 3 are the only primes dividing а, Ь or c, viz», (a, Ъ, с) =
A, -1, 24), A, -2, ±1), A, 2, 3), A, 3, 1), A, -6, 1), A, 8,1), A, 9, 1),
(I, -12, 1), A, 18, I), A, 24, 1), A, ±36, 1), A, -54, I), A, -72, 1),
A, 216, 1), <3, -1, 2), C, -2, I), D, ^1, 3), D, -3, 1), (9, -1, 8),
(9, -8, 1), B7, -2, 1).
T. Pepin74 stated that there is no rational solution of px*—36V4=z* if p
is a prime of the form a2-\-Q№, and many such theorems with 36 replaced
by new numbers, usually by the discriminant of the quadratic form for p.
Lucas stated and Moret-Blanc75 proved that x = 1, у = 0 and x=3, у = 2
are the only integral solutions ^0 of it*—5y*= 1*
Lucasr*° proved that either of 4V4—u*=3w*, 9v4—u4=8w* implies
U4 = V* = W*.
Pepin7* noted that necessary conditions for relatively prime integral
solutions of Au2=Bxi-\-Cyi are that ABt AC and —ВС be quadratic
residues of C, B} A, respectively, and that —ВС* be a biquadratic residue of
A. He proved that и1=Зу*—2х* is completely solved by the repeated
application of
where X, д, /, g are integers relatively prime in pairs such that
g : Х-Л*± >/3/4-2^ : ^+2/i2.
The same analysis gives the complete solution of xi—Gyi=z2 and
He treated other rare cases in wnich the complete solution is found:
ai4-|-7^4=8ii2 and 7x*—2y*^5u2, with the respective auxiliaries xi-\-2Syi=zi
and ^-
n Recberches but I'aDalyee indeterminee, Moulina, 1873; extract from Bull. Soc.
tioD du D^partenieot de Г Allierr 12,1873, 441-532. Bull Bibl Stafia 6c. Mat Tk^
lOr 1877, 239-58.
" Comptes Rendufl Parie, 78, 1874, 144-8; S8,1879,1255; 91,1880r 100 (reprinted, Sphinx
Oedipe, 5, 1910, 56-7); 94» 1882, 1224.
« Nouv. Ann. Math.. B), I4r l876r 526; 20» 1881» 203-5.
*• Nouv. Ann. Math. <2) t 16r 1877» 415,
*Atti AccaA Pdnt. Nuovi IJncei, 31,1877-8, 397427.

Сиар.ХХЩ ах4+Ъу*=а*. 681
A. Desboves77 employed the identity
B) (уЧ-2ух-:г')<+Bх-И,)я^B^
and that obtained by changing x to x2 and у to yi> to show that
C) it* -hay4=**
is solvable in integers if a is of the form Bz-\-y)x?y or 2z*-\-y*. By changing
X to £-|-p ni B) and making other simple transformations, he78 proved
that C) is solvable in integers if a= —&{хг+\?), ±y*~x*t ~x(x-\-l),
у(у=ь2**), x*Bx+y<), y*-2x2, -^(«"-ЛО^+у'-бхУ) with *=O in
tbe last case; and, by other identities, if a = —&(z*-\-if)t — x(x2-\-4), — x2—4.
If A) has solutions s, y, z, then Fermat's method conveniently applied leads
to the new solution7*
D) X=sDaV-3cV), K^yD&y-3cV), £=^4^*^-3@^-by4L],
of different type from Lagrange's solution when<r=c^l. For the examples
of Lucas73 not under Lagrange's case and for which D) do not give all solu-
solutions, we have (a+b)c a square, say Л Using fractional values, we may
set y = l. Then ac{xA-\)-\-v*=<?z7. Setting x = (t+l)/(t-l)9 we get an
equation for which Fermat'e method is applicable. If x} y, z is a solution
of A), then»
Zi=2ax*~ cz*, yi^2xyz> Z\=^-\-4a^i{cz2^ax4)
is a solution of я*+аЬсУ =z\ The latter becomes xA-{-u(i?-~u)yi=z2 for
ac=u, (a-\-b)c = tt. Hence A) is solvable if а=с=1, Ъ~и{&—и), as shown
also by the identity
E* Lucas81 obtained from one solution of ч№*-\-ру* = (\-\-ц)г1 the two
solutions
X=
wbere p=
Since the proposed equation is satisfied ifz=y-<?=zfcl,we obtain two new
solutions. 1Ъив 3x*—2^^^ has the solutions
33, 13, 1871; 28577, 8843, 1410140689.
If A) has the solution (so, Vo, z<>), it may be written in the form
He stated that his formulas above solve completely twenty equations of
"Comptea Kendus Paris, 87, 1878, 159-161. Reproduced, with pp. 321-2, 522, 598r in
Sphinx-Oedipe, 4,1909, 163-8.
"Comptea B^ndue P&r.'^ 87, lS78t 321-2,
п1Ш,> 522; correction, 699. Reproduced in Desbovet' Questions d'aJgebre, ed. 4, 1802.
<X Deebovee.11»
MNour. Ann, Math,, BV IS, 1870, 67-74. In Ьисййг expreaaioD for Z the coefficieot 4 of
the final term ehtmld be 16. If we adopt bla change of signs in m, we tnuat alter a sign
in bin Z,

632 History of the Theory of Numbers [Chaf. xxii
type A) in which a, b, с contain only the prime factors 2 and 3 [erroneous
for & - V=«f, Desboves»1].
Desboves* again gave B) and, by replacing у by v—z and then x2 by u,
deduced E). He noted that C) is solvable in the further cases a=z(y*—x)t
~xvK*+V), -Ф+Й, -xYW-V1)*. He again (W, p. 440) gave D).
He noted {ibid., p. 436-7) that A) has the solutions
If c=81aV—14a*kty+oby, and gave a simpler derivation of Lagrange's
solution of C). For ах*+Ъу*=а?, see Desboves.**
Solutions of it*-hy4 = 17^2are 1, 2, 1 and 13, 2, 41, neither of which can
be obtained (ibid., p. 495) from a solution x, yt г by the formulae D).
T. Pepinw gave the complete solution of 7x*—5y=2z* in integers.
ThenM-X^z, K=xytZ = Gic4-h5^4)/2^veall the solutions of Х'+35Г4=Я*
in which Y is odd; while those with Y even are all obtained by the method
of descent.
S. Eealis« noted that **~3^= 13s2 has the solution
if a4—3/3* ^ 1З72, and asked for the value of z.
Pepin* noted that in Euler's1" method of making a quartic V=Pz-\-QR
a square, not only a rational root of R=0 or Q*=0 or S=0 or F-0 leads
to an infinity of solutions of V= П, but this may be true of further roots.
The latter happens for lls*-^=z*, whence V= 11 -7^-Р*+дЛ, F=2{,
Q=*ll-\-7p, R^l—?. The complete solution is obtained by descent to
two irreducible solutions 1, 1, 2 and 2,1, 13 by four sets of formulas, among
them being an infinity of solutions which escape the methods of Fennat
and Euler. To obtain (pp. 42-48) the complete solution of ;t*+2(V=z*,
that of 5V—m*=^4P is found by descent» From one Bet of solutions x, y} z
of A) for с=а+Ь is derived,* by special assumptions, the new solutions
where /1: X~xy±z : ax*—bz*.
Pepin*8 obtained by descent all solutions of 13x< - lly4 = 2s2 and all of
8x*—3^5**, whereas Eider's144 method to make 40£*-15^O does not
give all solutions.
A. DesbovesM proved that, if (xt y, z) and {x't y', z') are solutions of A),
a new solution is given by
• Nouv. Ann. Math., B), 18,1879,434.
« Jour, de Math., C)r 5t 1879, 405-24.
«/Ш., E), 1, Щ5, 351-8.
" Nouv. Сопор. Mfeth., 6r 1880, 479.
"АШ Accad. F4>nt. Nuotj Iincei, 36,1882-Зг 4Й-67.
«1Ш 67-70- CfL"
Reodiu P*™, 104, 1S87, 84^-7.

where \ = ax*xfi—by*y'*, ^-xyzf-\-zxfy\ For a+b^ct we may set
and deduce his1M and Pepin's87 formulas. For a^c=l> xt^z/ = l, y'^0,
we get Lagrange's formula. He announced the empirical result that the
complete solution of A) in integers is given by as many systems F) as
A) has primitive solutions (xf, y\ z'). For 8z*~JV=5s2, Pepin's ten
systems reduce to the two systems F) with (xf, y% zf) ={1, 1, 1), B, 1, 5).
For80 the case c = a-\-bt set x=y=z^l in F) and drop the accents; we get
Х=а(а-Ъ)х*-Ъ(За+Ь)ху*-2Ъсуг,
while Y is derived from X by interchanging af b and xt y. He gave another
set of formulae of like degree. By finding a relation
such that Y/X is a function of yt x, involving only the irrationality
*, he obtained the quadratic formulas
Y=[2c*-(a-by~}cy+2c(a-b)zt
and stated that a like discussion may be made for
ex«+by*+«fey»«f, c=a+b-\-<L
Desboves" noted that, if A) is solved completely by F) when (ж', у\ *')
is replaced by {xh y\tz\) fort = I, 2, 3, in succession, then any one of these
solutions is called primitive if one does not get it when one determines all
solutions given by the other two and continues the calculations with them,
Desboves* stated that we can find, by a single system of formulae (not
given), the complete solution of ax4—by* ^2^ when a and b are consecutive
primes 8n+7 and 8n+5 or8n+5an<J8n+3.
T. Pepin" treated х*+2к*7у*=г* for k = 2a and 4a+3. He" gave a
detailed discussion of
E. B. Escottw noted that if in х*-\-у*=аг2 we set x=zk{l we obtain a
quadratic for z* which will be rational if (atL—Ba)\kyL = (amF)*, so that
the problem reduces to the pair of equations р*±2ая*= D (Cb. XVI),
Axel Thue* proved that xi—2inyi^l has no integral solutions.
Escottf* solved 4^+1=B*G by noting that the left member has the
factors 2A*±2A+1, whence Bi±lL1^0 (mod &).
A- Gerardin* noted that if (a, 0, y) and (A, B, C) are two eolations of A),
x = a-[-Aut у=0-\-Butz=y-\-8u-\-Cu2 pve a new solution provided a certain
N Compttt Rendu» Parw, lO4r 1887. p. 1832,
*Ibid.t 1602-5.
«Аввос. frans- av. ec., I67 I887r I» 175 (in full).
M Mem. Ace. Pont. Nuovi Linceir 4,1888, 227»
»lhid.t 9, I, 1893, 217-284,
m L'mtermMAire dee nath., 7t 1900, 109 (reply to 3, 1896, 130).
* Archiv tar Math, og NaturvidenskAbr 25, 1903, No. 3.
ьу Lrinterm6diaire dm math., 12, 1905r 165-6.
м ВиП. 8oc. PhUonmtbique, (lO)r 3, 1911, 234-6; Sphinx^edipe, 6, 1911,101-2.

634 Histoky or the Theory op Numbers. [Chap, xxii
quadratic equation in и is satisfied. Equating to zero the coefficient of u1
by choice of £, we get u rationally. He deduced R6ahV№ result.
A, Cunnintfiain* listed aU а*+Ь4=тпс2<10т, l+y*=™*, y<1000.
E. Fauquembergue100 proved Lucas'71 result that 3, 1, 2 is the only set
of solutions of x4—y*^5z4
W. Mantel101 proved by descent that x4-h2V+«2 unless ns=3 (mod 4).
H. C. Pocklington10* proved by descent the impossibility of
where p is a prime Sm+3, and indicated (p. 119) the solution of
R. D, Carmichael"* treated xi+my*=nz1* If there is a solution, there is
an integer p such that прР^з^+тР. Hence we are led to the equation
G) *+mtf-{*+**)*.
A special solution, other than the evident onex=*, y=t, z=lf is obtained
by setting z=pz+mq*. Then G) is satisfied if
A solution of this double equation is found by the method of Fermat:
By the method of infinite descent, he proved (pp. 19-21) that there is
no set of integers, all different from zero, satisfying either of the equations
:r*—4у< = ±Л Hence the area of no rational right triangle is the double
of a square; this implies that x4-hy4=«2hasno integral solutions all different
from zerOv
A» Ge>ardinlM explained three methods to obtain the complete solution
of ox*-hb|f*=cz2, given one solution.
A. Auric105 solved ая?+Ьу*=аРг* by eliminating z between it and the
auxiliary equation тх*+пу* = cdz and making the discriminant of the
eliminant a square.
M, Rignaux1" obtained an infinitude of solutions of x4—#*—az2; given
one solution. *E. Haentzschel10** discussed A).
made a square,
К Euler1*7 noted that in making F=ix^-\-jfcxV-hy4 a square there is a
lack of generality in assuming that F is the square of tf+tppjq or
M L'intenaediBire dee math,, 18,1911, 45-6.
1N L'mterai&li&ire dee math., 19, 1912, 281-3.
» Wiekundige Oreaven, 11, Idl2-#r 491^5.
-• РЮс. Cambridge РЫ1. Soc,, 17,1914, 110.
"• IMoi^aiitiiie AnalysU, 1915, 77-79-
>" Lrintftrm6diaire dee math., 22,1915,149-161.
Ы&. 23,1916, 7-8.
^/«^24191714
^/«^„24,1917,14,
»*• Sitzungsber. Berlin Math. GeeelL, 16,1917, 91
m Not» AcU Arad. Petrop., 10, ad annum 1792, 1797 A777), 27; Comm. Arith,, П, 188.

ах*+Ьу*+Aх*у* Made a Square, 635
x*+xyplq±y*. By a certain device he was led to the case к =/r*+2 fr
in which F is the square of y7+x?Jt+fy*. For 1 </< 100, he gave the least
integer у for which the radical is rational» For half of the positive values
of fc<100and for 30 negative values numerically <100, tables show values
of x : у for which F is a square.
Euler1M resumed the solution of zA+mx?yz+y*=z*. The resulting frac-
fraction for m can be given an integral form by use of a rational number a for
which * = <му-(я?±у*). Then т±2 = (ах2^2)(ау*-2). We may set
x=j?g, у=r«, a ^ 6/(pV), where ptg are relatively prime, likewise r,$. Then
Set b«J—2p2=cr*{ bs2+cr2=2n, bc^X. Then n2—p^Xy2, where yi is the
largest square dividing rf—pK Thus m = (Х$*Т2л)/|Л Conversely, for
assigned values of p, n, 5, the integer x=pq and the largest square y1 dividing
n*— p4 are solutions of the proposed equation with the preceding value of m*
In fact,
Euler gave tables of solutions with a slightly changed notation. In con-
conclusion (p. 498), he gave a more elegant method for the case т=\£*±а,
where «2-4=X^. Then x=0} y = 2$, г=р*±2а£* are solutions. Starting
with two sets of solutions or, ft and 2, 0 of the Pell equation, he derived the
solution
Since дк=1,А*-\В*=4> Thus for m=\p±A (f arbitrary), we get the
solutions x=B, у = 2/of the quartic equation,
Euler"» proved that wi4-H4™V-hn* is not a square if m and n are
relatively prime and m is even and n odd {excluding тга-^0, та —1), or if m
and n are both odd (excluding w=n = l). The question was reduced to
one on с?+3£?~П. By setting x^m*—nz, y=2mn, we see that x*+y*
and ac2-h4y2 are not both squares for x odd, у even Ф0. Another corollary
is that p(p-hg)(p+2g)(p-f-3tfL=D, so that four squares cannot be in arith-
arithmetical progression» Another corollary is p4—pV+ff^ ^ ^ P^+^+O,
and is derived by settingp = m+n, q = m—nioTp and #odd, and^+g=Jrt,
p—q—fi when one of p, g is even and the other odd.
Euler110 elsewhere stated that ^-x^-hl + D if х2Ф1 or 0. This was
proved by the editor of the 1810 English edition, p. 112, by showing in
the Appendix that p2—q7 and зЯ+Зд2 are not both squares.
С F. Kausler111 wrote z=x/y in Euler'B107 quartic F. The problem is
CI, or as a generalization P+bZ+eZ*^P*, Z=z**
Acad. Sc, St. Petertb4 7r &nn*» 181^6, 1820 A782), p. 10; (Ьц. Arith
Ilr 492. For miaprmte and errata Bee Cunningham.1**
1" Mlm, Acad. S^ St. Peterebourgr 8r aoDeee 1S17-18 A780), 3; Comm. Aritb.r U, 411 13-
Same «eulta by Vf A. Lebcegue, Nouv. Ann. Math., B), 2,1Й63, 68-77.
^Algebra, 2, 1770, art. 142; 2, 1774, p. 1Й9; Opera OmnU, A), It 403,
m Nova Aeta А<зА. Petrop., 13, Ad annoe 1795-в7 M€m., pp. 206-36.

6*36 Histohy of the Theory op Numbers. [Chap, xxii
= P*-f2. For a suitably chosen rational A, we may set
Eliminating P, we get Z = {2fA— b)/(e-A*). In our case, e=/=l, Ъ = к,
whence Z=(k—2A)/(A*—1) is to be a square г2» Thus к-ЪА^тр1,
A7—l=?n$*» Of the solutions of the Utter Pell equation, thoee are to be
selected which satisfy the first equation (a " solution " which he admitted
was imperfect)» By eliminating m and setting 2A = a, plq=*2n, we get
ft = a-b(c**—4}ti2, the case treated by Eufer108 at the end of his second paper
Kausler treated at length (pp. 219-236) the problem to make к integral
by choice of rational values of <*, n.
N. Fuss142 required integers m such that х4+тх1у*+у*=гК Set
m-2=ap, m+2=yb.
= pqty=rs, we have
Eliminating г and replacing x, у by their values, we get three linear equa-
equations between a, 0, 7, 5, which give
of which the last may be replaced by 7$=<x0+4. If p = r-l, then
7б = (а^—2)(of«2—2), and a, g, * may be ^.ven any values; as the values
of m<100 we get 2, 8, 12, 16, 17, 22, 23, 26, 31, ■ • >, 94.
B,. Admin118 proved by descent that г'+яУ+^фО. He and T.
Strong (p. 151) also noted that {х*+у2)*-ху=a2 requires that а2+хУ = D
and a*—3x*y2=n = (x2—yz)ii whereas cP-\-<f and a2—Z(f are not both
squares (Euler's Algebra, Second English transL, II, 481). H. J. Anderson1"
noted that we may take x and у positive and relatively prime» If x and у
are both odd, х*+х*у*+у*=3п+3 + D, Hence we may take x even, у odd.
Thus (г2+у*J—зУ is an odd square, whence x*-\-y*=p2+(fy xy = 2pq.
By an argument like that in Euler's Algebra, II, Art» 230, we conclude that
r2—s1 and r2—4s2 are odd squares, where s is even, and r, s are divisors of
xy y, and similarly that f—u2 and f2— 4v2 are odd squares, where u is even,
and f, u are divisors of r, *. Finally, we would reach odd squares u2—w1
and 1^4^, where jttf no longer has divisors» Hence the problem is
impossible.
A, M. Legendre11* found only two solutions of m4-4mW+r&*=p*7 viz»,
(ти, щ p) = A5, 4, 191), D42, 161, 364807). The complete solution, includ-
including B, 1, 1), was given by E» Lucas.11*
"* МЫ* Acad. Sc, St. Peterebourg, 9, 1824 A820). 159,
шТЬв MatK Мшу, New York, 1, 1825, Н7-1Б0. Cf. Genocchi'" acd
aJso Bdia-EddJD» of Cb XIV and Kaualer1' of Ch, XXVI.
"*ТЪбог1е<1ев1юшЬ«!в,еа.З,2, 183Or 127; Mwcr, U, 124. S«e Legendre47 of Ch, XIX.
i[i Recberches виг ГаавЛуне indeterminee, МоиДпб виг АШегг 1873» p» 97. Bull. Bibl. Stori»
Sc. MAt. Fie., 10,1877r 29Ь2,

на»» XXII] Ш^+Ь^+сЯхУ MADE A SQUARE. 637
V» A. Lebesgue117 noted that If xi+azty*+byi=z* has the solution x = r
= s, z=pt it has also the solution
x—r4—Ья4, y=2prSj
A* Deeboves118 remarked that this generalization of the result by
Lagrange** for a=0 is insignificant since it is made by replacing his initial
identity (the following for d=0) by
(uz-bt?¥+d(u2-b^Bvv+dt?)+b{2uv+dt?y = (u*+^^
which Lagrange gave in his addition IX to Euler'a algebra (French trans!
2,1774,040).
A» Genocchi11* proved by descent that х4-{-х2у2-^у4^О.
T. Pepin120 treated ^-f-fc^+l^ D,
E. Lucae121 deduced two solutions (Xt Y, Z) from а given solution
(x, y, z) of
For brevity, set
m= -
0
Then
A. Desboves123 noted that if x, y, z satisfy аа?+Ъу*+дхгу'1=сг*> then
satisfy Х*+аЬс*¥*+ЫХ*У*=22; while138
satisfy the initial equation i! q=axi—by*,f=:(P~4ab. CL Desboves.*0
Т. Рершш treated az?+2bxty*+cyi=n2> a necessary condition for which
is that the quadratic form (а, Ь, с) represent л3. If one such representation
is known, all are given by quadratic functions of two parameters. But in
returning to our quartic we are led again to the problem to make a quartic
a square.
Moret-BIanc1M found solutions of xi—Ъхъу*+Ьу*= □ and
(z«+yh)((z+y) = D
by Euler's method»
S. ReahV» proved that 2^—2y*+l — D only for y=0, 2.
"'Jour, da Math., 18r 1853, 84; Nouv. Aim. Math., B)r U^ 1872, 83-6» ^~
"«.Comptes Rendtis Para, 87,1878, Й25-
^♦AnnalidiSc»Mat. eFie., 6, 1855,302. Of, Adrain.»u
110 Atti Aocad. Pont, Nuovi Lmoei, 30,1876-7, 222-4» <X Euler, Algebra 2, Ch, 9r Art. 144
ш Nouv. Ann, Math., B), 18,187», 73,
**1Ш., B), 18,1879,384; fora=e^l, p. 437, Verification, B), 19, 1880,461-2.
mIbid.t B), 18, 1879, 440; implied, Comptes Rendua Fttb, 87, 1878, 522.
« Atti Accad. Pont. Nuovi Liocei, 32,187&-9, 79-128.
J* Nouv, Am. M&th.r B), 20, I881r 150-5.
ш Bull. Bibl. Storia Sc. Mat. Fia,, 16, 1883, 213; reproduced, Sphinx-Oedipe, 4, 1909,175-6
See рарегв 19-25 of Ch. XVU.

638 Histohy of the Theory of Numbers. [Chap, ххп
E. Fauquembergue12* gave the genera! solution of {x2-^yr)Bxi—y2) =2Л
A. Genurdm1* gave x, yt z=3/, 4/, 5/* md A/2, 2Л/3, 5tf/36.
x4+4x*-hl=Vab impossible in rational nmnbexs»m Cf. Pietrocola.
T. Pepin1» treated ^-usy+Sy4^** by the method of descent appli-
applicable only if у is even; then *=Х»-8У», y=2XYZ7 z=ZA~22X*Y\ For
у odd the equation is reduced to the pair pq=rs> p*—4g*-h4pg-h8a*—r*=0,
to which the method of descent is applicable. There exist only six sets of
solutions x, y, zf each +0, withy<104
C. Pietrocola1" discussed the equivalent equations
From one solution he derived another and proved the equation impossible
if ft=l. The last result had been proposed as a problem by P. Tannery.m
A. S, Werebrusow181 listed many values of m between —100 and +100
for which z'-hmzV-hj/4 =г* is impossible, and stated that it is impossible for
m positive or for m—$k+3 negative if m+2 and m—2 are primes,
A. Gerardm noted that the last statement fails for ro^99.
Gleizes and H. B. Mathieu116 gave special expressions for m for which the
equation is solvable.
A. Cunningham116 noted that the equation is solvable for m=60, 99,
—72, —96, contrary to Werebmsow,lte and for w*=91, —90, contrary to
Euler10* (p. 495, p. 498); and corrected various misprints on pp. 49(HS of
Euler's paper.
L. Aubry137 stated that WerebrusowVM theorem is true for a positive
я*=1,5ог7 (mod 8), and a negative m=*—(8&+5), but false for a positive
m=$k+3. Aubry (pp, 57-9) treated ^
by setting x?=p*u—cq\ y*=q*u+(b<f+pi)v and deducing an equation of
the initial form, whence one solution leads to two new solutions»
H. <X Pocklington1M proved that z'-sy-hy4, x4-hl4x23/2-f y* are neither
equares if x Фу. If N is not of the form 8n±3 and is not divisible by any
prime 4n+l, and at the same time JV^4 is an odd power of an odd prime
(including unity), then (x*+ysJ41Nx*y*=z2 is impossible in integers. For
N^l and the upper signs, we see that х*+х*у*+ур=г* is impossible. Also
x*—14«y+¥*=** is impossible. There is a Ust of values of л<100 for
which х4±ихгу2+У4=^1 is impossible. The complete solution is given of
h., 4,1897, 70.
»/№., 16,1909, 175.
»■ IWLt 1897, 20, 83» 203, 220; 1808, 89, 123; 1900, 87-90; 1909, 158; 1905,109,
"•Mem, Accad. Pont. Nttovi Uneei, 14,1898, 71-85,
"i Giornale di mat., 36,1S98, 77-80.
]« Lrmt6noedi&ii* dee m»tbA 1897, 20, 30, 203.
»»1Ш., 15,1908, 52, 282 (oomsctioiie); Mat. Sboraik, Moscow, 2fi, 1008* 599-617.
"I/mteim&li&Lre d<* math., 16,1909,154.
»Ibid., 15,1908,169.
Ж1Ш>, 17, 1910,201,
"/Ш, 1*1011, 208.
UiP№c, Cambridge PM. Soc,, 17,1914, Ш-Ц8.

Chap. XXII] QuAETIC FUNCTION MADE A SQUARE. 639
Cases in which axA+dx*yt+by* = П is impossible were noted by
Lebesgue,»'» Genocchi» M and PepinM of Ch. XXVI; by Desboves188 of
this Chapter. Solvable cases by Ptepin15* of Ch. I, Haentzschel of Ch. V.
QUARTIC FUNCTION MADE A
Fermat1* sought rational values of x for which
вЬяИ equal the square of a rational number, where a, • • •, e are integers.
The case in which a or € is the square of an integer is quite simple. For
a=o1, the first three terms of f(x) are identical with those of the square of
Comparing the terms with the factor x2, we obtain
X
Hence from a particular solution f(Q=c?f we may obtain new solutions
since/(£-hc) =ct*+bz-{ f-es* falls under the last case.
The same special cases were treated similarly by L. Enler1" and
A. M. Legendre.141
T. P. de Lagny142 made х*-Мкх?+26х*-7х+9 the square of х^-
for x=23/5.
L. Euler1*3 treated in a posthumous paper the equation
get c—M—2ad*=mn. Then
B)
This is satisfied if
Admitting fractional solutions, we may set y= 1. Then
For a fixed X, let p and q be given solutions. Let p' be the second root of
this quadratic in p, whence
p'= -p
Then p\ q' are corresponding values if
ш Diophanti Alexandrini Arith. libn Sex . . . Doctrio&e AnalyticAe Inveohim Nu;
CoII«ctuzii Л J, de ВШу ex vanjs Epietolie qu^a ad ешп . . . miait P. de Fermat, p. 30.
Frencb tractal., Oeuvn» de Fermat, 3,1896, 377-388 (the term i4 ifl omitted on p. 388.
™Afcebia,2,l770,Ck9>Noe.l2&-137; Etench transl., Lyon, 2,1774, ррЛ53^1в2. Opera
Omni*, A), l, 1911, 396-400. Sphitcc-Oedipe, 1908-9, 67-78.
ш Tbeorie dee nombroi, 1798, 458-9; ed. 3» 2, 1830,123; Maser, II, 120.
m Nowr. Elemena d'Arith. et d'Aig.f Parie, 1697, 436.
шМёт. Ac*d. Be. St. Petenjb., И, 1830 C17801 l; O>mm. AritU,, II, 418,

(И0 History of the Theory op Numbers. ЕСыл*. ххн
From p\ q*f we get p", q", etc. Any two consecutive terms of p, q, p\ g',
p'\ ••- yield a solution with y = l. Proceeding in the reverse order, we
obtain a sequence q> p, q^ piy q2, - • *, any two consecutive terms of which
yield a solution*
To obtain an initial pair of solutions, set y=l and let the quartic be
the square of ax*+bz—d or of ax1—bx—d; then
4bd y-2qd-c
To treat а&^Р^П, where a±0 is a square a*, set C-=(l+s)/(l—a:).
Then
which is of the above type. Euler treated in detail the cases
Euler144 treated V=A+Bx+Cx*+Dz*+Exi= □. If F can be given
the form P*-i-QR, where
then У=(Р+ОуУ, where 2Ру+О2/1-Я=О. The latter is also quadratic
in xt viz., Sx*+Tx+TJ=ft. From initial solutions x, y, we obtain141
x*= -z- Г/jS; then from x' we get y'= -y-2P*lQ\ ete. As in the pre-
preceding paper, we thus obtain two series of solutions of V = П.
If, for E=0, V=Pior x = a, we may take
p=ff Q^x-a, R=B+C(z+a)+D(x*+<ix+at).
For a general F, let V=p for z^a. When z is replaced by a+tf let
F become f*+<d+fiP+ye+№9 Then F=Qf
48 gave ten values of z for which
including
G. Iibri14r treated a^-hk^-hc^+ds-M^ with all coefficiente positive
(since we may replace x by xi-hu). Multiply by 4aJ and set
Thus
C)
1 Mun. Ac»d, Sc. St, Petereb», 11,1830 [1780;], 69; Comm. Aiith., П, 474. Cf. p
Нш method of eolving аду equation quadratic in x and in у was given by Euler also in
Мёш. Acad. Sc. BtP Petenb^ 11,1830, 59; Comm. Arith., П, 467. For application* to
rational quadrilaterals,aee Kummer™ and Schweringm of Ch. V. Cf.рарем 55, 143,
148,155; &1йо Fepin^ of Ch. IV, Guntache11' ш of Ch. V, On the relation of elliptic
functiona to an equation quadratic in % and in yt вее Q~ Frobenius, Jour, fur Math., 106,
i860,125-188.
Opera poetmnA,l, 1862, 266 (about 1782).
Jour, fur Math., 9,1892, 282.

Chap, ххщ Quabtic Function Made a Square» 641
A positive v cannot surpass a certain number L which makes every coefficient
in C) positive; hence we have only to try v—0, 1, • • •, L — 1. If v=^tf
where Q<t<x, let s be the least t for which 4a*(t+c)>№ and substitute
s+wfor —vm. C); we get an equation КкеАх2+Вх+4аЧ = (8+и>У with all
coefficients positive, whereas £*>£*= (s-fw)*; hence the only cases to try
are v=— 1, •••, —E — 1)» Finally, if v= —u9 0<u>xt let r be the re-
remainder <x on dividing и by x and n the quotient. Set
4a V = [2a V+(Ь -л}* - rp.
By*2>a*s*, we have b>n and need only try n = l, • ••, b—1.
C. G. J. Jaeobi14* stated that the analysis of Euler144^* to find an infinitude
of rational values of x, given one, making the quartic f(x) a square is the
same as that of Euler's"» (earlier) solution of the transcendental equation
Л ЫДх)
For the latter, Euler used a chain of л equations /(p, q)=0, Дд, т) = 0,
f(r,9)~0, •-, where
is symmetrical in p and g, whereas in the diophantine problem Euler's
canonical equation Qyi+2Py—R=8z2+Tz+U=Q is not symmetric in
xt yt as pointed out by L. Schlesinger,iw who discussed at length Jacobi'e
above remark. The latter had been discussed by T\ Pepm.wl Jaeobi
observed that the analysis of the multiplication of elliptic integrate D)
gives an infinitude of rational ^'s for which also V/W и rational, H a rational
x makes "ij(x) rational, and drew from the theory of abelian integrate the
conclusion1": If f(x) is of the fifth or sixth degree and if one rational value
of x makes V/(x) rational, there exist an infinitude of x's of the form
a-h&Vc, with a, b, с rational, for which V/S) =«'+&' Vc, with a', bf rational;
and the extension to f(x) of degree 2n-f 1 or 2n+2 and a's satisfying an
equation of degree n with rational coefficients, J. Ptaszycki15* remarked
tbat the last theorem follows at once from the representation of a rational
function by means of polynomials which enter in the development into a
continued fraction of the square root of this function. The generalization
of Jacohi's theorem has been considered by J. von Sz. Nagy.1"
The problem to make a quartic a rational square was proposed in 1865
as a prize subjeet by the Accad. Nuovi Lmcei of Rome.
L. Calzolari1" wrote a-hbiF-hc^-h^-heP4-^2 in the form
ш Jour, fur Math., 13Г1ШГ 353-5; Werke, П, 61-5,
™ tnethutionea Calculi Integrate, 1,1768, Ch. 6, Ftob. 83, 5 642.
» Jafarober. d. Deutaohen MatlL-Vetemig., 17, 1908, 63 (with hietory of /(*)«. D).
ш Atti Aecad. Pont, Nuovi liiwei, 30,1876-7, 224-37.
ш ^ Jftoobi, Jour, fur Math., 32,1846, 220; Werke, II, 135; Schwering!1* of Ch. XXI.
» Jahresber. d. Deutechen Mfttb.-Vefemig , 18, 1909,1-3.
™1Ш47 aN»CChX3OlL
, gy
« СИопЫе di M*t-f 7,1869, 317-50.

642 History of the Theory of Numbers, [Chap, xxii
Set Q = y[x, 2w=z/z, c'v+b'=utx. Then
u*=Ax*+By*+Cz\ А = Ъ'г-а'с\ B=-cr, C=c'e,
which can be given the form Ai(u2—??)=Bi(y2—г3) by choice of k. The
solutions for щ x, у, г are evident. Substitute these in the quadratic in
«, x, у obtained by eliminating v between Q=y/z, с'г+Ь' = и/х. For
example, if у*-2=^, then w2 = (vt+k7)-2ktf-№-2> Set &+к=и/х,
v=y/z, w=zfx. Then
for k = — 4, It has the solutions
«, г = К"Г±0*); 12*, 8y
Substitute these in их—кх*+у* (obtained by eliminating v). Thus
with coefficients quadratic in y, 5. Taking L=0 we get four &ets of solu-
solutions a, /3, 7, d; likewise four from N=0,
S. Bills14* made /=z4+4x5-f-8x2+7;r:+6= П by noting that /-4 for
z=— 1 and setting/-Q', О=д?я+2я?+А, where A; is chosen so that Q = ±2
for ж= -L
T, Pepinl№ made use of the notations of Euler,140 viz,, A) and
B{x)=f+gx+hx\ F(9)=f(M)-e{M)-a-f*+...+(e-h*)*.
Pepin took X}, a*, x3 arbitrary but distinct, and determined /, g, A, x by
E) в{х,) = п^Ш, *' = !, *-^=?-x,-x,-x, (г=1,2,3).
Then a?i, xi, x?> x are the roots of F(z)— 0. Hence if Xi, x2, xj are three
solutions of/(x) = n^then/, g, A are rational and x is a new solution. Next,
let xa=Xi; then F'(x{) =0, and E) for г^З is to be replaced by the deriva-
derivative of E) for t = L Finally, for Xj=x2=x3, we use E) for г=1 and its
first and second derivatives, and so obtain a second solution from a first.
Then the preceding case gives a third solution and E) a fourth solution.
Pepin1M noted that if a quartic f(x) can be transformed into a square
by replacing x by a rational function, then F=y*-f{x)^0 is a unicursal
curve and hence has three double points, whence the partial derivatives of
F with respect to x and у vanish, showing that / has a double root. The
problem is then to make the remaining quadratic factor a square. The
problem to make a product of two binary quadratic forms a square is
treated by means of a congruence. Conditions are given in order that a
reciprocal quartie shall never be a rational square for a rational value of the
variable.
A. Desboves1" noted that if x, y} г is a set of solutions of
* = cZ\
u Quest. Educ. Times, 22,1875, 91-2.
»»Atti Aocad. Pont. Nuovi Linoeir 30, 187G-7, 2П-37.
«• Atti Accad. Pont, Notm Lincei, 32t 1878-9r 166-202.
»»Comptce Reodue Pane, 88, 1879, 63&Ч0, 762 (correction). Cf, Deabovee1* of Ch. XXL

Chap. XXII] QuAKTIC FuNCTIOtf MADE A SaUARB. 643
formulas can be found giving in general four sets of solutions» In
consider ax4, etc, as coefficients; we thus have an equation of the first type
having now a+ \-g=c (an artifice due to Lucas81 for d=f=g=0).
After dividing such an equation by с and setting X = (p+x)f(p+l)f we get
an equation in p to which Fermat's method applies. The explicit formulas
for the two sets of solutions are very long (each furnishing two sets by
changing the sign of z).
F. Romero160 proved that x'-ht'-M^-bx—1— y1 has no positive integral
solutions. For, у is odd and the equation becomes
Thus x=4n-h2,and 4«H-3 would divide the sum of the squares of two rela-
relatively prime integers.
E. Lucas161 discussed f(x) ^ y*t where f(x) is a quartic with rational
coefficients. Set уф(х) ^F{x)> where Ф=х9+а1х9~1 + " * with rational a's,
while F is of degree p+2. Then F*=f4? is an equation of degree 2p+4 in
which enter 2p-h3 unknowns besides z- If we know 2p +3 sets of rational
solutions Xi, Vi of p*—f(x), no two of which differ merely in the sign of yt
and determine the coefficients in уф—F so that it shall be satisfied by these
2p-h3 sets, these coefficients will be rational. Then F*=f<f,* will furnish a
new rational x which leads to a new set of rational solutions of y*—f{x)*
We may take two or more of the Xi equal; if x^=xu we replace
by the derivative of ± V/(Si) —Р(Х\)!Ф(Х\)- Taking all of the x; equal, we
see that one solution of f(x) — y7 leads to an infinite sequence of solutions,
(Cf. Pepin.1**) lij{x) has a rational root a, we may take
If/has a rational quadratic factor q(x), we may take F—q^p and apply the
above method to 2p+l sets of solutions»
L, J. Mordell1" assumed that we have one solution of /=z2, where /
id a binary quartic with the invariants gZi g*. Then we can transform /
into a quartic with leading coefficient г2. The syzygy between its semin-
variants (ef. Mordell176 of Ch. XXI) is д2=4Н*-д*№-д^. Thus g/z*, kfz*
give rational solutions of
**^4s*—gia-gu
It is shown that the knowledge of all rational solutions of the latter le&ds
to all rational solutions oif=zt*
К Haenteschel1" treated tf=f{x)=a#*+-—+OA. Hrst, let f(x)-=Q
have a rational root v and apply the substitution
"• Nouv, Аид. Math., B), 18, 1879, 328.
шМоиу. Correep. Maih.j 5, 187», 183-6.
ш Quar. Jour. Matb., 45» 1913-4, 17&-18L
«Jour, fur Math., 144r 1914, 275-283.

644 History of the Theory of Numbers. [Chap, xxii
We obtain Weierstrass' normal form
F) vt=4s*-gis-gi=*(8-e1){s-e2){s-e3),
where glr g3 are the invariants of/; also y=*±if(r)vj(8—t)*. Euler27 of
Ch. XV discussed the problem to find s such that a—e< are squares for
г=1,2,3 {whence their product gives tfifi), but evidently restricted attention
to the case in which each e, is rational, Haentzschel showed how, from three
primitive solutions of F), to find four infinite sets of solutions by means of
Weierstrass f-function.
Removing the assumption of a rational rcot r, but assuming one solution
хъ, У а Qff=y*> be applied a certain linear fractional transformation giving a
quartic whose leading coefficient is a square.
G. Humbert1" stated that all the methods which have been proposed to
deduce rational solutions of o&-\ \-e-& from one or more initial solu-
solutions are identical at bottom, and gave the method in geometrical and
analytic form.
On х*±х?у+х?у2±ху*+у* = □, see papers 63-66 of Ch. II, Vol, I. On
ху{хг—у*) =Az*, see papers 11,18; also Congruent Numbers in Ch. XVI.
For other special quartics made squares, see papers 101 of Ch. I; 21,
92-4, 96-7,109,138-40 of Ch. IV; and 9, 72, 73, 77,92,133 of Ch. V; and
various papers of Chs. XIV-XX.
К Euler166 took A=p+q, D=p—q, C=r+s, B^r—s and derived
(l) pgfp+f)-
Set p=ax} q=by} r=kx} s=y. Then
If *=<*,««!, then у ^±a, C==fc^, jB==FZ>. Set therefore k=ab(l+z).
Then и*№=аЧЦ(&-1-гУ, where
Let Qbe the square of Ь2—1+/«+рг* and choose /, g to make the terms in
z, ^ agree* Thus
3b*-1 _364
} 9~ 8
} 2
Then x :y = ¥—l-z :a(P-l+/*+0e?). As examples, Euler took h = 2t
Ъ=3, and found the solution
4=2219449, B= -555617, C= 1584749, Z>^2061283,
and an erroneous118 one replaced in his next paper by
4 = 12231, £=*2903, C=10381,
>» L^ntermediure с!ед math., 26, 1918, 18-20.
m Novi Comm. Acad. Petrop., 17,1772, 64; Comm. Arith-r I, 473; Op. Om., {l), 1П, 211.
«• ТЫа error w« aleo noted in I'intermfeliaire dee math,, 2,1895,6, 394; 7,1900,86; MU
aie, 1889, 241-2.

xxiij A4+B* = С*+&. 646
Euler treated а*~Ь*=с*-& by Betting
Multiply the first by p, the second by q, add and subtract. Let <f—p*=A
Tben
B) V*=a*(p>+tf)-2c*pqf 2#pq=a**-bP(&+&.
1д Bi) take bs=a(q—p)+2p(a-c)x, whence
a \c=2pxXJrq ; 2px*-f2(g— p)x—q.
Taking a and с equal to these expressions, and multiplying B2) by z?lBpq/t
we find that
which is readily made a square since the first and last coefficients are squares*
For p^3, g=5, we have «=4 and
C) &~*£-В5х-2№*+102х*+9х*.
If we seek to make three terms of d* identical with those of the square of
2, we find that c*=a4. But
for z=ct+px, x-—E±i)h; also for z^x(txJrh)> x^—al($±t;). For the
special form C) we therefore get x= —15, 11/3, 1/18 or 5/22, each leading
to a permutation of the same values
a=542, 6=359, c-514, d=103.
EulerlM treated the following generalization of A):
pq(mp*+nq*) =T&(mrtJr ne2)*
Set q=ra, 8=pb. Then p* :r*=na*—mb :тЛ*—та. Set
Then
We may make C(l -jfe) = П by the usual methods for quartics. But we
obtain much simpler solutions by making C/(l— рг) = A+сЬ)г, viz,,
Taking 2d=3a+pf we get z= -
For m=n=l, Ь^З, we get a = 9/8, 0 = 1/8, d=7/4, г=-96/103,
p/r=25/193, and obtain the solution p=25f r = 193, q=291, 8=75 of A),
»7 Шт. Acad. Sc» St, Fctet*-r 11, 1830 (X780), 49; Comm. Arifch., U, i50. Euler wrote
с*-Ь<2* in hie second equation and ^—<? in Ыя third. Hie further formulas require that
<P be replaced by —d*, which would invalid&ta the conolusione, In the present report,
tP has been repUoed by —d1 at the outeet, so that the remaining developments become
correct aa they stood.
"• Nov» Acto Acad. Petrop., 13, ad annoe 1795-6, 1802 {1778), 45; Comm. Arith., II, 28h
To conform with the notations of Eider's first paper» the interchange of a with p, Ъ
with 9, с with r, d with t has been made. Abo, Opera postunia, 1, 1863, 246-9 (about
1777).

646 History of the Theoby of Numbers. [Chap, xxii
whence
D) 1W+594- 133<+1344.
For TO=n — 1, h=f/g, we get а=Р!(Г—д3)* In the resulting fraction
for p/r, take p to be the product of the numerator by g. We obtain the
solution of A).
The case/=2, g=l gives p = 275, g=928, r=626, «=550, whence
2379<+27<=729Ч577\
From one set of solutions of A) we obtain the second set
A. Desboves1M noted that
Desboves170 wrote &jq = m in A) and obtained p2+pg3—rrf<fr—mr*=0.
Regard m as a parameter. From the solution р=т, q=r = lJ we derive
by Cauchy's formula the new solution
q=(m2+l)(-m4+l$m*-l),
Replace m by f/g* The resulting solution is not new, as supposed by
Desboves,171 but1" is Euler's E). For/=1, g=3f we get D). For/-1,
g~2y we get Desboves1189 numbers.
A. Cunningham178 discussed the solution of the problem and proved the
impossibility of ^-fj^^^-f 4tj4.
R. Korrie,174 starting with an evident solution of A) took -p=pXi—st
r—pxi—q; thus
After making the coefficient of p zero by choice of x%\z\f we have only to
take — p equal to the ratio of the coefficient of p* to that of p3. After reduc-
reductions, we obtain Euler's E). The same method applies also to
A. S. Werebrusow17* gave 2394+74-2274-fl57< and Euler's solution D).
T. Hayashi176 reduced the problem to the solution of Зи4-^ f4—w*y from
one solution of which we obtain an infinitude (Desboves77).
)
p))
squares in eight ways and noted that the squares are biquadrates ш two
cases, giving Euler's D),
"•Nouv. Com*p. Math., 5, 1879, 279.
™ Аяюс. frang», 9,1880r 239-242.
ш N<mv. Сотяр. Math., 6, 1880, 32,
171 Noted by £. Faaquembeigue, Matheris, 9, 18&Q, 241-2; reproduced ia Sphinx-Oedipe, 5,
1910, 93-t
in Me^enger Math., 38,1908-9, 83-9.
""TJiuvenity of St. Andrews 500th Anniversary Mem. Vol., Edinburgh, 1911, 60-1
i™ L'ibtermediaire dee math., 20, 1913, 107; 19, 1912, 205.
»»The TAboku Math. Jour., 1,1913,143-5,
"Fteriodieo di Mat., 28,1913, 78.

Chap. XXII] A<+KB* = €*+№>*. 647
E. Fauquembergue171 gave the identity
8=4*4+132а*0+17^+13200»+ АР,
while by Fermat's method we may make e^D in an infinitude of ways,
e.g., «=8,0=25.
A. S. Werebrusow17* gave 2924+1934=256<+257\
J, E. A, Steggall treated xn—un-yn-vn by setting
Xx=l+ab, by = l+act Xii=a*-1-f&, \v=an~1+e,
wbieh determine a, Ь, с, X in terms of x, yt u, v* He discussed only the case
n=4, whence
This is satisfied if b+c=2a(l+t), and
A particular value making the left member a square is
whence we derive one of Euler's tentative solutions. The emaUest set of
integral solutions is said to be D).
M. Rignaux181 recalled [Euler18*] that A) is unaltered by the substitution
p^P+Q+R+8, q=P+Q-R-S, т=Р-<}+Е-8, s=P-Q-R+S.
He obtained (p. 128, pp. 133-4) various solutions of A),
A. Gerardin1*1* noted that A) has the solution
which is simpler than Euler's solution E)*
. Grigorief182 noted that
the latter being erroneous, He1M found an infinitude of solutions when
h=2t the least having eleven digits (from «=33, v=13), by making special
assumptions leading to the condition ЗгИ—21^=1^,
A, S. WerebrusowtM gave 1394+2-344=614+2 Ш\
A. Gerardin186 treated a4-f W=d+h& by setting a—c=m, d—b=x,
ee math., 21, 1914, 17 A8-19, bibliography).
i» Proc. Edinburgh Math. Soc.r 34, 1915-6,
ш L'intermeduire de» math.r 25, 1918,27-28.
^/bL2751
1B LrintermediAiie dee m&th.7 9, 1902, 322; 10, 1903, 245.
iU IbuLt 14, 1907, 184-6.
™1Ш.ЛП> 1910.127.
"»gpbinx-Oedipc,6r 1911, ft-7T 11-13. (XNome.

648 Histoby of the Theobt or Numbers. tCwur. xxil
a+c=p(d+b); thus
Equate to aero the coefficient of ЬК Then that of b is zero, and we obtain
m and h rationally in terms of p, ct x. In the special cases p=cx and
c-x = l, the resulting identities are simple. He gave solutions of the
systems formed by х*+тх*у*+у*=аг and various other quartics.
Gerardin1* gave solutions of а4+№=*с*+Л# for 26 numerical values of
h, and noted the solution a=2p*, c^2p; b,d=p=Fl; к=2р?(р*—1).
Sum of three biquadratic never a biquaphate.
L. Euler166-ш stated that this theorem was hardly to be doubted, though
not yet proved. Again helw stated u It has seemed to many Geometers
that this theorem (зж-Ь ynфгп, n>2) may be generalized. Just as there do
not exist two cubes whose sum or difference is a cube, it is certain that it is
impossible to exhibit three biquadrates whose sum is a biquadrate, but that
at least four biquadrates are needed if their sum is to be a biquadrate,
although no one has been able up to the present to assign four such bi-
biquadrates. In the same manner it would seem to be impossible to
exhibit four fifth powers whose sum is a fifth power, and similarly for higher
powers."
Euler1*7 noted that abc(a-f &-f c) = 1 has the rational solutions 4, 1/3,
1/6, and obcd(a+b+c+d) = l the solutions 4/3, 3/2, -1/3, -3/2. Hence
we cannot infer the impossibility of p*-f 5*4-3^=^ by setting a=p?fqrs,
b = tflpr8,c=7*lpq8; northatof p*-fg*-frs+^ = ^by setting a = pVgrsf, ■•■,
d=s<!pqrt.
A. Desboves1*8 expressed doubt as to the theorem and proved the im-
impossibility of p*±6pV~7<?*= О in connection with a study of
which has the solutions
L, Aubry189 proved that the fourth power of an integer ^1040 is not
a sum of three biquadrates.
A. S. Werebrusow1^ showed that no solution can be found by making
each term a biquadrate in Euler^s identity
StJM OP FOUB OB MOHB BIQUADRATBS A BIQUADRATE.
L. Euler190 remarked that it seemed possible to assiga four biquadrates
whose sum is a biquadrate, but that he had found no example, whereas he
"Sphinx-Oedipe, 8,1913,13. ——
"Operapcwtiima, 1.ДО2, 235-7 (about 1769). a. Euler.»"
"*Noav. Cotreep. Math., 6, 1S80, 32. a. Sphinx-Oedipe, 8, ШЗ, 27.
"• Sphuu-Oedipe, 7,1913, 45-4. Stated, Pzntermed. dee math., 19,1912, 48.
"•• Lintermtfiaixe da math. 21, 1914, 161.
*« Омпйр. Math. Fhy* (ed., Fuih), l, 1Ш, 618 F33), Ад«. 4,1763. See preceding topi*

. XXII] SUM OF BlQUADRATES A BlQUADHATE. 649
could give five biqufrdrates with a biquadrate as sum. He*7 again re-
remarked that he was trying to find four such biquadrates.
Euler1" gave an incomplete discussion of the u difficult " problem to
find four biquadrates whose sum is a biquadrate. Evidently
A4+B4+C*+Di=E*
for
A* = (р'+аЧ-г* -e*)/n, B* = 2ps[n, C2=2qslnf
These five functions are to be made squares. This will be true of the first
and last if
A) fo4-flM-H)/*-«4-*, 9>(n=2ab.
Then z?^2abn= D if 2п~сф, a=af*, b=pg*t whence s^a&fg. Next,
> * 2rs
4*
tfp=3?/(fg)tq~V*l(fg),r=z*l(fg)> Substitute these values into (li); we get
But no discussion of this final condition is given.
D. S. Hart1** employed the sum
of n consecutive biquadrates 1*, - *., n4, and
Thus (&+mL can be expressed as a sum of biquadrates if <r—t is» Evi-
Evidently n>S. For n=9, 5 = 14, m=*l, <r-£=
yielding
B)
Forn^20, s=30, m^4, 34< is the sum of the fourth powers of 1, 3, 4, 5, 9,
10, 11, 12, 14, 15, 16, 17, 18, 19, 30,
A. Martin188 gave B).
A. Martin1" started with the identity
But 96=34+2<—I4. Hence the new right member has six positive bi-
quadrates and the term — (m2L. For m=2, the latter cancels BmL and
we get
which, was communicated to him by P. S. Hart. For m=3,
where
i" Opem poetunm, 1, I862r 216-7 (about 1772).
«■ Mfttb. Qu«et. Educ. Times, 14r 1871, 86-7,
^Ihid.t2O,lB7Z,B5. Lrintcn»6diaire de* math., 1,1894, 26.
" Mftth. Magazine, % 1896, 173-184,

650 Histoby of the Theory of Numbers. [Chap, xxii
so that we get 6 or 7 biquadrates whose sum is a biquadrate. Multiplying
B) by 2* and by 54 and eliminating 304, we see that 75* is the sum of the
fourth powers of 8, 12, 16, 18, 20, 28, 40, 45, 70. Finally, he tabulated the
values of S = 144 +n4 for л^285 to use in seeking by trial to express
8-b4 as a sum of distinct biquadrates ^n4. Example in Martin,5* Ch.
XXIII,
E. Fauquembergue196 gave the identity
which becomes 5*=34+44-Н4+24+24 for x~y=*l.
C. B. Haldeman*» noted that а*+Ъ*+(а+Ъу=2(а*+аЬ+Ъ*)* [Proth**Q.
Hence on adding cF+c4, the sum will be a biquadrate if a2+ab+b*=de and
d*+e*= П. To satisfy the latter, take е = {Ф-4*I(Ш); then the former
condition gives
zz
Take t = d?-z2f whence #=*§(ЗЬЧ-5г*). Since Ъ^г makes d rational, set
b=y+z, and take d=2z^syjtt whence we find у and then b, d. Or we
may take d=2, z = l, whence t= Vl2-3fr'; set b = v+lt t^sv/t+S, whence
we get v and
C) SC2s2dbl2s^-6^L+2D22=Fl2^L+C54^L-{5«2+15^)^
Or, finally, take d=9, e=4, a2+o5-f^=4-37 since 2D-37)z-f94+44 = 154.
Since b = 6 gives a rational value for a, set &=6+т\ Then
by choice of r rationally in s, t. Hence the sum of the fourth powers of
&2+40$*-24t2, 6V-44s*-18i2, 1№-Ы-42Ь\ 9^+27^ 4^+12^ equals
(lW+ibP)* For 5 = 1, *=0, we get B), which is believed to be the solution
in least integers.
For six biquadrates, add e^-f/4 to each member of bis239 identity A) and
take ЗCа?-И*J = е/. It remains to make е^-КР^П, say the square of
1201 (ЗаН*2)/140, whence e = 7Ca2+F)/20. Or we may take the sum of
three of the six to be
and the others to be the fourth powers of 6, 12, 13 or 26, 27, 42 and the
sum of the six to be 154 or 45*.
For seven biquadrates, take Q*.b+dA+e*+BgL+gi = Cgy, Za?+bi=de,
whence ^+^=8^, which holds if e=+7d, g=—bdj2. Take d=y+a,
b=rylt+2a. Then у=2аG12-2Н)!(т2-7Р) and we have an answer. Or
use QQ. б+&т«+34 = 54, which is satisfied if За2+Ь*=4, 3d2+€2 = 16; taking
Ь = 2-а*//, e-i-dv/Zj we get a, b in terms of sf tf and d, e in terms of v, z.
Next, &)fc+Qdi,+24-l4=-34if3a4b*=4 = 3rf*+e2 (like preceding case).
M UiLtcrmediairt dee math., 5, 1898, 33.
i~M*th. Magazme, 2, 1904, 288-296. The editor Martin noted (p. 349 *nd щ his 1900
paper below) that this MS. had been long ш the editor's bands.

Chap. ХХЩ SUM ОГ BlQUADRATES A BlQUADBATE. 651
To find а вши of a biquadrates equal to a biquadrate for n=9,10, 11, 12,
multiply C) by a suitable biquadrate and eliminate one biquadrate by
use of one of the earlier results» Finally, given
2<-f64-f8<+2<-b7<-bl2< = 134, 2+6-8,
we can find a, b so that 2*+6*+Ъ4~а4+ЪА+(а+ЪL = 2(а2+аЬ+Ъ*)\ Thus
208-35*- -3^-36^+100=
whence we get y, bt a. Take *=2, ^1. Then 7y=-76, 76=-34,
7a=*58 and
A. Martin187 employed methods admittedly similar to Haldeman'e,
whose manuscript wag in his hands, but found many new sets of biquadrate$
whose sum is a biquadrate. For 5 biquadrates, take
which reduces to 2eCa*+&*) =y(y2-e?). First, take y=2e; then
which for e^2(e2+3^ leads to Haldeman's C). For y=%e, we get a
result equivalent to the last» The next solvable case is y=8e, giving
For y — lZey we get a similar formula. Next, let
The first becomes i^+z2—^2, whence take z=2pq, w~p*—qz} s=
The case p=2, q = l, leads to C). Omitting the discussions found to be
unfruitful, let p=r+2q, x = t+2q\ Then
Take A=0r whence t^llqrffy. Set y=qrmjn. We get g in terms of ms nt
whence
In the Congress paper, on the contrary, he took t=*—4q and found the special
solution 2*-H34+324-f34*+84'=854. For n biquadrates, n=$, 7, 8, 11,
he took
and found other sets by combination.
»»Deux. Coitgree Internet. Matb, 1900, Рлпа, 1902, 239-248. Reproduced with additi
in Mmth. M4gv 2,1910, 324-35?.

652 History or the Theory op Numbers. [Chap, xxii
E. Barbette1* used the final method of Martin1* to show that B) ie
the only sum of distinct Ы quadrat ее ^144 equal to a biquadrate, and that
is the only sum of distinct fifth powers ^ll5 equal to a fifth power»
R. Nome1" found (in confirmation of Euler'siw conjecture)
E) 3534-304-fl204-b272<-b3154J
by a series of special assumptions which lead to this smgle result. Next
(P. 77),
y
provided [bee
To solve the latter, set ti=rxi+2, v-lt x=rx1^-3J у=гхг+1. Hence
Equate the coefficient of r to zero. Then the equation gives r. For 6
biquadrates (p. 80), use
= {X4 - У
From the latter,
the second member being double the sum of 2*+2 biquadrates. Hence
(X^'+Y***)* equals a sum of 2^+2 biquadrates. Returning to the 6
biquadrate case, take x^u*, y = 2tt, whence x*-H6Y equals the value of
for Ь=и\ с=2Л Thus we get a biquadrate expressible simultaneously
as a sum of 6, S or 10 biquadrates. The sum of two of these biquadratee
has the factor u*+l№, which as before can be replaced by the sum of four
rational biquadrates. In this way we can assign a biquadrate which is a
sum of any even number >4 of biquadrates.
For 7 biquadrates (p. 84), take *= («•+»4+e4)/8 in
and set x=px—cf, у«2рд+Л z=2pq+p*. We get a relation between
biquadratesj one with the coefficient 2, for which we substitute the sum of
three rational biquadrates given by Gerardin's** C). Again,
where Т**(х+уУ+(х~у)\ But for any r^2, we can express T (in one
of its two occurrences) as a sum of г biquadrates and hence obtain а Ы-
"* Lee flommee de p-ieme» ршавалсев dietinctee egalee A ime p>ieme ршюапсет liege, 1910,
183-146-
1H Univereity of 8t. Andrews 500th Anniversary Memorial Vol.» Ediubmth, 1911, 80.

Chap. ХХП} EqUAJ, Sum» OF BlQUAl>HATE8, 653
quadrate expressed as a sum of r+5 biquadrates. In fact,
where т-&-Ъ%, 2-*J+»i-J \-st
Equal sums of
A, Martin*» tabulated various sets of numbers having equal sums of
fourth powers, as 1,2,0 and 3,7,8; 1, 9, XOand 5, 6, 11; 1,11,12 and 4, 9,
13; 1,5,8, 10 and 3, 11.
C. B. Haldeman201 noted that the sums Q,A ь *&d Q*. « of three biquadrates
are equal if За2-ЪЬ*^ЗсР+е*. Taking e^b—v, we get bt e rationally in
terms of a, d} v and see that
ш unaltered by the interchange of a and d. Fora = l, d=v=2t we get
Next, let
whence V~N*f{4*)f №=д*-з*-12а*&. Taking
we get a and d rationally. Or take JV^iP-*1, whence &^W+#f set
a*=2s+y bxl& solve as usual. Again,
Q-l»+l' = Qrfl.+3< if Зл»+^^7, 3<?+^-3.
Take a**l+x,d=\+y and solve as usual. Fmally, to find a sum of four
biquadrates equal to a sum of three, employ his*19 identity A) and equate
the left member to Q«t n. The resulting condition, 3(№+e)*=2v*+n*
ш satisfied if
. Cunnmgham1^ noted that X4+Y*+a<+s<e-Xri+*i+0i+*i follows by
ining a solution of each of X<+r< = Xj + ^,^+Jf}+^^^}+V?+^-
A.
combi
Again, х*+у*+2и1=х1+у\+2г? follows from
(solved, Сшшш^т^)^1Ъ«=Л2+3^1;1^^?+З^ЛВ^Л1^ь whence
A, S. Werebmsow*1 gave an incorrect proof of tbe impossibility of
=Zu4 in relatively prime integers.
»»»Math. Мадоше, 2f 18QQ, 183.
*«/Ы,, 2, 1904, 286-8. Рог the notation ft вве ЩИетлп1" D).
** Menenger M^tb., 38,1908-0,109-4.
• Liddii d h., 16,190S, 28b Cf. 16, 1900, 55, 208; 17, 1910, 279.

654 History of the Theory of Numbers. CChap. xxn
F. Ferrari2" noted the identity
while IF. Bini (ibid.) gave the identity
with the plus sign. A* G&ardin (ibid., 19, 1912, 254) stated that the eiga
should be minus and gave other such identities. Welsch (ibid., 132,184)
gave another method of correcting the signs: retain the plus sign, but change
the final term of the first member to — b(c+3d).
A. Cunningham** found numbers expressible in several ways in the
form а^+у'Н-г4 by use of x4+v4=2ixt—г4, и=а?+ху+уг, z=x+y, and
expressing this и in the form A*+3Z?* in several waye.
E. Miot** stated that [ЧЬе цд^ &—c of Ferrari's901 identity]
A) Dрд)Н(Зр«+2рд-^+C^-2рд--Л4«2C^+Й|
and noted cases when a sum of three squares equals a sum of three bi-
quadrates and a вшп of three eighth powers. Welsch2^ stated that Miot'e
solution is erroneous and noted that
always implies that
A. Gerardin™ noted cases of two equal sums of three biquadrates and
gave four methods of finding particular solutions of
B) x*+^
the fourth leading to the solution
[It is expressed by the next identity with k=l, l=qt&ndp replaced by 2p.]
He gave 16 identities which follow by a change of variable from
In conclusion, he gave
C) (p:^
A. Martin2" gave A) and C),
R Miot510 noted the solution 37, 17; 35, 26, 3 of B).
"* L'intermediaire dea math., 16,1909, 83.
** Math. Queet. Educ, Timea, B)t U, 1908, 83-4, Same in Mew. Math., 38,19О&-9,101-2.
"• L'mtem^diaire do» malh., 17, 1910, 214.
*»1Ш~, 18,1911,64,
•«Авзос. frang., 39, 1910, I, 44-5S. Some in Sphio^Oedipe, 5, 1910, 180-6; 6, 1911, 3-9;
8, 1913, 119.
**• МбАЬ. MeeaaDe, 2,1910, 351.
>" L'intermediAire dee math., 18, 1911, 27-28.

Chap. XXII] EQUAL SUMS OF BlQUAPRATES. 655
R. Norrie511 gave several methods to solve
D)
First, takux=TXi-\-a, y=TXt-\-btz=rx^-\-ctu=Txx—ai v=rxz+c, w^
We obtain a cubic in r whose constant term is zero. The coefficient of r
will be zeroif xa=xi+2i1a1/F2—c1). Then —r is the ratio of the coefficient
of r* to that of r*. Second, he noted that
equals identically the sum derived by interchanging the subscripts 1, 2.
Replacing xu yl} xit y5 by their reciprocals and multiplying each root
by (xtfiXiy*)*, we obtain a new integral function which is added to the
former. Hence
is unaltered by the interchange of the subscripts 1, 2. Multiplying
by the identity derived by interchanging the subscripts, we get two equal
sums of five biquadrates. The third method is really Haldeman's2*1
remark that Q9.x=Q*,u if 3y*+a?=3i?+u*. The general solution of the
latter is stated to be
x, it- {CXl±l)*+C\*=Fl)y}/<2X),
where X is arbitrary. Again, x'-rV+te+yL is unaltered when x is replaced
by Cx—5y)/7And у by Er+8g)/7. Changing the siga of у and subtracting
the new identity from the former, we get
Finally there is given the identity, in which r=/i2ca—ХЧ»2,
Х{Ьс3(Х/1г+2^кг4) И+^дА^-ХЪЧL
^Х{^(Х^г-2А^)}4+^{2^^^+Х2ЬЧ)
К we replace vxk by 2J"Ji\xJ—2£Ik#J, we get a solution of
In the last, Norrie made the restrictions that s=r, ъ^vif whence л<=
A. Gerardin213 noted the identity
E. N. Barisien*11 noted the identity A).
Gerardin*14 quoted his*09 solutions of B) involving two parameters with
3=z-puand noted that C) is simpler than Ferrari's2** formula, which fol-
follows by taking a+c=p, h+c^—q.
™ University of 8t. Andrews 500th Annivefsaiy, Edinburgh, 19 Ц, 83-75.
m BnD. Soc PHlomAthique, A0), 3, 1911, 236,
^ Nttiv. Ann. Matb^ D), 11, 1911, 280-2.
ш Uintermediftire do maih, 18, 19U, 200-1, 287^8.

656 History of the Theory of Numbers, [Сил*, xxii
"V. G. Tariste15 noted that C) is derived from Bini's™ fonnula by
equating to zero one of the six biquadrates.
O. Birck'" stated that C), viz.,-
gives the most general solution of z+y—z+u+v=Q with either B) or
гг+и*+&т He noted that
A* S. Werebrusow*17 gave equal sums of three biquadrates involving
many parameters and derived Gerardin's** formulas by specialization.
He'ia gave 374+384-264+424+254 and eight more such sets-
E. Fauquembergue2" gave the identity
where v=4a*-*a>p+lZ*l0i-Zea0*+2*p, and found five sets making t>=a>
all giving trivial solutions of B). A. Tafelmacher2* drew the same con-
conclusion from a complete study of the identity derived by replaciag a. by
L. Bastien221 stated a solution of x\-\ Yx\^y\-\ \-yU n^2%
R. D. Carmichael322 noted that x*+y*+4z*=P has the special solution
xf t = p^2(TAt y=2p*<r, z=2p<r*. Solutions involving two parameters are
given for л^+ау*+аг4=^4 and ^+^+02*=at*, if a=2 or 8, Abo,
A;^=-i,of C). By
A. S. Werebrusow223 tabulated all solutions, each ^50, of D).
E, Miot2^ gave a solution of D) involving a parameter; likewise for
two equal sums of 4 or 5 biquadrates*
Werebrusow*** noted that
for
b =
ш L'mtenn^diaire dee math-, 19, 1912, 183-Ч.
M Ibid., 265.
toI7Wd.,20r1913,105-6»
Ы1Ш., 68; error in fourtb set, p. 301.
«/Ьй.,245.
"•/Wrf.,21, 1914,59-62.
n Sphmx-Oedipet 8,1913,154-5.
~Amer. Math, Monthly, 20, 1913, 306-7.
"I/uiterm&frure dee math., 21,1914, 153-5.
«ibwL, 165-6.
** Ibid., 23,1916, 223. Math. Sbornik.

т* ХХП] BlQUADRATJJfl AND SquARBS. 657
where
Relations involving both biquadratics and squahes.
Biopbantus, V, 32, treated xH-y'+z4**1 by setting v=a?-k. Then
tf = (kx-tf-z()KZk). Take k^tf+*. Then s'^V/tjf+z1)- Hence
y*+*J equals a square Л For у^З, s~4, we get k=25, x=12/5. Dio-
pbantus' method thus leads to the identity (cf. Fauquembergue1*)
Taking y=od, г«=Ь<?, to=ac, we get [Korrie,111 p. 91]
E* Waring18* reproduced Diophantus' argument with к eliminated.
F. Prothw recalled that any prime N of the form &r+l is expressible in
the form N=a*+¥+ab. Thus 2W=a*+bf+(«+&)*• By multiplication,
РС4, whence
It is stated that if N is of the form fc+1, whether prune or not, 2N* ie a
sum of three biquadrates [incorrect, Kempner41 of Ch.XXV, Diss., p* 44].
If N is expressible in two ways in the form af+b*+eb, as
we get a number expressible as a sum of three biquadrates in two ways:
& Realis** noted
where 8, p
G. Dostor128 gave the identity
S. E^ahV» noted that t^+a^+y1=2^ is satisfied if
^«391^-27870^+2641^-2057^
t- B*+2/3)(Ш**-73Оа/Н-3910*),
whence for а «^1,^=0 or 1,
l«2 -57128^
«*M«iitatianee AJcdvaica^ 1770, IW; cA 3, 1782, 325.
*Nv. Comep. Math,, 4, 1878,1T&18L
., 60, 1877,445.
b, ^ 1880, 338-9. Miequofed, C. A. Lriant, Algebre, 1805, 221-2.

G58 History of the Thkort of Numbers. [Crap, xxil
From a given solution is deduced a second by long formulas, whence
14+ЗЧ-104=2'712, 7Ч-7Ч-12<-2- ИЗ2, 14+14+2*=2-3*.
A. Martin2*1 gave 9 biquadrates, 720*, - - >, 3120*, whose sum is a square.
Martin412, assuming that the sum of the fourth powers of x, x~ayt
х-Ъу,х—су, is a square, obtained */y=a/0, where a and 0 are polynomials
in a, bt c, and took x=<x, y —0. By the same method, he2" elsewhere found
199<H-271<+343H559« = 344162*.
Martin and R. J. Adcock234 repeated the solution by Diophantua and
stated that Diophantus' result 12<+154+20<=4812 gives the least solution
in integers,
E. Fauquembergue2** noted that, if
These two formulas were given also by A. Martin,23* To find n biquadrates
whose sum is a square, the latter took their roots to be xt x—ay, х~Ъу,
z-cy, pxy, - • ■, v*-*y. Then shall
say the square of 2x2~2a-sy+J{62a2-BuJ}#2. Thus x/y is determined.
E. B, Escott^7 noted that
» Fauquembergue288 gave identities including
С. В. Haldeman2" found four biquadrates whose sum is a square:
Take s = d?+v, 3a2+62=tt7. Then vt bz, 5 are determined rationally in
terms of d, gt a. Take y-2, a=3/7. Then V=4t<Pi7-27/49. Since b is
rational for d=l, take d=y-\-l and equate Ь to r^/£+l/7, thus determining y.
Then
For r = l, t=0, we get 2*+4*+64+7< =63*. Next, let the sum of the initial
biquadrates equal 2a2. The condition is evidently satisfied if
4v
»l Arnmla of Math., 5, 1889ЧЮ, 112-3.
~/btd., 6,1891-2, 73.
*" Amer. Math. Monthly, 1, 1894, 401-2.
*bid., 279-80,
m Lpinterm6diaire dee math., 1, 1894, 167 ф
" Math. Magazine, 2,1898, 210-1.
" L'intenn6diaire dee math,, 6, 1899, 51.
*»/Wd., 7,1900,412.
*" Math, Magaane, 2,1904, 285-6*

СЬл*.ХХП] BlQUADEATB» AND SQUARES. 659
Take <P=2vf 3aT+62-((+6)f- Thus Ь, dt v are found rationally in terms
of a, t, whence
(I) Dai)HCaf+2^-^)HCal-2ai-O'+(ea1+2B)*=2CCatH-?)l}t-
For a = ltt=2, we get &+&+84+и4=2 1472.
A. Cunningham**, to solve x*+yl+*i=2u**, took as и any number of
the form а^+ЗД1, whence uin is of the form А*-\-&В* and a solution is
x=B-Af y-B+A,z=2B.
A. Gerardm*1 noted that A+паУ+(туL+{тж)*^ (l+2mx)* if
Its diecriminarit must be a square, say BSx)*t whence x4 —y*—z4=2S3. Set
S=zUt i^-bb'-x2. Then *уЧ-4(**-1)г*=*7*. Hence tbe problem re-
reduces to a " double equation/' that of making the two binary quadratics
squares.
E. K* Barisien^ noted the identity
Mehmed-Nadir*** gave two special sets of solutions of
A. Cunningham and E, MiotJ44 obtained solutions by use of the identity
A. GeMrdin*44 solved X4+Y*+Z4=A*+B* by use of tbe identity
settingq=qP$p=2btf* ItieinainBtosolveab(/I+2^=Zt. Fora=b = l,
we may take/~т*~2п\ Х=т*+2п*, g=2mn. He noted (ibwf., p* 90) that
В- Norrie^1" pp. 90-92, would derive a second solution of
from one solution a{H hai=a2 by setting X^rXi+аъ Х^,
and making the coefficients of г and r* zero by choice of t/, x. To obtain an
explicit solution when n>4, take ^tf+sp+ifin {?+*)* ^Р+{ггL+2№,
whence 2f ^х*+у*+(H-yL, But x*+^ can be expressed as a sum of r
biquadrates P< if r>2 [Norriej1» end]. Hence
E- N. Barisien** wrote Proth's*» identity in the form
- Mf**nger Math., 38,1908-9, 101,103.
*" Bull. Soc. Phibmathique, A0), 3,1911» 239-240.
*" Nwv, Ann. Math, D), 11,1911, 380-2.
^ I/interroediaire dtf maib.t 18, 19X1, 217.
bkLf 19,1913,70-71.
»*• Spbin*Oedipe, 6r 1911, 21-23.
•**ММЫД4Lт413

660 History of the Theory of Numbers. [Chap, xxii
R. D. Carmichael showed that one solution of х4+оу*+Ьг4= D leads
to a second.
E. N. Barisien noted that N= (a1+**)(€*+#) (aV+iftJ1) equals
Let Nf be derived from N by interchanging с and d. Then NNf is a sum
of nine squares in four ways, in two of which two of the nine squares are
biquadrates.
See papers 178, 188, 206-7, 219-20, 287-8, 292; also Gerardin, p. 38;
Lucas** of Ch. XXIIL
MlSCELLAHEOtTS SINGLE EQUATIONS OF DEGREE FOUR.
С Wolf*48 treated xY+a^+y^ EL First, make xV+^-П, i.e.,
y*+l^t?=(*~y)\ whence y = (f—l)f2L Since zV+^2=^V, it remains to
make яЧ>Ч-£*= П, say (z~vzJ; we thus obtain x.
L. Euler1" made F~(p2—fl^fe2—r2) a biquadrate by setting p=
q—a+Ь, r=a—6, whence P=16abc(a-|-&-|-c). Consider therefore
Take 8*= (s+y+*)y. Thus
Dt^ix+y)??, D(x+y+z)=xy(z+y), Ds*
Set x=ntf, y=nr*t nqr—p=k{<f-\-r*) and eliminate p.
k{~2nqr+k(q*+r*)\
For n=2kt F=2(q—ry!(tf+r*—4qr). As Euler omitted the factor 2, it is
not sufficient to make the denominator a square. Next, let n=fc. Then
р=(^+гг^grr)/(^f—t)\ Equate the numerator to the square of q+rf/g*
Thus q : r=ga—/*: if+3fip. Or we may begin by taking p = 2xyj(x+y),
whence &=2ху{х+уJ/(х—уУ; taker=2g3, q**r* to make 2xy=Q.
Euler250 treated V+V+tf+W1 e п ЬУ setting
ТЪив 2»=i*+|f+l. Take y = 2z. Then ^«lW+1, wbich is satisfied
by (ж, *)=B/3, 7/3), B/9, 11/9), F, 19).
Euler3»1 treated Ll=D$ where L-A+ife+C!z*, Z-a+k+0^ Take
Ы=рЧ?> ТЪеп i=p4, wbich can be solved if one solution is known.
J. К Lagrange** treated the more general problem to solve
where / is of the fourth degree and e of the second. It F(p, q)=0, set
"• Dioph*niine Analjwt 1915, 44.
^ Ncmv. Ann. Math, D), 16,1016, 390-1.
"• Elements Matheeeoe Uwvereae, Hal&et 1,1742, 38a
*» Opera, poetuma, lr 1862,239 (about 1760). Extract in Bull. Soc. Philomathique, A0), 3,
1911,240-3. Cf. Еи1«г,ш G^rarfm,*- КощшегеИ »
"♦Орет* poetuma, 1, 1862, 215-6 (about 1774).
"/Ш.р 218-ft (about 1777).
•»Nouv. Mem. Acad. Sc. B«riin, «inee 1777,1779; Oeuvre*, IV, 397.

Chat. ХХП] SINGLE EQUATIONS OF DeGBBE FOUR. 661
я—р-И, y=q+tz. After dividing by tt we obtain B+Cz+1Q=Q) where Q
is quadratic in t and г, while В and С are constants* From the solution
t=0, *= —В/С of this cubic, we obtain a second by the tangent method.
Euler*1 treated as two separate problems the solution of
V *=
Tben
The left members will be squares if
&Г 2pr ' yv 2qe f
whence
2 2pr ' 2 2qs '
From A) we obtain x2/^ and у*/г* by multiplication and division* Hence
we have values a, b, c, d for which x^at, v=btty=cu,*=du* Then B) give
2pr '
By eubtractioB^ we get t/u. Hence we take
and obtain x, y$ vf г. Cbangmg the sign of n, we obtain a second set of
solutions* Rational solutions result only when the product of the right
members of A) is a rational square. For the upper signs, take p—^fgf
T=*p~Q*7q=2hky $=h*~k\ Then the condition is
It is the square of 3mnfg(f~g) for
See Euler* of (Ж XVI.
Euler*4 used the preceding F+=F to finds2, •■-, tfeucb that
sball be squares. We have
Hence we seek solutions of F=Q. Solving the latter for x2 we get
for 2=5, tr*=3, whence ^=16^-
у=B25+g)/(8Q.
AetaAcad.Petrop.t2,ntl781 A778), 85; Comm. Arith ,П, 366; 0р.0ш„ A), Ш, 439.
Opera poefcima, 1, I862t 257-8 (aboat 1782). For sums, instead of diffenmce*, eee Eulcr*
ofCh-XVl.

662 History of the Theory or Numbers, EChaf, xxii
Taking t=5, we get $/ = 25/4, T ^20, x=39/4. Multiplying the unknowns
by 4, we get the solution z=39, y=25, г = 20, r=12. Or we may solve
F=0fort*and get ^гЯ-х2-^-*2, Sl=xV+«%1+!^8- SetS
Then
Then v2 is a complicated function of degree 6 and was not treated. A
solution is said to result from $=185/153. For £=13/3, x=5, j/=4, we get
the above solution z=39, etc.
C. F. Kausler156 treated the problem to find all rational numbers з, у
for which N^(x2^l)(y2—1) is an integer. Set y—pjq, where p and q are
relatively prime integers. The numerator and denominator of the resulting
fraction for z* are (N—l)q*+p?*=mP* and pf~<?=тО*. For m = If the
latter gives р = (Аг+В*IA, q=(A*—Br)fdt where А, В are relatively prime,
one even or both odd according as d = 1 or 2. The first condition tnen gives
N which is an integer for d=l if Р±2ЛВ is divisible by (Л2—B*)\ For
m>l, p+q=7nQ, m or Q2, the last two yielding (as far as numbers <100)
only the same values of N as above. For p+g—*nQ, then p^q—Q and,
dropping the common factor Q/2 in p, qf we have p=w+l, g=m— 1, m
even, tf=m(P*-4)/(m-:iJ. Tnen ^^2=^(^-1)*, whence
G. Eisenstein^ considered a binary cubic whose coefficients are variables*
Its discriminant D is a quartic in these four variables. Given one solution
of D—constant, we can find an infinitude of solutions by means of the
formulas for the coefficients of the cubic obtained by a linear transformation
of determinant unity.
Y. A. Lebesgue267 noted that
is satisfied identically by
t=x(by2-cz2)f «
with s the product of the binomials, and by
Several2*8 found two numbers whose sum equals the difference of their
fourth powers. Let the numbers be (n±l)x. Then z~Dn7+4)~ltl is
rational if n^zfcl* Hence set n=m+L Tnen x^N~1!zt tf ~ (pm+2)8 if
p=2/3, w=9/2.
E* Lucas25* stated that the difference of two consecutive cubes is never
a biquadrate. Moret-Blanc76 noted that 3x*+3x-\-l <^z? since 4zl~l
D. S. Hart280 found rational numbers a, ht x for which
*» Nova Acta Acad. Petrop., 15, ad annoe 1799-1802, 1806,
■* Jour, fur Math,, 27,1844, 76.
>" Compile Aeodue Paris, 59, 18&4, 1069,
"" Math. Queet. Educ. "Rmea, 2, 1865, 77; сГ B)р 4, 1903, 68-9.
*•* Recherche лиг ГйпЫуее indetermin^e71^ 1873, 92; extract in Matheaie, 8, 1888,21.
** Math, Queet, Educ, Timee, 24, 1876, 35-36.

, ххщ Single Equations of Degree Fouk. 663
Take Bz*+ax)*-(<zx—Ъ)г. We get x rationally and a condition on a, b,
which is solved for a. Take Ъ = — m*/2, whence a follows rationally.
A. Desboves2*1 gave identities yielding an infinitude of solutions of
cp4 for certain values of e. He*1 noted that aX4+bY*=cZ* for
and gave long formulae yielding solutions of aXl-hbY*=eZi when с is
represented by a certain form of degree 20. Further, X4~Y*=cZ* is
solvable when с is of one of the forme
S. Re'alis*1 gave various quartie equations not having a rational root, as
/5+0 (mod 5).
Several284 solved xHy* = (z-yL. Set x+y=u, х-у=г. ТЪеп
Set r*=(8/±3J. Hence there are two types of solutions.
R. W- D. Christie116 made 12abc(a+b+c) a square, but not a biquadrate
as claimed. A. G^rardin1" noted that it is a biquadrate for (а, Ъ, с) = A, 2,
6), C,4, 9), etc.
E. GrigorieP7 noted that ll'-l^+lF+W. P. F. Teilhet*8 gave
cases of х*~у\+у1+у1 for x=Z, 10, 17, 20, 29, 36, 43, 55, 62. He** noted
that
*-three such sums.
Kbmmerell170 gave as the positive integral solutions of
where j/i, zi are without square factors, <Fyi is relatively prime to e*zi, and
)
|/1B/j
A. Hurwitz271 proved tbat х*у+^Н-^=0 is impossible since
и7-И7+юг=0
is impossible.
« Nouv. Aon. Math., B), 18,1879, 408.
«/Ьй44(М
~ 1Ш., C), 2, 1883r 370; 4, 1885» 376, 427-31; Mfttlicaie, 7, 1887,96; Jour, de math, spec.,
1888, 90 (and questions 66, 67). Reprinted, С A, Laieent'e Algebra, 1895, 224-6.
** Zertecbr. Math. Naturw. Untenicht, 20, 1889. 264-5,
~Edtic.'nmee,49, 1896.
«• Bull. 8oc- PhiJomathique, (Ю), 3, 19П, 244.
m L'intermediaire dee math., 9, 1902, 319,
ш Ibid.t 10, 19ОД, 170-1,
■birf,,ll, 1904,18.
»"Math. Naturw. Mitteilungen, Stuttgart, B), 7, 1905, 74-8. a. Brehm,*
шЬо papers 12, 22 of СЬ. V.
m MMh. AoDAlen, 65,1908, 428-30. Generalisation, BurwiU," Ch. XXVI.

664 HtSTOBY ОГ THE ТНЕОВТ OF NUMBEBS. [Сиар. XXII
F. L. Griffin and G. B. M. Zerr*7* made a sum of n squares a biquadrate.
A. Geiardinrs noted that a* is divisible by *s, where
For/=1, the quotient is —4175. E. Fauquembergue noted also that
A* Cunningham*4 expressed numbers in the form (я?+я*)/{я^+у*) in
several ways.
A. Cunningham,*7* found certain types of solutions of
and (pp. 111-2) of e(x, y)=&(x, г), &ху=П, y+z. He™ gave various
criteria for the solvability of ±N=x*—2yJ=a£—2yJ, tf=±l (mod 8)«
He discussed (p* 108) ffi^gi^D, where gr=^J+yr. He*77 proved the
existence of an infinitude of integral solutions of F(x, y)=F(x\ y1) for each
а/к, where
If (k+10a)/ (к—6a) is a rational square, F(xt y) is a product of two factors.
If (pp. 94-95) either of Bат2Ь+с)A2а-2с) is of the form -o*-/F,
is usually solvable in integers* Certain numbers (pp. 39-40) can be
expressed simultaneously in the forms
and ДГ;/3, ДГ1/8, ДГ^З, ДГ1/3, where ^^(й*-»;1)/^-»!), etc. He»1 con-
considered numbers expressible in two or four of the forms ±(x*—2y*),
±(a*—2y*). He17* showed that certain binary quartic functions of four
pairs of variables are equal for an infinite of set of values, by use of the
b^fc )
He*> solved *1+ЛГ»=Я'Н-Л, where ЛГг = (х^^)/(хг-^)^
He2*1 gave a method to solve xfy—яу* «=a.
H* B. Mathieu20 noted that each triangular number which is a square
yields a solution of x9+y*=zi. Thus, A*#=35* gives
™Amer. Math. Monthly, 17, I910r H7-8.
^Sphinr-Oedipe, 1006-7,159-160.
m Ммм. Mftth., 39, 1909-10, 97-128; 40, 1910-1 lt 1-36.
™ Math. Quest. Educ. Times, B), 16,1909, 76.
**IbuL, B), 17,1910,66-7.
1G7 Ли*., B), 19,1911, 27-28.
**1Ш., C), 23t 1912, «-41P107-9; 23,1913,
тЛй.р С2), 21,1913, 89-90, l(»Ht.
"»M, B), 26, l9Ht60.
ш /«Л, B)t 27, 1915, 74r^5.
w» I/mterm&ikire dee m&tb, 19,1912,129.

Chap. XXII] SlNGLE EQUATIONS O? D&GHEE FOUR. 665
L. Aubry and H. Brocard» solved 2^+1=^+^+3* for
Auhry** gave a solution involving three parameters of
Brehm*5 solved zyz(z+y—г) =f* in integers. Set tq=xyp, where p and
q are relatively prime integers. Then tbe equation &ves s(x+y—z) =rp%
ys~r<fz, where r and s are relatively prime integers. Hence zt y, t are
expressed in terms of t
E. Swift»1 proved that x*—tf=z* is impossible for z prime to y.
R. D. Carmichael287 noted that if z*, Уъ> «0> v0 give a sohition of
we can deduce a second solution [after performing the operations]:
F* L. Csrmichael1** obtained the solution
u=u\-hv\+ay[—
where
vt=2rnn+2n*;
also two simpler solutions, аз well as solutions when а/Ь^П, a = 0or
L. Bastien and Ъ. Aubry** found the general sohition of
Several^ treated *-&+
A. GSrardin»1 discussed z*+tf+*y
A. Cunningham^ treated а'+б^^+г*?, for b and с given.
L, Aubry»1- solved (**-У)(**+2у*) -л1-V-
Gerardin1*7 of Ch. IV solved ^+6i:V+^=e^+6«2^+^ On equa-
equations quadratic in x and my, see note 145- On pq(mj?+ntf) ^^(wr^+ne1)»
see paper» 168,170, 174, 181.
To FIND П NUMBEBS WHOSE SUM IS Л BQUAKB AND BUM ОГ SQUARES 18 A
BIQUAPRATE.
For the case n=2, see papers 37-63.
G. W. ЪелЬтхш considered the case n=3.
, 19r 1912,157-&7з (for«ресЫ eoiutioufl).
w. Mitt., <2), 15r Ifll3t 20-21. Cf. Kommefefl."
Amtf. MMh, МооШу, 22,1915, 7<Ы.
«" Diopluntme Azu^yre, 1915, 4в-8,
«Amer. Math. Monthly, 23,1910, 321^.
™ L'uiUnij^diaire detf matb> 23,1916, 86^8.
"Ibid., 123-4; 34,1917, ЗД 88,133-4
~7Ь«^ 2*, 1917,32.
**1ШЯ, 143-4.
•>*Л*Л, 23,1919,150-2.
"• MS. in Bibliotbek Нлшютег, about 1676. а. ТУ. Mabnke, BIbliothe» Math.r C)r I3r
1912-^30. J. Wall^C^era MAth.r3 Л e^^l^ quoted alette from b^mj to ОИео-
Ьагь Oct. 23,1674, in wbkb tbk ptoblem w meotlooed (ВаП. Bibl, Bto^ft Se. Mat. e
Fk., 13,1879, 519).

666 Histobt or the Theory of Ntjmbebs. CCha*. xxn
L. Euler2*4 required four positive integers whose sum and eum of squares
are- biquadratics. He took them to be х=а*-Ъ-Ьх+с?—<P, y=2ad, z=2bd,
v=2cd. Then Хх%=Gк1гO. Set a=jfi+tf+i*-*t Ъ=2р*, c=2qtt d=2rs.
ItremamstomakeZs^Q1. Takep^*—g-J-$r. Then
which for g=r-M will be the square of 3r/2—и if
(M-3«+3u)r=и*-
=r=2, s=9, p = 10, we get x~409, y=24, s = 160, »=32, 2з=54,
2л*=21А Euler gave a similar treatment of the problem in five integers.
Eider" (first paper of 1780) treated the problem for n=3, 4, 5 and
obtained the sets 8, 40, 64; 320,400,961; 16,48,104,193; 32, 32,88,137;
16, 16, 32, 72, 89; 64, 152, 409; 17424, 108864, 580993, the last two sets
having also the sum a biquadrate.
J. Cunliffe2"* took x2, 2xy, 2y* as the n=3 numbers, the sum of their
squares being (х*-Ъ-2у*у. Their sum is the square of ry—x if y
()iX2&l5
,+y()
Walmond and Mason28* wrote x4 for the biquadrate. Take r = V4s—5,
2x—1 and хг—2 as the n=3 numbers, their sum being (s-J-l)*if r—3 = 1,
whence s=21/4. For я=4, take r= V6s-6, 2—2, x-1, x*— lt whose
sum =(x+lJif r-4=a,s-31/6. Forn=5, taker = V4a>-12,s-H,x-l,
2x-1, x2-3r whose sum =(z+2I if г-4=4, л = 19.
S. Bills1 employed the identity of Aida» of Ch. DC:
(г=2, »•-,?»)•
The remaining condition щ-\— • +«л= П becomes a quartic in xH which
is equated to the square of xl-\-2xn-lxn-\-x^-\-- * -H-xLi- Hence
Xm=T-$Xn^i,
where r—xi-\— * + xn-z-
A* B. Evans*7 used the numbers x, a\y, • —, a._ty and wrote
m~at-\ f-a*~i, v=a]-\ Ьа!_|.
Take x—a1—py. Then ^-f (aj+f)j/l=a4 determines у rationally. Hence
a"^(ui+i'+p*)*Ca;-Hai+™)!/]= (<*J-i-pai-l-&)*, 6=jwnH-t>—Jp1, determinee ai
rationalry.
D. S. Hart1" used the numbers px*—ax, px*+ax, - * -, N^—Zx^ Nx*+Zx
and, if n is odd, Sx2. Equating the sum of their squares to (xmfnL, we get x2.
"* Opera poetuma, lr 1862t 255 (about 1782).
**■ New Series of Math. Repoeitory (ed., T. Leyboum, 3,1814r I, 79-^80»
"* Ladiee' Diary, 1827, 36-7, Quest. 1452. Reference waa made to *F&unac, Bull, dee Se.
Math., Ш, 276.
*» Math. Quest. Educ. Tunee, 18,1873, 104-5.
«" №t 22,1875, 69-71.
к£., 24,1876,55-57.

Сиар.ххп] Sum op Squabbs a Biqtjadrate. 667
Examples for n=^4, ■ ■ -, n = 7 are deduced. To proceed otherwise when
n=S, employ the numbers 2mp, 2rp, m24-r*—p*. Their sum is the square
of т-т+р if m=(p*—2rp)/r. Then the sum of their squares equals
(m*+r*+p?)z and is a biquadrate if
whence p=4r, m-8r, and the desired numbers are G4r2, 8r* 49r*. A.
Martin employed 2a,*(t—1, >■ -, л-l), a?H Н^—s*as the n numbers
and wrote wi*=aH \-a^u Then shall
Take s = A-£, 2ai+2m-2s=A +В. Then either of the preceding equa-
equations gives oi.
R. Goormaghtigh™ discussed (x+y+z)i=z*+y2+z*=Mi.
Miscellaneous systems or equations op degree four.
Diophantus, V, 5, found three squared such that the product of any two
added either to the sum of the same two or to the remaining one gives a
square (cf. Fermat100 of Ch. XIX).
J. Ptestet** found three squares such that the product of any two added
to the product of a given square a7 by either the sum of those two or the
remaining one gives a square. For a=Z, he found 25, 64, 196.
Beha-Eddin*01 A547-1622) included, among seven problems remaining
unsolved from former times,
Prob. 1:
РтоЪ. 5: х+у=Щ *+-=*.
у х
Fermat3OTri0tedthatx4-y4isacubeandx-^ = lif^^l3/22>j/^ -9/22,
while positive solutions can be found by setting x—z-\-lZj22f у=г~9/22.
L. Eulei*03 required three numbers z, yf z such that k
+, y+y+\ у\ +Л ЗГЧ, V+Ч^,
ЧУ+* shall be all squares. He took z*=&+tf+1+2<&. For y^+
we have k*=w\ z*=4w, where w-z2+x+l. Now w=\3 = (t—xI for
& l)/4l) Then the solutions are
_2tz+2t+2
Z
X~2t+V y
2t+V y~2t+V Z~ 2t+\ '
Euler304 treated the three problems to make (i) AB and AC squares;
(И) ВС a square; (iii) В and С squares, where
"• L^intermediaire dee math., 25,1918, 17-18.
««ElemeDfl d«a Math., Paris, 1675, 33 L
101 Efflenz der Kecbenknnet von Mohammed B«ba^ddb hen Alhoasain див Amiut
u. detitecb von G. H, F. Neetehn&nn, Berlin» 1843, 56-6. French trams], by A. Marre,
Nouv. Ana. Math,, 5, 1846,313. a. A. Genoccbi, Annali di Se. Mat. eFie., 6, 1855,207.
tm Oeuvre», lt 300-1; French гхлхай., Ill, 248-9. Obeervation on DiophantUH, ГУ, 12.
«■ Novi Сотш. Acad. Petrop., 6, 1756, 85; Com*»- Aritb,, I, 258; Op. Om., A), II, 428.
wNi(>A^Pt20l776A77JL8Cfc

668 History of the Theory of Numbkbs. [Chap, xxu
It suffices to treat the case in which x and у are relatively prime, also t
and «• For problem"* (i), AB is the square of Axy(j?+<?) if
Then С is found to have the factor Ay so that АС=П if
Taking Q=2pgs* - (p1 - g*)xy - 2pgy*-f«y*, we have
a(« - 4pg)y* - 2a(p*- f)xy+*pq{a- pg)*1 = 0.
For a=4pg, we obtain the solution
For a=pqt we obtain a similar solution* For a = =F2p*, we get
For problem 5i)rBCis a square if x=3, у=5, *=11, «^45, or if
For problem (iii), we apply the last solution with ml+»1= П,
Eulei** required four numbers the four elementary symmetric functions
of which are squares. For the numbers Mob, Mbcy Mcd, Mda the conditions
reduce to
abcd=U, М(а'+с*)+йс(Н-<*J=П, М - (ab+bc+cd+do)/A
Finding the second condition impossible if h/d=2 or 3, Bute took Ь/d=j^/g1.
Then must p*g2(a1+cI)H-ac(pl+g2K be a square, say that of pqa+cm{n.
Thus a/c is found, and we readily form the condition that etc and hence
abed shall be a square. By trial Euto found the two solutions** a=64,
b = 9, d=4, c-49 or 289, Af=1469 or 4589; also one of another type:
a=^16, 6 = 5, c=5, d=4J=3, jlf^21. He discussed at length the problem
to find fr, <2 such that the initial second condition can be satisfied by choice
of a, c.
Euler*08 treated **+^+^= □, *У+^Ч-2/Ъ*= П. The first is satis-
satisfied if z=p?-\-qi—r*i y=2pr7 z—2qr. The second then becomes
A) (р|+дЧ(р|+$2-
Set n = (p—r)!q and eliminate r. Then shall
Set ^ = A-^)^+271^+^ The terms in pV cancel if
"• F. van Scfeooten hhd proposed to find ratiomd vdn of & trimngb P^ai tbe b«tt e, altitude
& and ratio m: n of the other sd» (irz, iu). Thus Ъ^Ътпху, л-(^-п*)(^+Л
i»-(^+^)C(«^n^+(»=F»)Vl Wling under problem fi). The smpteot eofetioo
is x~3, y~5, m-Z^n-n, a-23, b-28,
"• Novi Couun. AcaA Ffetrof^ I?, 1772, 24; ObmnL Arith., t, 450; Op. Om.t (I), Ш, 173.
*" ReprodQced by A. Ger&nnn, lrmterm6duUre dn nmtb, 16,1999, 105-6.
«• Ada Ac*d. Petrop.r 3,1,1782 A779)^30; Comm. AiitiL, П, 4S7; O^ Ohl, а

г. xxiij Equations of Degree Four. 669
From the linear relation between p* and p*qf we get
p:g=Sn(l-n*) :5-10л2+л4.
J. A. Euler10» treated bis father's*8 problem. Multiply
V-iW-V+iI
by 4$* and the like identity in q by (p*-H)* and add. Thus
Hence we have three squares whose sum is a square. The sum of their
products by twos is 4g* times
This is to be made a square. Set Д«(р"+1)', £=4p(p*-l). Then
(g2—lOA*+q*B* is to be a square, say (Atf+v)*. Then q*~(A2-t?)fd,
where d^2A2~B*+2Av. Take i^A2-£*. Then d is the square of
Л+* NA2B2(<6p4I}2» Hence
В
Hence, after multiplication by (p*—IJ, we have the solution
For р=^2, we get 35, 72, 96. Next (p. 47) let z^om, y-&m, г=сл, where
Ь*=с^ m=2pq, n=p*-g*. Then
The latter is the square of т(а*-\-Ь*) if p=afq=b. Then
is a solution. It may be obtained by using bis father's notations and
assuming that pz~\-g*=c*. Then the condition A) becomes
which is satisfied if r=cpfq, since the left member becomes j
The problem*** to find four integers whose sum is a biquadrate and sum
of any two a square reduces to finding a biquadrate n4 which i& a sum of
two squares in three ways. Take as n a product of two or more primes
J. CunlinV*6 noted that the problem to find three positive integers
whose Bum is it square and sums by twos are biquadrates is evidently
equivalent to that to find three biquadrates half of whose sum is a square
and the sum of any two exceeds the remaining one. Half the тип of the
fourth powers of m-\-n+9V,m~\-rv, n-J-*(r-M) \aA*+2Bv-\ boV, where
"AcUAcad. ft*rop., pro шю 1779, I, 1782, Мёац pp. 40-48.
»* New Series of Mmth. Bepotfttary (ed., T. Leybourn), 1, 1306,1, 59-61,
01 ThuLj 2,1909, Ц 178-9. If we wave the ooodition th&t the numbeta be poeitive,
tfa bidt *г п\ {m+n)\ half of whoee sum ie {?\+*)*

670 History of the Theobt of Numbers. [Chap, xxn
A = т'+тпп+п*, B=s(m+nL^V-l-n<(r+e), a = г*+гв+Л Equate it to
the square of A+vBfAzhaii* to get v rationally.
Several'10 found 7 numbers in arithmetical progression the sum of whose
cubes is a biquadrate. Let nx—Зх, пх—2х7 -«-, nx-\-Zx be the numbers.
Equating the sum of their cubes 7п*х*+%4пх* to m¥, we get x. Or uae
x, - -f7xy the sum of whose cubes is 784т1.
To find a rectangular parallelepiped whose edges, sum of edges, and
sum of faces, are rational squares, several*" took x3, jr*, z* as the adjacent
edges, and яЧ-Н+^—(*+У~z)\ whence z=xy!(x-i-y)- Then
S^2x*y*+2xV+2yh2= D
if х*+ху+уг= О = (тх-у)\ which gives x/y. C. Wilder took 8=
printed S=4wi*], and x*=myzB-<t)l{2a). Then
Xx*
if
у
Eliminating m from the assumed expression for x3, we get у in terms of xt
zf a, 6, which are arbitrary. [The solution is false as it satisfies neither
of the proposed equations, but only the combination of them which was
employed.]
To mid three positive integers the sum of any two of which is a square
and double the sum of all three is a biquadrate, R. Maffett and D. Roberts111
took a% b2, c* as the sums by pairs. Then shall a*+V+t? be a biquadrate.
Take а^ЗС^+г1), Ъ=4(р>-г*), c=Spr. Then Za'= (Sp'+Sr2I, which
equals B&r*J for p=2r.
To find two integers whose sum, sum of squares, and sum of cubes,
are all squares, and sum of biquadrates is a cube, J. Whitley*11 used the
numbers x=2rs, у=г*—&*, whence xs—xv+jf2 = D if r=*te. Call X, Y the
products of x, у by 23=8+15. Then X-23-843, Г-15-23** satisfy the
first three conditions. Also -X>+F4=23W, where f=23(84+15<), will be
a cube if s=t. C- Gill used x=b sin А, у = Ь cos A with the sum a*. Then
xM-H-a'PU-sin A cos A) =c*
if c = aKl-isin^)? cot *Л-4, whence z=86/17, » = 1»/17. By their
sum, b-17a9/23. The fourth condition is satisfied if a=23l-5472L
E. Lucas'18* proved that 2i^-«4=w4J 2i*+i*f=3** imply
E. Iionnet^4 desired a number Л^ which, as well as its biquadrate, k
the sum of the squares of two consecutive integers. J. LissenQon wrote
«♦The GeDtlemurt Diary, or Kfatb. Repository, London, No. 76t 1816, 39, Quest 1043.
«"The Meth. Diary, New York, 1,1825.125^7.
« LaW Di«y, 1833, 35, Qnest. 1512.
»" The Lady's and Gentleman1* Diary, London, 1S54, 52-3, Quest. 1857.
"• Nouv. Aim. Math-, B), 16,18T7t 414.
™ N<wv. Ann. Matt, BI19,1880^ 472-a. Bepeated is Zeitochr. Math- Natunr. Unterridit,
13,18glr 268.

Chap, XXII] EQUATIONS OF DEGREE FOUR. 671
ЛГ=а1Н-(а+1I, whence
Then 1=A—B gives а(а-\- 1)*(а—2)=0. The only answer, given by а —2,
L. Bastien*1* solved the system у1+г*+Р=2х\ y*-\-z4+ti=2z* by
eliminating x. Thus уг+г*—Р=±2уг, y^z = =kt Let у=г-Н. Sub-
Substitute thb value of у in the first equation. We get zt = (z-\-t+x)(z-\-t—x).
Hence set z = ab, t=cd, z-\-t-\-x=acf z+t—z=bd, 2b—c—hd, h — 2c — ha.
The solution is now evident.
A. Gerardin816 gave special cases in which a*—z, $*—у, s4—z are all
squares or all cubes, where s=x-\-y-t-z.
L. Aubry817 proved the impossibility of the system
Gerardin1* solved the system лЧ-х^^а2, y*+z2y2=Ь*, х-]-у = <?. М.
Rignaux119 noted that a=axf Ъ={1у, whence the system reduces to
and is easily solved.
E. Fauquembergue310 discussed the system xt—Ay4— П, xi-\-hy*= D.
A. Gerardinni discussed the system 2P4=2 Г/4, Р^Д - U VW.
A. Cunningham812 solved X4—Z=A2, X*+Z=B* by taking any odd
integer a and any even integer & and setting X=o2+/52.
Euler," and Euler8! of Ch. XVI, made x^^z7^ A2TyV, yV=FsV
squares. Petrus12 of Ch. XV made р2-!-^, t2+fl*P jw^ squares. Woepcke4*
of Ch. XVI treated о*+фо*=<г*+ф<г1^а. Gerardin185 of Ch. XXII
treated xi+mz2y*-\-y4=a* with other quartics.
^Sphinx-Oedipe, H, 1913, 173.
ш LJmterm6diai*e d«e math.r 23, 1916, 150, Ш. R, Goormaehtigh and A. Colucci gave
solutions, 24, 1917, 134-5.
"ЧШ.,23, 1916, 12»-131.
111 /Md., 24t 1917, 65-6-
«• /Wd, 39.
111 /5tf., 100-1.
ш Math. Queet. and SoL, 4, 1917, 4-5.

CHAPTER ХХШ.
EQUATIONS OF DEGREE n.
Solution of /= const., where / is a binary fobm.
J. L. Lagrange1 noted that, in seeking integral solutions of
А =Bt++Ct*-*u+ ■ ■ ■ +Kun,
where A, - • •, К are given integers, we may take « relatively prime to Ay
and thus find integers 0, у such that t=uB—Ay. Inserting the value of ty
we see that Вв»+Се*-1+-+К must be divisible by A. If such an
integer В exists, the proposed equation reduces, after division by At to
F{u, y)=:Pu»+Qu»-*y+>*-+Fy-lf
where P, ■■«, Vare given integers. Set u/y=x, F(x,l)^z. Thenl/y*=*.
The problem of solving F= 1 in Integers reduces to the examination of the
real values о of x for which z is zero or a minimum (whence dzjdx=O)w
For such an a, Lagrange employed the continued fraction for a and two
series of convergents and proved that ujy must equal one of these conver-
convergents ijLj whence u«s±J, y = ±L. While a root of z^O may lead to an
infinitude of solutions, a root of <hjdx=0 furnishes only a limited number.
A. M. Legendre* reproduced this method of Lagrange's, developing
into a continued fraction each real root of F(x, 1) =0 and also the real part
of each imaginary root and forming their various convergents p/q. The
least of the F(p, q) is the minimum of F(u, y) for integral values u> y* In
case the minimum is ±1, we have a solution of F(u, у)=^±1 and hence a
solution of the initial equation A=Btn-\ .
H. Poincare1 noted that the problem reduces to the case of the repre-
representation of a number N by a form in which the leading coefficient is unity:
xm-^-Azm^Jy-\ . We first solve the congruence £**—A£*"~4 ^0 (mod
N) and then determine by Hermite's method whether or not two decompoa-
able forms in m variables are equivalent under jn-ary linear transformation.
G. Cornacchia* gave a method of solving in integers
when Co and C* are positive and a root x9>P/2 of the corresponding con-
congruence 2СлЕ*~*=0 (mod P) is known. Take уй such that x&q^±1
(modP). Apply the g.c.d. process to P,xo, and let Xi,xt, ♦■ -,xm = l be the
remainders. Let j/L, ---jy^^lbetheiKJiTespondmgremamderafromP,^.
Then if A) has relatively prime integral solutions а, Ъ such that 2ab<P,
this solution is one of the above pairs zf, y*+\-{ or is a pair obtained
1 Mem. Aead. Berlin,24,annfe 17U8,1770, 236; Oeuvrea, П,662, 675. Forn-2,
fCLXIL
> Tb^orie dea nombrea, 1798, 169-180; ed. 3, l$30r I, 179; Germin franeL, Maeerr I, 179.
' Comptee Rendus Fans, ДО, l$glt 777. Cf. Ршб"
* Gioniale <h Mat.r 46r 190Й, 33-90.
673

674 History op the Theory of Numbers. [Chap, xxiii
similarly from another root of the congruence. The process is simplified,
applied to x*-\-gy2=m and compared with the method of binary quadratic
forms.
Conditions for an infinitude of solutions of f{x, y) = 0.
C. Runge5 considered an irreducible polynomial f{x7 y) with integral
coefficients (L e., not a product of such polynomials), and the algebraic
function у defined hyf(x, y)= 0. By one system of conjugate developments
of у according to descending powers of x is meant those obtained from a
single development by replacing the single algebraic number, in terms of
which all the coefficients are expressed rationally, by its conjugate values
and the fractional power of x by all its values. He proved that if the
various developments of у foim more than ове system of conjugates there
is only a finite number of integral values of x for which/(x, y) =0 is satisfied
by rational values of y. Also that/=O has an infinitude of pairs of integral
solutions xy у only when x, у become infinite simultaneously and when the
developments according to descending powers of one of these variables
form a single system of conjugate developments. Hence necessary (but
not sufficient) conditions for an infinitude of pairs of integral solutions
xy у of f(x, y) = Q are: (i) If / is of degree m in x and n in y, the coefficients
of xm and yn are constants a, b. (ii) The algebraic function у defined by
/(x, y)=0 becomes infinite with x with the order of W". If aftf is a
term of/, then np-ymo ^mn. (Ш) The sum of the terms for which
np-\-mv = тпп
must be expressible in the form
MI(/-e^) tf-1,2, ---.п/Х),
where TL(u—dff) is a power of an irreducible function of u.
A. Boutin8 raised the question as to the types of equations such that,
if tft, yi (i—n—I, n—2) are two sets of integral solutions,
A) s*=as«_i+0z,.^, Уп=<ху*-\+&Ул-2
are also solutions. E. Maillet* treated the properties of one or two recurring
series xn+p=aixn+p^i-\- ^ —h***» with rational (or integral) coefficients and
proved that the only equations F(xy y) =0, where F is without a rational
divisor, with an infinitude of integral solutions given by a formula of
recurrence A) of the second order are either linear, quadratic
Ax*+Bxy+Cy*±H=0,
or (Wy—tim)*—(ra'i-«n'y)«(^-^)"=0,
where p, q are relatively prime integers. If we consider rational solutions,
we obtain an analogous result.
E. Maillet* proved theorems concerning arithmetically irreducible equa-
equations
• Jour, fur Msth., 100, 1987, 425-35.
»L'intenn£diairede8m*th>, 1.1894, 20-21.
• Mem» Acad. Sc. Toulouee, (9), 7, l8»5, 182-213-
• Comptes Reodue Pa™, 128,1899, 1383; Jour, de Math., E), 6, 1900,261-77.

Chap. ХХШ] INFINITUDE OF SOLUTIONS OF f(z, y) -0. 675
where 0, is homogeneous and of degree j. (I) Let фп(х, у) be arithmetically
reducible; let d be a simple real root of Ф*A,с) =0of degree X; let ^*A, c)
be an irreducible factor of ф„, of degree к (k<n) and with the root C\.
Then F=0 has, on the infinite branch whose asymptote has cx as angular
coefficient, an infinitude of solutions only if one of the 0,A, Cj), i=n— 1,
• ", n~k> is not zero. (II) There exists no irreducible equation F(xf y) = 6
with integral coefficients having an infinitude of integral solutions on an
infinite branch of F=0 such that the angular coefficient of the asymptote
is rational and not zero, if this coefficient is a simple root of tf*(l, с) —0.
If the real angular coefficients of the asymptotes of F=0 are all rational, not
zero and distinct, then F=0 has only a finite number of integral solutions»
By amplifying the case k=2f he obtains a complicated third theorem; also
one onF{xf y, z) = 0.
A. Thue9 proved that, if U(xy y) is an irreducible homogeneous poly-
polynomial with integral coefficients and с is a given constant, U(p, q) =c has
only a finite number of positive integral solutions pt qy when the degree of
U exceeds 2.
A. Thuew considered homogeneous integral functions P(x, yO Q(x, y),
R(xt y) of degrees p, g, r, with integral coefficients, P(x, y) being irreducible.
If p>g} p>2, P = Q does not have an infinitude of pairs of integral solutions
x, y. If p>q>ry p<q+r, P+Q+R—0 is not satisfied by an infinitude of
pairs of relatively prime integers x, y.
E. Maillet" completed a lacuna in the proof by Thue9 and gave the
following generalization of his theorem. Let фх be a homogeneous poly-
polynomial of degree г in x? y. While the coefficients of фоу ■ - -, фш need not
be rational, let фг (r>s) have integral coefficients and contain a term in xT
and one in y'. If
фт(х, у)-ф.(х, у)-ф^(х7 у) ^0=0
is irreducible, it has an infinitude of integral solutions x, у only when s
exceeds a specified quantity depending on the reducibility of tfr=0. When
Ф, is irreducible, this quantity is rx—2 or n~ 1, according as r^2ti or
Maillet11<a gave a practical method to find an upper limit to the absolute
values of the integral solutions xt у of an equation of type B), subject to
certain conditions on #n which imply that B) has only a finite number of
integral solutions.
Rational points on the plane curve f(x, y} z)=0.
D. Hilbert and A. Hurwitz12 treated homogeneous polynomials/{Xi, z2, xt)
of degree n with integral coefficients such that the curve/=0 is of genus
(or deficiency, geschlecht) zero» In view of results by M. Noether,18 we
'Jour, for Ма1Ц 135, 1909,303-*. «. Maillet.u
'< Skrifter Yidemkape. Kiistiania (Math.), 1, 19U, No. 3 (Gerauux).
11 Nouvr Ann. Math., D), W, 1916, 338-345.
"'M. D), I8r 1918,281-92.
"Ada Math., 14, 1890-1, 217-24.
u Math. Annalen, 23, lSS4r 311-358.

676 History op the Theory of Numbers. [Chap, xxiii
can decide by rational operations whether or ikot /—0 is of genus zero and
if so we can find by rational operations »—1 linearly independent ternary
forms ф< of degree n—1 with integral coefficients such that for arbitrary
parameters X<the curve/=Oiscut by the curve
in n—2 points varying with the parameters X>. Set
Ф>=\н4>1-\ +X^!0^i (t«1, 2, 3),
where the X<> are arbitrary parameters. Transform /= 0 by
Vi '-Vi :уз = Ф1 : Ф* : Фз.
The result is д{у\, уг, у*) =0, where g is an irreducible form of degree n—2
m the y's with integral coefficients» Now give to the parameters X* such
integral values that g remains irreducible. Since our transformation is
birational, every rational point on /=0 corresponds to a rational point on
0=0 and conversely. Hence the initial problem is reduced to the equation
0=0 also of genus zero, but of lower degree by two units. Ultimately we
reach an equation of degree 1 or 2. For a linear equation l(ul7 ub uj) = 0,
we can evidently find three linear functions qj< of the homogeneous param-
parameter t\ji\ such that wi : «2 : u3 — о>г : о>г : wj gives all rational solutions of
Z=0 when £i, tz take all integral values. By applying the inverses of our
transformations, we get the mitial/=0 and solutions Xi : x2 :xj=pi : p2 : pz,
where the pi are forms of degree n ш t\, *2. The only missing solutions are
those, finite in number and found rationally, which correspond to rational
singular points of/=0, where our transformations cease to be birational.
Second, if we reached a quadratic equation, it can be transformed rationally
into a\u\+atu\-\-azu\=ftf the a's without square factors and relatively
prime m pairs. It has integral solutions if and only if the a's are not all
of like sign and if — <hfli, — e3ai, — a\a^ are quadratic residues of Oi, aiy at,
respectively (papers 114, 116, 119 of Ch. XIII). When these conditions
are satisfied, the conic has rational points and can be transformed bira-
tionally into a straight line; we proceed as before.
M. Noether14 had earlier proved that a rational curve can be trans-
transformed birationally into a straight line or conic; a curve of order 2n with
a Bn—l)-fold point is counted as curve of odd order.
H. Poincare* proved the above result that any unicursal curve with
rational coefficients is equivalent to a conic or a straight line, two curves
being called equivalent if one can be transformed into the other by a
birational transformation with rational coefficients. A curve/^0 of genus
1 (bicursal curve) with rational coefficients is equivalent to a curve of
order p (p^3) if and only if/=0 has a rational group of p points, i. e., a
set of p points such that every elementary symmetric function of their
coordinates is rational.
w Math, Annaten, 3, 1871,170.
" Jour. <fe Mafch.r E), 7,1901, 161-233. For a special case, von S«. Nagy1" of Ch. XXI»

Снлг.ххш] Equations Formed from Linear Functions. 677
J. vod Sz. Nagyie proved that any curve of genus 2 with rational coeffi-
coefficients is equivalent ш general to a quartic curve and contains an infinitude
of rational groups of two points.
J. von Sz, Nagyir cited the known fact that a curve (Z of order n and
genus p>\ has ш general no birational automorphs besides identity, and
never more than 84(p—1), and concluded tbat we can derive at most a
finite number of rational points from one. The birational automorphs
of non-hypereUiptic and hyperelliptic curves are discussed. An example
shows tbat from a rational point we do not in general obtain all other rational
points by means of the birational automorphs of the curve.
J. vonSz. Nagy18 wrote Qn for the g.c-d. of «and 2p—2and proved that
a curve C£ of order n and genus p contains infinitely many rational groups of
hQ* points if Л is an integer for which hQn>p~1; it is equivalent to a
curve C* for m>p+l if and only if it contains a rational group of m поп-
singular points. In particular, they are equivalent if m is a multiple of QKj
and hence if m = 2p~2, p>2, and the curves are not hypereUiptie.
E. Maillet1*1 considered a polynomial f(x, y) of degree n>2, irreducible,
with integral coefficients, and such that thecurve/=0isunicur8al (of genus
0). If there are at least n—3 simple rational points, there is an infinitude
corresponding to the rational values of a parameter t, and x^ft(t)ffi(C)y
V^MfylMt), where the fi are polynomials with integral coefficients having
no common divisor, of degrees n^n, one being of degree ft (cf. papers 12,
15). The curve has an infinite number of points with integral coordinates
only when /i is a constant or of one of the forms a(Mt+N)n, with «, M, N
integers, or a(Mt2+Nt+P)nf2, where n is even and N^—AMP is positive and
not a square, while a, My N} P are integers. There are extensions to cer-
certain equations f(x, y)=0 of genus >0 and to certain unicursal surfaces.
For cubic curves of genus unity, see Levi** and Hurwitz311 of Ch, XXI.
Equations formed from linear functions.
For related pepers, see Lagrange,142 Rados1M*; papers 313-23 of Ch.
XXI; and Ch. XX.
G, L. Dirichlet" stated a theorem, which he regarded as remarkable for
its simplicity and importance: if an equation
A) «•+O8*-|+...+0H-ft^O
with integral coefficients has no rational divisor and if at least one of its
roots or, p, * ■ •, w is real, and if we set
Ф(«) =
then the indeterminate equation
B)F(x,y, ■•■,«)
'♦ Math. Natunr. Berichte аоя TJngarn, 26,1903 A913), 186 A68-195),
17 JiihreAericbt d. Deutechea Math.-Veretiiiguii& 2lr 1912r 183-191.
" Math, АшЫвл, 73, 1913, 230-240, 600.
«•Comptee Rendue Paria, 168,1919, 217-20; Joar. Boole Polyt., B), 20,1919,115-56.
» Comptee Rendue Pane, 10, l840r 2S6-8; We*er I, 619-623.

678 History of the Theory or Numbers. [Chap. Хлш
has an infinity of integral solutions. Application is made to functions
considered by lAgrange14* which repeat under multiplication. If such a
function can take a given value, it takes the same value for an infinitude of
sets of values of x, — -, 2, under the assumption that the algebraic equation
to which the function owes its origin has no rational divisor, but has at
least one real root.
G. Iibri* stated that the conditions imposed on A) that there be a
геч1 root and no rational factor are not necessary, it sufficing to have A=± 1.
J. Liouville21 proved Libri's theorem false. For, if A) is «^-h 1=0,
then B) is {x+yi){x—yi) =z2H-&l=l, with only a finite number of integral
solutions.
Dirichlet22 noted that hie theorem remains true if A) has only imaginary
roots, provided n>2. The problem is that of the units of an algebraic
domain.
P. Bachmann11 treated the solution of N=1, where N is the norm of the
general algebraic number determined by a root of an equation of degree n.
H. Poincare*24 noted that, for F defined by B) by means of any equation
A), the problem to find integers fr such that F{fiiy - - -, fin) shall equal any
given integer N reduces to the problem to form all complex ideals of norm N.
In the solution of the latter one considers the congruences s^H-tw"-!-* —^0
(mod ц), p. any divisor of N.
E. Meissel2* considered the product, extended over the roots of <P=1,
V=(x, y, z, «, t>)-n(x+fyp+tep>+^p4*V), p= VI.
By the reciprocal solution of V= 1 is meant 1/V= (a, b, c, rf, e) — 1, where
6V KA dV
—, 5Ae—> bAd=~, 5Ac=~, 5АЪ=—.
дх' ду dz' ди' дь
For 2<A^7, he gave two primary solutions Vi=l, Vi = l, accompanied
by their reciprocal solutions. He stated tbat two primary solutions always
exist and deduced the solutions V^V\. He conjectured that, if p is a
prime, the corresponding Pell equation of degree p has %(p—1) primary
solutions.
A. Thue26 considered a homogeneous polynomial F{xi, - —, xn) of degree
n — 1 such that F=0 can be given the form
C) РЛ"чР-1-«!«1---0^1,
where Pif Q> are linear functions of xh • ■ ■, xn with integral coefficients. Set
D) aiPi = a2Qi, ajt>t ^a*Qb -> a«-iP^i = aiQ*-i,
where the a'& are any integers without common divisor. Then D) if
independent give xt-=£A^ (i = l, - * -, n), where Д» is a homogeneous
» Comptes Rendua Fkii% I0t 1840, 311-4, 383. —— _^^^^^____
ЙМ„ 381-2. Bufl^deeScMfttb^ B)^32,1,1908,48-65.
B Bericht Akad. Warn. Berlin, 1842, 95; 1846, 103^7; Weike, I, 638-644.
**De unitatum сошр1кхагшп theorie-, Dke., Berlin, 1864.
* Comptes Kendus Pane, 92, 1881, 777-9; Bull. Soc. Math, France, 13, 1885, 102-194.
*Brit»g ш PeD^echen Gldchtm^ hfiheret Grade. Frogr., Kiel, 189L
« Det K«L Nomke Yidendcaben Sebkab» Sbriftcr, 1806, Na 7 (German).

Chap. ХХОД PRODUCT OF CONSECUTIVE INTBGEKS NOT A PoWEtt. 679
nomial of degree n—1 in at, - - -, a*-t- Finally we choose к to make these
x*8 integers.
If F^O can be given the form C), every set of integral solutions of
Pi=Ot Qy=O (*, i-1, •' ■, я—1) is evidently a solution of F=0. Con-
Conversely, if a certain number of integral solutions of Pi=Qj=0 satisfy
F=0, then F=Q can be given the form C). In fact, if a polynomial
F(xi, - -, xn) of degree m always vanishes simultaneously with the products
U-Pi'-'P^ V=Qi-Qq of linear functions of xL, ■ >«, xH, such that
not all the values for which any two are гею make a third zero, then
F*zAU+BV, where A and Л are polynomials inxi, - - >,**.
A. Palmstrom* extended the preceding method to the equation
Pti Pn ' * ' Pi n-i
Pll Pt2 " ' " P* »-l
E)
=0,
where the P's are linear homogeneous functions of Xi, ■ ■ ■, xn. For every
set of integral xJs satisfying E) there exist n-1 relatively prime integers
«i, ■• -, a»^i satisfying
F) «iPu+ejP^H +a—iP,- »-i-0 (г-1, - - -, n-1),
and conversely. From the latter, х</хя-Х/Ап, so that we may set х, = £Д;
(j= 1, • • •, n) and choose к to make the xJs integral. Here the a's have
any values for which Ai, ■ • •, Д„ are not all zero. In case the A's are all
identically zero, so that only p of the equations F) are independent, we
can assign arbitrary values to n—p—l of the z's and determine the remain-
remaining x'b by p linear equations. He130 gave a detailed example.
G. Metrod*7" found the number of ways to decompose a given number
into a product of n factors (including unity).
Product Pn of n consecutive integebs ^not an exact power.
Chr. Goldbach28 argued that a P% is not a square since its root would be
a multiple of m and a divisor of (m-\-\){m+2), whence m=l or 2»
J. Iiouville* proved by use of Bertrand's postulate [Vol. I, Ch. XVIII]
that m(m+l)*-~(m+n—1) is not a square or higher power if at least one
factor m, • • •, m+n—1 is a prime, or if n>m-5. The latter was proved
similarly by E. Mathieu,ao who verified the theorem for any n when m^lOO.
In particular, m\ is not an exact power, a fact proved in the same way by
W. E. HeaL31
Mile. A. T)*2 proved that a P, is not an exact power.
"&knfer tJdgivae eJ VkfemduibsttlakAbet, Christum*, 1900 A899), Math.-Naturw, El,,
No, 7 (German).
*»- L'mtmnediaire des math., 2в7 1919, 153-4. <X MinetoU^ of СЯь UI, and Ceeiro»" of
СЫХ; aleo Index to YoLl (шккт "Number," including«=aV)-
rtCorreflP- Math- Phye- (ed^ Fun), 2, 1843, 210, letter to D. Bernoulli, July 23, 1724.
» Joar. lie Math., B), 2, 1857, 277. (X- Morewi."
~Nouv. Ann. Matb., 17,1858, 235-6,
л Math, M&gasme, 1, 1882-4, 206-9,
M Nottv. Ann. Мж11ч 16, 1857, 288-290, Prop<»ed by Faure, p. 183.

680 History of the Theory of Numbers. [Chap, xxiii
G, C. Gerono8* proved that ft=|=D by setting (m+l)(m+4) =2p,
whence (т+2)(то+3) = 2{р-Н), while р(р+1)фП. "К A. G,"M gave a
proof by use of
Gerono16 proved that Ph P* or Pi is not a square.
V. A. Lebesgue* proved that P6 is not a square or cube.
A. Guibert* proved that, if 8^n^l7, P»+D, while P« or Pi or a
product of any three integers in arithmetical progression is not a cube.
A. B. Evans*8 and G. W. ШР proved that P« Ф П.
D. Andre40 proved that, if n>l, Ря*уп or y*±l.
А. В. БЬгашв41 proved that P^ P« or Pi is not a square.
H. Bourget41 proved that Рь + П.
R. Bricard48 proved that P6 + П by use of a Pell equation.
L. Aubry44 proved that ft is not a cube by treating the case in which
a single one of the four numbers is divisible by 3 and the case in which two
are divisible by 3, necessarily the first and fourth, and examining in the
second case the residues modulo 9 of the four numbers.
T\ Hayashi** proved that Pt or Pt is not a square or cube, Pt^xn} n^2.
Also (p. l«),y(¥+l)B*+l)+*l',*S2.
S. Narumi48 proved that x(x+l)- -(x+n)^ П 40 is impossible if
»<202.
T\ Bayashi* proved that Рь Ф D.
Further properties of products of consecutive integers.
J. Iiouville4* proved that, if p is a prime >5,
Berton" verified that P~a(a+h)(a+2h)(a+3k)*pA since
Hence the area ^P of an inscriptible quadrilateral whose sides are in arith-
arithmetical progression is not a square.
* Nouv, Ann- Math., 16, 1S57, 393^.
"ltod.t 17, 1858,98.
»Ibid., 19r I860, 38^2.
»Ibid.t 113-5,135^6.
" Ibid., 213 [4002l B), I, 1862, 103-9.
" The Lady's and Gentleman'* Diary, London, 1870, 88^-9, Quest. 2106.
w ТЪе Analyst, Dee Moinee, Iowa, 1, 1874, 28-29-
«■ Noav. Ann. Math., B), 10,1871, 207-8.
« Math. Quest. Educ. Times, 27,1877, 30; 44,1886, 65-0.
«Jour, de math. ^6m.r 1881t 66.
^L'intermediaire dee math., 17,1910, 139-40.
«Sphmx-OedipeT 8,1913,136.
«Nouv. Ann. Matb, D), 16,1916,155-&
• Tfthoku Math. Jour., 11,1917,128-142.
л Notnr. Ann. Blath», D). 18,1918,18-21.
«Jour, de MaflL, B), 1,1856, 351.
«■ Nomr. Ann. Math., 18, 185», 191-

Сна*. XXIII] PROPERTIES OF PEODTTCT8 OF CONSECUTIVE InteGEIIS 681
C. Moreau» repeated the first remark by Iiouville.M
H, Brocard*1 asked for values of x making Ц-zI a square. He52 sugges-
suggested that the only solutions are 4, 5, 7.
E. Lucas" noted that the product P of the first n primes is not of the
form ap±&*, where a and Ь are positive integers and p>l> P>2.
E. Iionnet" stated that no product 1 ■ 3 ♦ 5 ♦ - * of consecutive odd numbers
is a square or higher power. Moret-Blanc55 proved the last statement by
Bertrand's postulate»
Moret-Blanc56 solved y(y+l)(y+2) =z(z+l), proposed by lionnet.
Adding 1 to the product by 4, we are to make 4y*A~ 12^2+8#H-1 = П, say
(my—1)\ The discriminant of the quadratic ш у is to be rational. Thus
m=2n, n*—6V—4n4-l = Q, which holds for n=3. Thus solutions are
l-2-3 = 2-3, 5 6-7 = 1415. G. C. Gerono (p. 432) noted that, since
2x+l = 2ny—1, the initial equation becomes y*—(n1—3}^-hnH-2^0 and
proved that л=3.
E» Iionnet proposed and Moret~BIanc*r solved the problem to find N
such that both N and N/2 are products of two consecutive integers, the
smaller factor of N/2 being a product x(x+l) of two consecutive integers.
Thus
Euler's process to deduce new solutions from x=l leads only to # = 0 or
fractional values»
E. Lemoine*8 asked if the product of three consecutive numbers (besides
2J 3, 4) is of the form p^t where p is a prime. H. Brocard (p. 304) noted
that the problem reduces to у*—у*=рх*, took y-p and concluded that
x-27 y=3. Several replies (p. 369) show readily tbat 2} 3, 4 is the only
solution.
E. B. Escott" proved that x(x-H)(*H-6) =j= D.
G. de Kocquigny*' proposed for solution
E» B» Escott61 noted the solutions x = 1 or —6, y=&, besides the evident
solutions я=0, —1, - - >, —5. P. F. Teilhet*2 proved that tbese are tbe
only solutions by noting that tbe left member becomes (z-4)z(z+2) for
(l
»Kouv»Ann.M*th.p B), 11,1872,172.
« Nouv. Correep. Math., 2,1876, 287; Nouv. Ann. Math.r C), 471885, 391.
»MatIi«aiflp 7,1887, 280.
w Nouy. Comsp. Math^ 4r 1878, 123; Throne dee потЬлв, 1Й91, 351, Еж. 4. Proof by R
Bachmann, NEedeie Zahlentheorie, I, 1902, 44-6,
« Nouv. Ann. Math^ B), 20,1881, 615.
* Ibid., C), 1, 1882, 362- Invalid objection by G. C. G*ronor p. 820.
»Nouv. Ann. Math., B), 20, 1881, 431^2. Seme, Zeitechr. Math. Naturw. UnterHcht, 13,
1882, 451.
" Nouv. Ann. Math., C), 20,1881, 375-
" L'int«nn№ure d«a math., 2,1895,15.
7Ы7 *»
«/Ы-, 9,1902,203.
«/Ш.,10, 1903,132.
-/i«,, 12,1905,116-8.

682 History op the Theory of Ntjmbbbs. [Chap, xxin
P. F. Teilhet*3 stated for m=Z and several proved that, if m is a prime,
n(n+l)(n±2) =mA* is impossible.
A. Gerardin64 remarked that if lH-x!=yl has solutions other than
x^4y 5j 7; y = 5t 11, 71, then у has at least 20 digits.
Sum of птн powers an гстн power»
Euler (Ch. XXII, paper 187 and the one preceding it) expressed his
belief that no sum of four fifth powers is a fifth power.
E. Collins" noted that if Л7"=1+п+т*Ч l-ft* is divisible by a prime
p, then p=l (mod k)t since n*«=(n—1)ЛГ+1. Henceforth, let this ^bea
prime. Then, if Л is any integer not divisible by 2V, Aq is congruent to a
power of n modulo N, where q = (N~l)/fc, since A" is a root of xk = l
(mod N), and its roots are powers of n. Hence if aT-f \-al=*A*t while
uit ■->QB are not divisible by the prime N7 the difference of some two
of the at is divisible by N. For example, if n = 2, fc=3, then N=7, q^2y
whence if a sum of two squares (each prime to 7) is a square, their difference
is divisible by 7. Again, let N = l+5+&=31; then g^lO and, if a sum of
five tenth powers (not divisible by 31) be a tenth power, a difference of
two of the powers is divisible by 31» He verified that q>n except when
к = 2, or к = 3, n = 2. He conjectured that a sum of n numbers each an eth
power is not an eth power if n<e.
F. Pauletw announced that no nth power is a sum of nth powers if n>2.
A committee reported adversely, citing the known formula 6а=33-Ь4|4-5*.
O, Schier47 made an erroneous discussion of xn+yn+z*=un. First, let
n be an odd prime. Then x+y+z = u+nd. Subtract its nth power from
the given equation. The new left member has the factor y+г which is
said to be divisible by the factor n of the new right member. This admitted,
the given equation would be impossible for n a prime >3 and hence for
any n>Z. Only special sets of solutions are found for и —3 and n=2.
A. Martin68 found by tentative methods (Hartus and Martin11* of Ch.
XXI)
Barbette198 of Ch. XXII noted that the first result is the only one in-
involving fifth powers each :Hl2\
Martinw would by trial express lrt4-2rtH \-xn-bn as a sum of distinct
nth powers each ШХ*. For те=5, х = 11, b = 12, we get hisw first result.
M Lpinterm6dUiire dee math., 11,1904, 68, 182^4.
« Nouv. Ann. Math., D), 6, 1906, 223-
«■ Mem. Acad. Sc. St. PetcrabouiK, 8, аппбеа 1817 et 1818, 1822, 242-^6.
M Comptee Rendus Para, 12, 1841, 120, 21L
" SiteuPgBber. Akad. Wiee. Wien (Math.), 82, 1IP 1881, 883-862.
M Bull. Phil. Soc. Waeb,, 10, 1S87, 107^ in Smithsooiati MieoeL CoU., 33, 1888.
" Math, Quest. Educ. Times; 50,1889, 74-5.

Chap. XXIII] STJM OF ПТН POWERS AN ПТН POWER. 683
Martin and G. B. M. Zerr™ multiplied the numbers 4, 5, • * •, 12 in the
formula just cited by 42* and obtained six numbers whose sum is a fifth
power 42* and sum of fifth powers is a fifth power.
Martin71 multiplied his*8 first formula by 2* and replaced the new third
term 12* by its value to get a formula for 245. There is an analogous longer
formula for 50*. Again,
Martin72 found sets of fifth powers whose sum is a fifth power.
G. de Rocquigny" proposed for solution (х—г)т+хт-\-(х+г)ш=ут*
H. Brocard7* noted x=4, r=l, y=b O*=3], and E. B. Escott74 noted
3 = 1, r^2, t/ = 3, for m any odd number. Cf. Gelin," also Escott*1 of
Ch. XXI, and Bottari1" of Ch. XXV.
A. Martin74 found sixth powers whose sum is a sixth power by the tenta-
tentative method of expressing p*— <f as a sum of distinct sixth powers #$*,
or S— b* as a sum of sixth powers ^n\ where £=l*-f f-n*. By each
method he found his71 example, also that the sum of the sixth powers of
1, 1, 2, 5, 9, 11, 12, 13, 15, 18, 21, 22, 23, 24 is 29s [false] and that of 1, 2,
2, 4, 5, 6, 8, 9, 10, 12, 14, 15, 18, 19, 27, 33, 49 is 50е (each with one repeated
term). By combining these, he found eleven new sets of 29, 31 (seven), 32,
46, 47. He tabulated the values of n* and 14 \-n6 for n ^228.
C. Bianca" noted that & = €%-] \-cd-i is a pth power if
where d»+c*=d*. For, if а1=кЬя, ■ >>, then s = (№)'.
A. Martm77 reported on sums of nth powers equal to an nth power.
* N. AgronomoF proved that xf^H h^+l =^0 is solvable ш integers
if fc=4n+l and n^m> He proved the identity
where S> denotes the sum of the <27n+l)-th powers of all the sums of
2?л+2 parameters taken j at a time. A. Filippov78fl gave an account in
French of this paper, with details for the case wi=2.
7*M»th, Queet, Educ. Timee, 55, 1891, 113.
» Qiwr, Jour. Math^ 26,1893, 225-7.
n Math. Papers Internet. Congree* of 1803 at Chicago, iSdd, 168-174. RepubliahedT
Math. Mag,, 2,1898, 201-8, with the following corrections: In Ex. 18, p. 173, insert 16»;
on p. 169, fourth line up, delete one 3l; on p. 174, delete the final equation. Id Part III
(combining earlier seta) he added a new set of n fifth powers for л -17, 2\, 24, 26, 28, 36,
42, 48, 52, 63, 67, 72 and three seta for n -33,
» L'intermediairc dee math., 9,1902, 203.
uIbid.y 10, 1903, 131-3.
" Math. Mag., 2, 1904, 265^-271.
"II Pitagora, Palermo, 13,1906-7, 65-6.
» Рпж, Fifth Intern, Congress of Matb., 1912,1, 431-7.
л bv. Fi* Mat. Obe. Kazan (Bull. 8oc. Phye. Matb. Kaean), 1914,1915.
"•ТбЬоки Matb. Jour., 15, 1919, 135-40.

684 Hibtort or the Theory of Numbers. [Cbap. xxrn
T"WO EQUAL SUMS OF ПТН
A. Desboves™ noted that w5+t^=sb+K^ has the complex solution
u, v=2xy±(x*-2y2); 9,
J. W. Nicholson80 recalled that, if $—
Thus ll»=9*+8n+5*-6"-3n-2» for n = 2 or 1 [Euler2, Ch. XXIV];
etc.
Several writers81 determined the signs so that
A» de Farkas82 proved it is impossible to find two different sets x> and
yi such that for a and q are arbitrary
N. Agronomof83 argued the existence of integral solutions of the equation
But, as shown by Filippov78* for the case p = 5, h=g^= 4, the method leads
only to the trivial solution xY= = —х$, Хг~ —Хг, у\= —уъ, уг= — у*.
С, В» Haldeman88* gave special rational solutions of s3 = 34 and 8*=&n,
where sn denotes a sum of n fifth powers»
Од x*+vn = y*+uni g^ gteggallieof Ch. XXII.
Miscellaneous results on sums of like powers.
J. Hill84 noted that the sum of the cubes of x*/2r 2z2/3, 5x*/6 is a sixth
power x\ Cf. Emerson62 of Ch. XXI.
L. Euler^ stated that no sum of three biquadrates is divisible by 5 or 29,
which alone are exceptional. Cf. Gegenbauer"8 of Ch. XXVI.
R. Elliott" noted that 154 +nb=U if F^\{2nJ+2n-l) = D and
took n=x+l. Then 9F=6a2-hl8z+o, = n = (<Lr-3)* determines x> The
anonymous proposer solved F—a3 for n; the radical must be a rational
number 3c. Take а^р+Я, с=р—д. Then p*— lO^+^^l, whence
^= П, whose solution is known.
G. Iibri87 expresst^ as a trigonometric sum the number of sets of sohi-
tions of z*+ - »'+x!+l=O (mod p), where p is a prime an+1
Cf. pp. 22^-5 of Vol. I of this History.
" Адаос. franco ft, 1880, 242-4.
» Amer. Math. Monthly, 9,1902» 187, 211,
" Math. Quest. Educat. Timea, B), 13, l908r 110-111.
« L'intermediaire dee math., 20r 1913, 79-S0-
M Tihoku Math. Jour., 10, 1916, 211.
•*• Amer. Math. Monthly, 25, 1918, 390-402.
"Ladiee' Diaryr 1737, Quest. 192; Leyboum'a M»th. Quest. L. D,, 1, 1817r 254-5.
Math. Queat. Educ. Timee, 66, 1897, 120.
M Opera poetuma, 1,1862,186 (between 1775 and 1779).
« Ladiee' IMary, 1796, 40-1, Queet, 992; Leybouns'e M. Quest. L. D., 3,1817, 296-7.
" Mem. diver* savanta acad, ec, de Tlnstitut de France (math.)a 5? 1838T 61-63.

Сиар.ХХпг] Sums op Like Powers. 685
V. Bouniakowsky8* obtained the identity
from f\{z-\-ay-(x-ay}dz by setting a = 10A*.
E. Lucas884 stated that the sum of the cubes of the first n (odd) integers
is never a cube, fifth or eighth power (cube, fourth or fifth power). The
sum of the cubes of three consecutive integers is never a square, cube or
fifth power, except for Р+2э+За=-62, зз+4*+5*=бэ [correction, Aubry**
of Cn* XXI]. The eum of the first n biquadrates is never a square, cube
or fifth power. The sum of the first n fifth powers is never a cube, fourth
or fifth power.
E. Lucas8* asked for what values of n the sum of the fifth powers of
the first n odd numbers is a square. The problem reduces to
whose complete solution was given by L. Aubry.M
Lucas*1 asked for what n's the sum of the fifth or seventh powers of lp
• * »f п is a square» H. Brocard*1 noted that the sum of the fifth powers is
in2(n+lL, where ^Bгс2+2я-1)/3. То make t^y\ we have
which must have 9 as its final digit, whence y=10wid=l. He noted the
special solutions у=n=l; y^ll, n^!3, Cf. Moret-BIanc,M Fortey.*
H. Brocard92 noted that the sum п2Bл2—1) of the cubes of the first n
odd numbers is a square for n= 1,5,29,169,985, • ■ -. As to Lucas'8*1 theorem
that the sum s of the squares of the firet n odd numbers is not a square, cube
or fifth power, he stated that this is evident since s = Bn—l)Brc)Brt+l)/6.
Lucas (p. 247-8) noted that this proof would require extensive develop-
developments; if p is a product of three consecutive numbers, p/6 is not a square
if the first of the three numbers is odd, and also if it be even except for
/622480/6 Ш*
/,/
Abbe Gelin" proved that (x-iy^+s^+tx+l)'11^*» is impossible
and tbat the sum of like even powers of 9 or 12 consecutive integers is never
an exact power (stated for 9 by Lucas, p. 248), The proof is by use of
various properties of 2{N), obtained by adding the digits of N, then adding
the digits of this sum, etc., until there results a sum with a single digit.
E. Lucas stated and H. Brocard, Radicke and E, Cesaro9* proved that
"Bulh Acad. Sc.St.PgterabouigCPbys.-MftthO, 11,1853,65-74* Extract in Sphinx-Oedipe,
5P 1910,14-16.
"•Recherchee виг Рангуне indetermin&, Moulins, 1873, 91-2, Extract from Bui!» Soc.
d'Emuktioo du Departement de ГАШв», 12, 1873, 531-2.
" tfouv. Correep. Matb4 2,1876, 95.
*• L'intenii^diaire dee math., 18P1911, 60-62. a. 16,1909, 283.
« Nouv. Coireep. Math., 3, 1S77, 119-120. a. 4r 187S, 167.
*/W*,3, 1877,16^-7.
*Ibid.t 388-390 (extras frtjia Le» Mondea, July 14P 1877),
n 1Ш.Г 5r 1879P 112P 213-5; 6,1880, 467.

686 Histoby of the Theohy of Numbers. [Chap» ххш
is always a square, but never a bi quad rate,
Moret-Blanc*6 found the x*s for which (Lucas*1)
Hence (Zu2—l)/2=u2 or Cw—2vJ-6(v—u)*=l, whose solutions are given
by the convergents of odd rank in the continued fraction for V6.
E. Catalan" noted that, if p is an odd prime and j is an odd integer
^p-l, the sum of the £(p-l)th powers of j integers relatively prime to p
is not divisible by p.
A. Berger*7 proved that, if sr mf nf glf - -, gb are positive integers, and
ф(п) is the number of positive integral solutions of gixT-\Г
L. Gegenbauer95 proved a generalization of Catalan's5* theorem. If X
is one of the numbers 2, 3, 4, and if p is a prime =1 (mod X), and r an
integer prime to X and <plft, where i is the largest integer ^(X+l)/2,
then the sum of the (p—1)/Xth powers of r integers relatively prime to p
is not divisible by p.
H. Forteyw found that 14 \-n*=\3 for n~l, 13, 133, 1321, -,
by use of 3y2-2xi= L Cf. Moret-Blanc «
E. Lemoine100 said that A is decomposed into maximum nth powers
if A=a\-\ НС where a", aj, aj, —* are the largest nth powers ^A,
A— a\, A—ai—a*, --, respectively» Similarly, consider the decomposition
А=а"-а*2+<х] ±aj, where ai is the least integer ^VAand/?ithe
remainder a*—A, a2 is the least integer ^^/Si and R* the remainder, a\
the least integer ^ ^Sj, etc., and call yv the least number requiring p
powers. Then, for n=2, yi = l, Ti = 3, 7s=6 = 3s-22+l2, y^-i = bj+b
For n = 3, he101 gave elsewhere the possible forms of the iinal power aj.
L. Aubry104 proved that -P+33—5*4 Kto—I)8 is never a square,
cube or biquadrate.
Welsch and E. Miotiw noted cases in which ал-Ь(а-|-1)лЧ +(a+k)n
is of the form J2—m% and hence is a sum of consecutive odd numbers of
wbich the least is 2m-|-l.
C. Bismaniw noted that a sum of like even powers of n9+4 numbers
can be expressed as the algebraic sum of n2+5 squares of which only one
is taken negatively»
»Nouv. Ann. Math., B), 20r 1S81, 212.
"Mem. Soc. R, Sc, de Iidge, B), 13,1880, 291. Cf. Gegenbauer."
"Ofversigt K, Vetenekapa-Akad. Fdrhand., Stockholm, 43, 1886, 355-66.
«Sitsungsber. Akad. Wisa, Wien (Math.), 95, П, 1887, 83^*42.
•• Mftth, Quest, Educ. Times, 48, 1888,30-31.
i">Aeeoc. fran$.,25 1896, 11^ 73r-7. Forn=27eee рарвл 20,21 of Ch. IX.
" L'intermediwe dee math., 1, 1S04, 282.
"Sphinx-Oedipe, 6, I911r 3S-9. E. Lucaa, Nouy. Correep. Mathv 5, 1879V 112r bad aaked
for Bolutioos.
i« L'mtermediaire dm math., 20, 1913, 47-18.
i" Mfttoesie, D)r 3,1913, 257-9.

Снар.ххпг] Product of Factors (x+1)/x. 687
T. SuzukiJ0S noted that there are at least (p-2)(p—I)»-* solutions of
«?+-••-ha^=O (modp),
if two of the a's are primitive roots of the prime p. Also there are solutions
if at is a primitive root and if not every a<=l (mod p) for i=2, - • •, n.
Rational solutions or x*^y=~
L. Euler106 set y=tx and deduced xt^1=L The graph is composed of
y=x, a branch asymptotic to the positive x and у axes, and an infinity of
isolated points. Among the rational solutions are (z, y)^Br 4), C2/28,
3V28) D'/3* 4W
D. Bernoulli" noted that, for x^y, the only integral solution is 2, 4;
but that there is an infinitude of rational solutions.
J, van HengellW remarked that fH-~>(r+n)T if г and n are positive
integers either one ^3. Thus if 0^=6*, it remains to treat the cases a-1
or 2. If a=2, b>4, whence Ъ=2-\-п> we apply the above remark.
* C. Herbet10* noted that 2, 4 give the only solution in integers.
* A. Flechsenhaar31* and R. Schimmack111 discussed the rational solu-
solutions.
A. M. Nesbitt112 and E. L Moulton118 discussed the graph of x*=y\
A. Tanturri1" proved that 2, 4 give the only solution in integers.
Product of factors (z+l)fz equal to such a fraction*
Fermat21* proposed the problem to find in how many ways (n+l)/n can
be expressed as a product of к such fractions, citing the case n=8, k= 10, as
suitable to be proposed to ali mathematicians of his time. Tannery noted
that of the decompositions of 9/8 the difference of the factors is least and
greatest in respectively
90 80 88 87 86 85 84 83 S2 81 9+1 9*+l У+1 9^+1 9
89"88'87'8б'85'84'83"82'8Г80' 9 92 ' 94 '"' 9^ >»-l*
V» Bouniakowksy118 noted that an irreducible fraction afb less than
unity can be expressed in an infinitude of ways as a product of fractions
of the form x/(x+l). We may often find fewer than the b—a fractions
м Tdboku Math. Jour., 5, 1914, 48-53. Cf, paper* 2Й5-6 of Ch. XXVI,
i" Introductio in amUyain infin,, Kb. 2P cap. 21, §519; French tranel. by J. B. Labey, 2.
1797 and 1836, 297.
«"Correep. Math. Phj». (ed., Fuag), 2,1843, 262; letter to Goldbachr June 29, 1728.
"i Beweis d« Satwe, dae unter alien reelbn pouitrren ganten ZahJen nur dae Zahlen Paar 4
und 2 fur a nnd b der Gleichung a* =-^ genugt, Progr. Enamerich, 1888.
i« UnterrichtebL fur Math., 15,1909, 62-3.
"•Ibid., 17,1911,70-3-
ш/Ьй.г18,1912, 34^5.
»«Math. Queet. Educ. "Hmee, {2), 23,1913, 77-S,
w Amer. Math. МопШу, 23,1916, 233.
^«Pteriodk» di Mat., 30,1915,186-7,
" Oenyrea, I, 397. Quoted by Tannery, l'interm^diaire dea math, 9, 1902, 170-1,
» MAn. Acad. Sc. St. Petenbourg (Sc. M»tk Phys,), F), 3, lS44r 1-16.

688 History of the Theory of Numbers, [Сна*, xxiii
used in
a_ a a+1 b—X
Ь~а+Гл+2 Ь~ш
Set
a p и
whence
bp—aq
Consider the case bp —aq = l and let p=«, q=0 be the least solutions.
Then
a a a{3
Proceed similarly with <*/0. Many numerical examples are given.
A. Padoalir noted the equivalence of
n x у
Hence if n is given we obtain all couples x, у Ьу finding all pairs of positive
integers whose product is n{n-\-1), and adding n to each factor.
J. E. A. Steggall11* found positive integral solutions of
W x у г '
by noting that a^mnst be divisible by x-\-y-\-l =a, and hence x(x+l) by a.
Hence for any integer x, determine a factor a>z+l of z(z+l); Uien
y=a— x—1, while z=x—b where b=x(x+l)/a* T. W. Chaundy (pp. 74-5)
deduced (x— z){y— z)= z(z+l) and set z=pqt x—z = piq} where pt p\ are
relatively prime. Hence у—г=р$ь piQi—pq+1*
G. Ascoli and P. Niewenglowski119 gave solutions of A).
A. M. Legendre120 evaluated, up to 10=1229,
24610 w-1
Optic formula —I— = -; generalization.
x у a1
An anonymous writer121 noted that, if three regular polygons of x, yf z
sides fill the space about a point, then l/s+l/y+l/z = l/2. If there are
four regular polygons of x, y, z, z sides, then l/x+lly+2/z^L The
number of solutions is found, also for 5 or б polygons.
117 L'iniermediflire dm math., 10, 1903, 30-31.
in Math. Quest Edue. Timee, B), 20, 1911, SO-L
"'Supplem. al Periodico di M^t,, Hp 1911,101-4, 116^7.
"> ТЬйопе dea nombra, ed. 2, 1808; ed. 3T 1830. Table IX.
m Lbdiat Dtaiy, 1785, 40-1, Quest. 829; Leyboum'a M. Quwt. L. D., 2,1817, 132-3.

. ххш] Optic Formula 1/х-Ы/у = 1/а; Generalization. 689
D. Andre122 deduced х—д=
e— — a of a1 being excluded.
S
а —e, where
Ztige128 gave s^
a*, the pair of divisors
where
g g у,
pq=a. F. Schilling114 noted that Zuge's solution is incomplete and gave
that due to Andre with a geometrical mterpretatioa of the optic formula.
A. Thorinm asked if I/a = 1/ai-HA** has integral solutions besides
a — mn, a1 = m(n+l)> a%^ wm{n-j-l).
A. Palmstrom, J, Sadier, and C. Moreau1* each gave the solution
a=\mn, ai
and noted that
has the special solution
Dujardin1*7 stated that, if n^2, all solutions are given by
a2
ai = a+— (X a divisor o! e2),
while A) naay be written Ла^аСНаи+С), where A=ai* • -a^i, and Л, С
are integral functions of аь *--, an_i [with C^A]. Then Ba=A—ACfb,
where X=^J?aft+C. Hence give to Oi, * * •, an-i any values and choose a
divisor X of Л С Take as В and a two integers whose product is A— AC/\.
If X—С is divisible by B, we get a solution.
M. Lagoutinsky1*8 stated that if n=*3 the complete solution of A) is
given~by formulas involving 13 parameters*
V» V. Bobynin!W discussed the expressing of fractions in the form Hl/Xi
in the papyrus of Aihmim (Achmtm), about the seventh century, and in
the Liber Abbaci of Leonardo Hsano,
A. Palmstrom1** treated, as an example of a more general type,*7
++
which may be written in the form
—Xi
—x2
—x7
—X2
Xt—Xi
X*
0
0
0
0
Xi
0
0
xt
0
0
Хь
0
Xi
-. 0
-. 0
<.. о
• •• ^1
ш Nouv. Ann. Math,, C), 10,1871, 298.
«Zeitachrift Math. Naturw, Untcrricht, 26,1805,15-16,
**1Шг1491-3.
ш L'mtermediaire dee math., 2P 1395, P- 3.
Ч3,1806,14.
«JKA, 4, 1897,175.
Ui Abb. GewWchte Math-, IX, I-13'(SuppI. Zeite^i. M^th. Pl^ys., 44,1809),
1MSkrifter Udgivne af VMenakabttebkabet, Chrietiano, 1600 A809), Math.-Naturw. Ш»
No. 7 (German), Lr'intermediairt doe math., 5, 1898, 81-3,

G90 History of тки Тнвовт of Numbers. [Chap, ххщ
For integral solutions Xi there exist relatively prime integers а< satisfying
and conversely. Hence
к being chosen to make the g'e integers.
M. Lagoutinsky1" treated A) for the case in which a, ai, • ■ * have no
common divisor. Call their Le.m. A} and set Afa = kt Л/а<=&<, Thus
k = Zki. Hence we take fti, "^^ to be any integers without a common
divisor and find the Lc.m. A of these к/в and к = 2Jfc<, Then the solution
is a=A/kf ui=Ajki.
Zuge18* solved аху-\-Ъх-\-су+d=0 by multiplying by a. Thus ax+c « P,
ay+b^Q, where bc—ad**PQ* For integral solutions, select the factors
P, Q so that P=c, Q^b (mod a). For the special case xy^a{x-\-y)% the
result by Andr6123 follows.
P. Whitworth1** noted that each divisor of N% = {x—N){y—2f) yields a
solution of lfx+ljy = l/N.
P. Zuhlke1" gave, for llx+lfy=2fmt 2x—m=p, 2y—m=qf pq=m*. If
m is odd the resulting x, у are integers»
E. S6s18* noted that the general solution of l/x^l/si-l-l/^ is
where ylt уг are any relatively prime integers. Calling such a solution
irreducible if k = l, and setting s=p?- * -jC*, where plf • • •, pp are distinct
primes, we find that there are 2^1 essentially distinct irreducible solutions
belonging to a given x, with xs, x{ counted the same as zh х%\ in all^
essentially distinct solutions belonging to x. For the complete solution of
2K—1 parameters j/» are introduced.
S6s1M noted that, if the a's are given integers,
has (not the only) solutions z^ax, zi=a^ti> if B) holds. The complete
solution in positive integers, with g.c.d. unity, is obtained for C), The
method is similar to that for the case n=2. Set Zi = ZZi, zt^ZZi, where
Zi, Zi are relatively prime. Then
1X1 Lpintenn6diaii« dai math., 7,1900, 198.
^Arcbiv Math. Phye.r B), 17,1900, 32&^2.
'« Math, Queet. Educ Timee, 75r 1901, 85.
^Atchiv Math. Phye., C), 8, 1905, §8.
"Zeitochrift Math. Natuiw. Unterriciit, 36, 1905, 97.
»1Ш., 37,1906, 186-190.

Сиар.ххш] Single Equations or Degree n>4. 691
bet f—p/q, where pf q are relatively prime. Thus Z is a multiple zxq of q
and z=*zlp, Zi=zlqZb z2=z1qZtr
A. Flechsenhaar1" and E. Scbulte discussed 1/а-Н/Ь = 1/с. Е. S6s
(p. 113) treated B). W. Hofmann138 discussed the integral solutions of
1 1^1 I^I^I^l
a^b c' abbe
G. Lemaire1*' transformed given decompositions SI//of 9/10 into others.
R, Janculescu14* noted that in l/x+l{y=l/z, г will be integral only when
the g.c.d. d of x and у is a multiple of x/d+y/d.
D. Biddle141 solved each of l/(a±b)+l/(*±a) = I/a.
Miscellaneous single equations of degree n>4.
J. L. lagrange14* noted that, if a is a fixed nth root of unity, the product
of two functions of the type
is of like form» Hence if we replace а Ьу the different nth roots of unity
and form the product of the functions so obtained from pt we obtain a
rational function P of t, tt, • • -, zt A such that the product of two functions
of type P is a third function of type P. We can find P by eliminating u>
between
then P is the term free of I in the eliminant. For example, if n = 2,
P=fi—Аи*. An application is to the solution of
A) г"-Лвт = д".
We seek to express each factor r—a$AlIn as an mth power рш, where a*-l,
and p is the above linear function. Then
Hence г=Г, » = -l7f X=0, ••-, Z=0. Thus A) is solvable by this
method if X=0, * * -, Z=0 are solvable. Although only n—2 equations
in n variables, they do not always have rational solutions. For details on
the case n=3, m=2, and Lagrange's extension of the method in his addition
IX to Euler's Algebra where an=l is replaced by any equation of degree n,
see papers 161-6 of Ch. XXI; also Ch. XX,
Lagrange1" treated the problem to make y^p/q an integer when
Ъ', д=.а*+Ъ1х-\ are polynomials in x- By eliminating x,
 Uatemchteblatter Math,, 16,1910,41,41-2.
"■/Wtf.,17, 1911,14-15.
U1 L'mtermMbire des math., 18,1911, 214-в.
M Matheais, D)p 3, 1913, 119-120.
ш Math. Queet. Educat. Tim«r <2)p 25,1914, 61-3.
"■ M&n, Acad. R. Be. Berlin, 23, aanee 1767, 1769; Oeuvree, П, 527-532. Exposition by
A. Deabovea, Nouv, Ann. Math., B), 18,1S79, 265-79; applications, 39&-11О, 433-444,
481-499; abo by R. D. Carmichael, Diophanttne AnalyHia, New York, 1915, 35-63.
a.Dirichlet»; also Iibri«^ of Ch. XXV.
«• Addition IV to Euler's Algebra, 2, 1774, 537-533. Oeuvw d<j Lagrange, VII, 95-8.
Eulerb Opera Omnift, (I), I, 579.

692 History or the Theoky of Numbers. [Chap, xxiii
we get O=A+Bp+Cq+Dp*-I . Replacing p by qyt we see that A
must be divisible by g. Hence we take for q the various factors of Л in
turn and solve q=at+b1x-\—- for rational x's. A special treatment is
necessary when q reduces to the constant л1. G. Libri1" eliminated x
between the congruences p=0, q=0 (mod q) and obtained D^O (mod q),
where D is a function of the coefficients of pf q. Next, seek the integral
solutions of q=d for each divisor d of D in turn, and then solve y^pjq*
As another method he suggested (p. 317) the use of series.
A» J. Lexell14** found values of p, q, rt 8 for which
L, Euler1" treated vW+AsY*2- Q, where
To make e have the factor r, set
z=agz-\-(f+bff)y> v = (f
Then *lr=p-\-(ac—bi)gt=L The proposed equation becomes
which iB of type B) of Euler1*, Ch, XXII. The case b~0 was treated in
more detail.
G. Libri14* treated a»z»+bztlr*1+ ■ > *+q^z" with all coefficients positive.
Set s=os-K, whence e»-l(na"-Je-b)H +(e"-g) = 0. Seek the least e
for which all the coefficients are positive and the greatest e for wbich they
are all negative* For each integer e within these limits, seek the positive
integral solutions x. If the coefficients in the given equation are not all
positive, set x=A+y and chooee A so that the coefficients of the resulting
equation will all be positive.
Iibri147 investigated the integral solutions ^0 of ф(х, у, •••)-() for
which x<a, y<b, * * -, where а, Ъ, • •• are given positive integers. Set
liet F^O be the result of eliminating z, y, - • • between ^-0, X=0, Y=0,
• • -. If the equation of condition F=0 is satisfied, take the equation, вау
Xi(a:) ^0, in one variable, preceding the final stage of elimination. Then if
X% is the g.c.d. of Xl and X, all possible integral values of x occur among the
roots of Xj^O; similarly for the other variables. The same method applies
toacongruence4=0(moda). Foraa prime р,Х=з*— х (modp),r=2/*—у
(mod p)* Since
if 2b,.. 2for
cos h^srn—
m m
"* Jour, fur Math., 9,1832, 74-76. ~~ — - -
^Euler^e Opera poefcuma, 1,1862, 487-40 (about 1766)»
"M&n. Acad. Sc. St. Petereb., 9p 1819 C17S0114; Coram. Aritb., П, 414.
"• Memorm ворга Ь teoria dei numeri, Fineose» 1820, 24 pp.
1Л M&norie вит 1& theorie dee zxombiee, M6m. divers Sav&nte Acad. Sc. de 1'Institut de Fnmoe
(Math. Phye.)p 5, 1838 (praented 1825), 1-76.

Chap. XXI1IJ SlNGLE EQUATIONS OF DEGREE П>4. 693
according as rc is divisible by m or not, the number of roots of #=0 (mod m) is
U.J.
gM«bJL
m
When applied to ф=х*-\-с, this formula leads to Gauss' results on trigo-
trigonometric sums. Again, хг+Ау2+В=О (mod p) has p±l sets of solutions,
Iibri1** noted that the number of sets of positive integral solutions of
Ф&, V, ■ ") ^0 and the number of sets in which x, y, • • • take the values
h ' - *r n—1 are approximately
respectively. To apply the method of the preceding paper to the linear
congruence ф=Ax —1=0 (mod p)y A not divisible by p, we use x1^1—1=0,
or (Ax)*-1—1. Since the division of the latter by Ф is exact, we get x = Л*.
Next, for ^=x2+gx+r=0 (mod 2p+l =prime), we divides2*—1 by фand
require that the remainder be divisible by 2p+L Thus the conditions for
two roots ctt & [neither zero] are
which by use of symmetric functions can be expressed in terms of q and r.
For the case x2—e=0 (mod 2p+l), the first condition is satisfied and the
second reduces to sp—1=0. Forx^x+l^O (mod бр+l), the first condi-
condition is equivalent to (—3)*>=L
V. Bouniakowsky1" noted that there is an infinitude of solutions of
x^Xn-\-ymYn^zmZnJ
where mr n are relatively prime. Determine a, fi so that ma—тф = 1. bet
a and Ъ be arbitrary and с - а+Ь. Then a solution is
x=a\ у=Ъ\ z=c\ X=*W, Г-Л*, Z=aV.
New solutions follow from the integral form of
a^/a*'+b^fV"=€*"!(?*",
Similarly, if p, ?, r, — * are without a common factor, we may solve
by use of XAidi=0, pa=h^± — -^lf replacing а^ by a?****"", throwing
negative powers into the denominator and clearing of fractions.
G, C, Geronolw noted that if r is the radius of the circle inscribed in a
triangle with sides л, Ь, с and area Д and if х=а/г, у=Ь/г, z^cjr, Heron's
formula for A, and A*=i pr, where p is the perimeter, give
Call the factors 2X, 2У, 2Z, respectively. Let x, y, z be positive integers.
"* Mem. Accad. 8c. di Torino, 28, 1824, 2724); Jour, far Math,, 9T 1832, 59.
|« BuU. Acad. Sc. St. HtcrBbourg, 6P 1S4S, 200-2, Cf. Hurwitz^ oC Ch. XXVL
ua Noay. AnD. M&tb^ 17,1858» 360.

694 History of the Theory of Numbers. [Сялр, xxin
Then X, Y, Z are positive integers for which XYZ=X-\- Y+Z. If X is the
largest of Xt Yt Z, thenXYZ<3X, YZ=2ot 1. We may take Г=2, Z = l.
Then 2=5, у=4, x=3. See the next two papers, and 341 of Ch. XXI.
Houeel161 proved that the sum of n distinct positive mtegers equals their
product only when the integers are 1, 2, 3.
J. Murent151 discussed the positive integral solutions (оъ -, a*) of
One solution is (n, 2, 1, • ■ «, 1). Always at least two a'$ exceed unity. If
n>2, at least one a is unity; call i the index of a solution (ai, - • -, atf, I,
- *f 1) withci>l, * -,й(>1. 1Ъеп2*-г^п; if ^n,thenai= * »-^a^^2.
If n^5^25-3, there is a single solution B, 2, 2, 1, 1) of index 3, while the
only remaining solutions are C, 3, I, 1, 1) and E, 2, 1, 1, 1) of index 2.
P. di San Robert1** noted that F(x, y7 2) =0 can be solved by use of the
slide rule only if reducible to X{x)+Y4$) = Z{z), a necessary and sufficient
condition for which is
dxdy * ~dx ' By'
S. Realis1" noted that
is not an mth power, being between a*1 and (a+l)m, and that mQ is not
divisible by (а-Н}т—am.
E. Lucas1*5 noted that х*+х+к?=у* is impossible if h is odd.
S. R6alisiw noted that, if zy+0, бх^Зл^Ч-У4) Фг8 or 4z*. The impossi-
impossibility (p. 524-5) of
7
x«-I-#•=92-1-7 or 7z+5, S^J^9^+8
is easily verified by use of remainders modulo 9 or 7. M. Rochetti16*1
expressed
as a sum of three cubes»
A. Markoflfiw gave complicated formulas for all positive integral
solutions of x*+y7+z*^3xyz*
К Fauquemberguelw proved that 14-3+3*4 ЬЗП=^* only when
n=0, 1, 4, by using the powers of a4-bV^2 to treat
ш Nouv. Ann. Math., <2), 1, 1862» 67-69.
ш Ibid.f B), 4, 1865, 116-20.
*w Atti della R, Aoc*d. Sc. Torino, 2,1866-7,
"* Nouv. Ann. Math., B), 12, 1873, 450-L
"• Nouv. Comep- Math., 4, 1878, 122, 224.
«• Nouv. Ann. Math., B), 17,1878, 468.
»"/Ьй^B), 19, 1880,459.
ш Math» Annalen, 17,1880, 396. Cf.
»■ MatheriB, B), 4,1894,169-170.

Chap. XXTIIJ SlNGLE EQUATIONS OF DEGREE n>4. 605
G. Cordone1M investigated polynomials U, V in x which satisfy
P.(*) V-+P, (x) U"~lV+ ■ • • = R(x)
identically in xt where the Р*(х) are polynomials in x.
E. MaiUet3W considered recurring series «o, uu ■ - - of rational terms with
the generating equation f(z)=z*'\-aiX4~l+' • -+д3—Oand law of recurrence
B) Un-K-i-aiUa-f-r-iH |-««и*-0,
where czi, ■■•,a< are rational. An algebraic equation with rational coeffi-
coefficients is irreducible if and only if all the recurring series of rational terms
having the equation as their generating equation admit the corresponding
law of recurrence as an irreducible law. To apply this to diophantine
equations, let
become F(uKi иц+ъ - - -, «*+«-i) when «*+*«-*, - • ■, «*+, are expressed in
terms of ttn+*-i, • ■ ■, un by means of B)* It is lmown that the law B)
is reducible if and only ifAg{0) = 0. Hence F(u0t ■ ■, i*^i)-0 has rational
solutions if and only if f(x) = 0 is reducible. If щ, -- >, u^i give a rational
solution, the same argument shows that uH) • • • t un+^i give a rational
solution for n arbitrary. We get all the rational solutions by taking in
turn all the maximum divisors x(#) =х*Л \-Ci, with rational coefficients,
off(x), i. e., a divisor not dividing any other divisor off(x)f and forming all
the recurring series of rational terms having х{я) ^ 0 as generating equation
and any rational numbers as the first t terms щу Ui, • - >, Ut-\. Among the
recurring series which together give all the rational solutions of F=0,
those which give only a finite number of solutions are the ones whose
generating functions are divisors d(x), with rational coefficients, of f(x)t
such that 6{x) = Ohas as its roots only distinct roots of unity. For example,
let$=3 and/(x)^=x8-7. Then
Let у be the cube of a rational number B, so that / is reducible. The
maximum divisors are x~b and a^+te-i-S2. To the first correspond the
solutions u0, Вщ, &hio, where щ is any rational number. To the second
correspond u0, ulf — B(ui-\-&u0), where щ and t*! are any rational numbers.
If 7 is not the cube of a rational number, there is no rational solution of
F^O. Let B) be an irreducible law for ш,щ, • " and let a« = ±l. Then
Дд@) =0+0, F{u#f • * >, Uti+^i) = Jzgt so that we have rational solutions of
the latter. There are similar results for integral solutions when the a's are
integral.
D. Hilbert1*1 treated tbe diophantine equation D = ±lf where
»♦ Giontale di Mai.r 33, 1895, 106, 218»
"• Abkkj. inns. tiv. ec.> 24r II, 1895r 233-42,
ш Gfittingen NAcbricbtcn (Math,), 1897, 46-62. Cf. Eieenetein^ of Cb. XXII for n-3.

096 History or the Theory of Numbers. [Chap» xxiii
is the discriminant of x9 tn+xitn~'1-\ hx*^0, with undetermined coeffi-
coefficients, and roots tu - —, tn. By use of xi = 0, ■ ■«, z^* = 0, it ig readily
proved that D = ±l has rational solutions* The main theorem is: For
я>3, D= ±1 is not solvable in integers; the only equations with integral
coefficients and with the discnminant zfcl are Q={ul-\-v)(u4-\-iP)=z0 and
the cubic Q[(u+u1)t+v+vl2=Q, where u, ult v, v1 are any integers for
which w1—ti4>=±l. The proof employs the theorem1** that the dis-
discriminant of an algebraic domain is always distinct from d= 1 and the lemma
(here proved by use of ideals): If an equation with integral coefficients is
irreducible in the domain of rational numbers, its discriminant is an integer
divisible by the discriminant of the domain determined by a root of the
equation.
C. Stormer1" noted that, if A, B, Mh 2V> are positive integers,
AM?- -ЛО-BiVr•■ -A^-il or ±2
has only a finite number of sets (if any) of integral solutions xif yh and
that these can be found by solving a finite number of Pell equations.
E. Fauquembergue1" noted that Зз*~4д*—z* has no solutions with
yf z relatively prime, since (x+&)*—(x—£)* = {2уг)г gives х=*&, у=Л
On х2^-^, see Pubs11 of Ch. XXI.
G. B. Matbews1* noted that xy(x+y)=z* has no solution if n=Zmy
while if n^3w±l the general solution is (X^f, X"% X*ft, where (£ ij, f) is
the unique solution in which x[y equals a given irreducible fraction, and
the g.c.d. of x and у is not divisible by an nth power.
A. Cunningham1" solved in integers NiNi = NtN4, where #ш=*4
also
He solved[MXMS^MJdAi where 3fr-{
S. 0. Satunovskyiw discussed the solution in integers of
as'""-i-aix""'~14 +атл = Ьу*, Ъ = ±а/см.
P. F. TeilhetIW gave, for m-1, recurring series leading to all (an infini-
infinitude of) solutions of x?m—y2m=z1*ym~l and asked if there are solutions
when m>l other than z~y = l.
♦ ILKuhne1* noted that if the system of n functions z<=4;(t«, ■ -,{R_i)
is equivalent to the system of n functions f»=/»(^o, * * *> ^»-i)i the coeffi-
coefficients of the tf's and/s belonging to the same domain, there exists between
the x's and the £'s a connection (Verknupfung) and these connections
have the group property. This concept leads to a process of solving all
*• Mmkowaki, Geomctrie der ZahleD, 1896, 1Ж "
»• Comptee R^ndu» Paris, 127, 1898, 752.
ш L'iDtermediAire dw math., 5, 1898,106-7,
» Math. Qoeet. Educ. Time», 73, lflOO, 37. For f=1, Euler1» of Ch» XXI.
»1Ш., 75r 190X, 43; B), 1, 1902, 2fr-7r 38-9-
™ Zap. mat. otd. obec., Odeeea, 20,1902, 1-21 Otuaum).
ш L'mtermedkire dee math., 9, 1902, 318.
"' Math. Naturw. Blitter, 1, 1904,16-20, 29-33, 45-68,

. ХХШЗ Single Equations of Degree n>4. 697
diophantine equations m n unknowns such that all the unknowns are
expressible rationally in n —1 parameter?. An instance is the method of
solving x84-yf4-^+wt^0 used by 3chwering7J and KuhneM of Ch. XXI.
A. Cunningham170 found solutions of
C) (f+ffi(x*+Y4-e+f
by expressing л = (з*+0*)/(:Н-у) in the form t*-h3u* in one of the three
waye: Их—у)*±Щх)* for x even, (х-$у)г+Щу¥ for у even,
- У b^h odd,
and by expressing N = (*4 TV)/(X+ Y) ш the form T*+W\ Then
But Л*Ч-3#*ю expressible in the form (?+п*)К£+ч) in one of three ways.
Hence C) is reduced to (х+?)(.ЗГ+У) *=£+*■ R. W. D. Christie1" noted
the specie! solution
He172 noted that 104302^C'+72)C2-M*)- Cunningham noted that
is satisfied if Л = Л?, а^л2, A\=o?c*bd, B~o?d±bc.
• P. S. Frolov171 found the least solution of D) for x =1.
A. Hurwitzir< discussed the positive integral solutions хъ - *-,хп of
D) *J+ • ■ 4ЙВ»А' ■ ■*», "^3,
where x is an integer. If {= (x, xh •• ■, *л) is ft solution, then evidently
£'^(z, a;!, ^J; ••', xn) is a solution when z\+Zi=xx% - ■«„- Similarly,
{/р«=(л, я?1, ж», a?i, ■ •-, zn) is a solution when Xt+xt^3ZiXi* • xn> Call
these solutions {', ?', - - •, $(ж> " neighbors " to f. Build the neighbors to
each of these, etc. Then all such solutions are said to be " derived "
from £♦ Call fa" fundamental" solution if no one of its n neighbors
has a smaller sum XiA J-xw* It is proved that { is a fundamental
solutionif and only if 2$*^xxt< - it» for i=l, -—,n; that every solution is
either a fundamental solution or can be derived from another one; that
there is no positive integral solution of D) when s is a given integer >n;
that all positive integral solutions with x=n can be derived from
*!*= ■ ■ * *=s«^l (the case n=3 being due to Markoff^O- If n>5 and if
x>*u '' •» z« form a fundamental solution of D) withxi^xi^- * * =xni the
last n—2—к of the х/б have the value unity, where к is determined by
E. B. Escott17* cited two numerical equations x7+rz*+8X*+tz+k=0
with rational roots [see Ch. XXIVW3. <£Charbonier^ A8, 1911, 62-3)
employed tbe roots a, ht — а—Ъ, с, d, ef —c~d—e.
^^ Math. Quest. Ediic. Timce, B), 6,1904, 76. К* 27,1Й15,17-18J
« 7Wrf 100
,(),,t
17> Vert, opytn. fiflid (Spactneki'e Bote MalhO, Odeeea, 1898, Nee. 419-20, pp. 243-56.
** Aichiv Math. Vbs*., C), 11,1907,185-96. a. paper? 173, 186, 1H, 19&».
lT» L'intomediaire dee math., 16,1000, 242.

698 History of the Theory op Numbers. qChap. ХХШ
E. N. Barisien17* noted that z=f(n), у = ф(п) give solutions (but not
necessarily all solutions) of the equation F(x, y) = 0 obtained by eliminating
n. Similarly when xt y, z are functions of n, m. A. Cunningham177 gave
the least solution 3, 4, 5, and the general solution of
E. B. Escott177' noted that, if X^
A. Thue178 considered solutions x, y, z, relatively prime in pairs, of
Ax*+By»+Czn-xyzU{x, y, z)=0,
where U is a homogeneous polynomial of degree n—Z whose coefficients,
as well as A, B> C, are integers» Let n be odd. Let p, q, т be integers, not
all zero, such that px-\-qy+rz=Q* Then
ft
xy
with two similar equations derived by permuimg x, y, z and p, qf r. Then
ax=Brn—Cq*, by=Cp*—Ar", cz=Aq"-Bpn.
Hence Aax+Bby-\-Ccz*=Of so that we have a second linear relation. Also
uyn~l — bx4 = Eij with two similar, equations. Let и be the greatest of
яг, yt z numerically; X the greatest of p} q, r; I of Л, В, С; m the greatest of
the coefficient» of U, and $=^(n—2)(n-l)m+B»-l+l)l. He proved the
following theorems. If АВСФО, n^3f and if py q, r can be found such that
\n~1<ul(lbI then a—b = c=0. If our given function of degree n is irre-
ducible, we can determine a function K^U of A> Bt С and the coefficients
of I/, such that no numbers p, q, r exist for which \*~1<ulK. If
has relatively prime solutions and if n is odd and >1, there do not exist
solutions p} g, т not all zero of px-\-qy +rz=0 for which X" <i*/j Bn~l+lOJ2 j >
G. Candido179 considered a polynomial f(x, y) with the factors L=z-\-<xy
вл&Ф(х1у)> where a is rational. Setz-\-ay^zn, ф=А. Then/(ж, у) =^Azm
has the solutions
х-&л(р, q), у = Ьрмл(р> q), z=\+<*vt р^2ХЧ-л^ д=Х2+лХр,
where щ, vk satisfy (lvk)*—(\p2—q)ul=qk. Similarly, if / has the factor
Q=x2+Pxy-\-yz2J where ft у are rational, take it as z\ Each method is
applied in detail to solve LQ^AJ; in the particular case x*+y*=Az*, the
solutions are those obtained by Lucas198 of Ch. XXI.
A. Cunningham190 proved that if 4z*—<^=Zx%yzt in positive integers,
then x=yt z = l. He discussed (p. 28) zb-\-y& = 1?+utt a necessary and
i" Sphin*Oedipe, 5,19ШГ 7&ТТ.
17' Math. Quest, Educ, Times, B), 15,190Й, 49; B), 18, l0l0J 101-2»
»"*/W., B), 17,1910,57,
i7» Skrifter Videnskapmelflk. КНвОашл (Math.), 2t 1911, No. 20.
1» Periodico di Mat., 27,1912, 265-2ГГЗ.
» MOi. Queat» Educat. Time», B)r 22,1912, 69-70.

Снлг.ххпц Single Equations of Degree n>4. 699
sufficient condition181 being that x+y and N=(z*+ff)f{x+y) be 67.
Since N=(x?—3xy+y*J+5xy(x—y)\ set x=£*, у=5»^and makex-\-y^ E3.
E. Miot182 took x*4-y*=2*p$r*, where p is a prime 4n+l, whence 2*р=^Ч-^
and multiplied the initial equation by q*> L. Aubry obtained an infinitude
of solutions by setting
x—l=an, y~l=bn, t~ l=cn-\-dn*f u~l**en+fn*.
" V. G. Tariste И noted that, if xt у, г are < 10,
х*+у*+г*+хуг = 100x4- 10у+г
hold» only for ft=&and then x, y} г are the digit» of 370, 407 or 952. A. H.
Hohnes1** obtained special solutions with n*=\ or 2 by assuming that
yz = lWorxz=\Q.
A. Cinmingham185 noted that every prime p^Xn— Y*t with я = 12то-|-7,
can be expressed in the forms (x?±if)l{x±y). Cf. Cunnin^iam.J87
G. Frobenius1* proved that 2?+y*-\-zl=kxyz is solvable in positive
integers only for fc=3 and &=1, while the latter case reduces to the former
by the substitution x=3X, y=3Y, z=ZZ. Cf. Hurwit*.1*
Cunningham187 noted that, if n>3,X*—Yn=z*+xy Ч-^has an infinitude
of positive integral solutions. He noted B4, 1913, 85-6) cases when
x8—y1 otx7—^isexpressibleintheformQ2-hl. Heexpressed B6,1914,50)
the product of two numbers of type z*-\-x+l and B7, 1915, 102) the
product of three such factors in the form A2+ZB1 in several ways.
T. Kojima1** proved that if a rational function of several variables
with integral coefficients equals an nth power for all integral values of the
variables, it is an exact nth power.
H. Brocard189 stated that x—y = l is the only integral solution of
х^Ч-^^^Ч-^, and that x*+yv=xy has no positive integral solution. These
problems were proposed by G. W. Leibniz.IW
A» Cunningham1*1 gave several solutions of И{х1+х+-\-1) =z*.
E. Fauquembergue1*2 notsd that the only solutions of Dя*~ 1)Dх—1) = yt
in integers are #—0, 1, 2; ±#=1,3,21.
M. Riguaux188 gave two identities a^+y1—z*-\-w*.
W. Mantel1^ proved that xt-\-y*+z*=rly2P is impossible in integers;
that, if n=2, 6, 9, 11, 12, x(-i-« —\-x\**XiXt- * -xn has no positive integral
solutions, and gave the least solutions for n=3 C, 3, 3); n=4 B, 2, 2, 2),
lU Bepubliahed, rintermedLaire des math,, 19, 1012, 227-8.
™Ibid.t 119-120.
«"/^133.
>«Ашег, Math. Monthly, 18,1911,69-70.
вд L'intermediaine dee matb-r 20f 1913, 3. PWxrf by Aubiyr p. 120; by Wefach, p» 184.
'«Sitzungaber. Ak&d. Wiee. Berlin, 1913, 458-87.
*» M»th. Quest. Educ. Times, 23,19J3t 31-32.
i" T6hoku Math. Jour., 8t 1915, 24.
i» L'intermediaire dea math*, 22,1915, 61-2; 21,1914,101.
lf# Opera omoia <«L, L. Duteoe), III, 85-6; letter to Oldenbouig, June 21,1677.
ш L'intermedmire des math., 23, 191&, 41-2.
*" Ibid., 24, 1917,41-42.
ш 1Ш., 25r 1918, 7. For *■+*<«*■+«>•, вес G4rardin- of Ch. XXI.
** Wiskundige Opgavenr 12,1917, 305-9»

700 HiSToHT or the Theory of Numbers. [Chap. ХХШ
n = 5 A, I, 3, 3, 4), n = 7, 8, 10. He stated and L. de Jong proved that the
g.c.d. of solutions x, yt z of x2-\-yi-\-z2^xyz is 3, and listsd seven sets of
solutions. Cf. Hurwitz17*
G. RadoB1*4" proved that if a polynomial F{x) of degree n with integral
coefficients decomposes with respect to every prime modulus into n linear
factors with integral coefficients, then F(x) decomposes algebraically into
n linear factors with integral coefficients.
A* Korselt1"* argued that, if f(x, y) is a homogeneous function of degree
d>l with no multiple root, f(xt у) = гп is solvable in relatively prime
integral rational functions xi y, z of any parameters if and only if d=2yn
any, or rf=3, n=2>
" V. G. Tariste " ststed and RXoormaghtigh196 proved that xv-yx^x-y
has only the integral solutions x = y-\-l = 1, 2, 3.
M. Rignaux195e proved by the theory of quadratic forms that
holds, when с = 1, only for #=3. Cf. Hurwitz174.
F. Irwin1*6 gave a method to find the integral solutions of
axr—bxy-\-y—c=0-
For (хп~1)/(х-1) = а, see Landau, p. 57 of VoL I of this History.
On рг(р*-г*) :qs(q7-^)f see papers 67-77 of Cb IV, Euler*1 of Ch.
XVI, Euler18'" of Ch. XVIII and Euler2*3 of Ch. XXII.
For p+4ktiv= D, where fc-U2+l)(^+l), see Haentzschel144 of Ch. V.
By Hilbert*4 of Cb» XIII an equation /=6 may have no rational solu-
solution, while/=0 (mod p*) is solvable when p is any prime. From one solu-
solution of F(x, y, z) = Q, Cauchylw of Ch. XIII found another. For
(Z*-fcW-A4) = D,see Euler28 and Gerardin** of Ch. XV, Ward" of Ch.
XIX On/(^) = DseeJacobi,151etc.,of Ch. XXII. Brunei*8 of Ch. XXI
solved x"-\-X2 = F, where F h a cyclic determinant of order n. Euler187 of
Ch. XXII noted rational solutions of abcd(a-\-b-\-c+d) = 1.
Miscellaneous systems or equations of degree n>4.
C. Gill and T. Beverley1M found numbers whose sum is a 4ttth power
and such that if the square of each be added to their sum there results a
square. Take px2*, qx2*, < -< as the numbers and x** as their sum. The
final conditions give j^+l *= D, g*+l = D, r*+l = П, • - -, which hold if
P~ 2yx» f q~ 2yx«
Tomakep4-g4- x%\ takey = (++)/(), /+/
J. Iiouville11>r stated that, if there be a finite number of sets of positive
i*" Math. 46 term&. ertesHo (Hungarian lead, of ScO, 35, 1917, 20-30. "~
»»АгсЫт Math. Phys-, 27, 1913, 181-3.
"* L'intermediaire dee m&tb.t 25, 1918, 30, 95.
wibid 13Ь2
t
* Ашег. М&ХК МопШу, 26r 1919, 270-1.
The Gentleman*» Math. Companion, London, 5, No. 2S, 1825r 367-9.
' Jour, de Math., B), 4, 1859, 271-2. a. Gegenbauer.**

. ххш] Systems of Equations of Degheb n>4. 701
integral solutions of f(xt> "•,xlt)=0, --^Ffai, •♦ -, a>) — 0, and we set
xi=dihi in all possible ways and write 17 =+1 or —1 according as d\* ■ -d0
is a product of an even or odd number of primes (equal or distinct), then
2ij is the number of sets of solutions of the given equations in which each
Xi is a square*
H. Delonne'*8 noted that the system x2*=ayu+l, xi^1 = by2^1-\-c is
in solvable if a+1 and с are divisible by 3, while b id not [since impossible
modulo 3].
A. B. Evans119 found four integers (ax*t - • -, dz*) whose sum is a sixth
power and the sum of any three a fifth power. Take a-\-b+c+d=z~
Then the conditions are x—a=pt, -- -,x—d=s** Thus x = |(рЧ-дЧ-г*+г*)
is an integer if p = Zmt q=3m+lt r=3m+2, s=3m+Z, and then a, b, c, d
are also integers.
A. Desboves*00 called a a congruent number of order m if the system
has integral solutions. For m=2, the quotient of the expression found
for a by 16 is xy(x2-y2){x4-^Y+y4){2. Taking *=2, y = lt the latter
becomes —21. The least congruent number of order 2 is 21. A. GerardinMJ
remarked that it seems more logical to call a a congruent number of order m
if x^zLatT = П hold simultaneously. Cf. papers 210, 222, and Ch. XVI.
L. Gegenhauer403 considered a set of positive integral solutions x^ • * •, s°
of the system of equations J\{xi, ■•■f^JI)=0, "*,fr(xi, • • *, xj =*Q, and
any divisor &t of x\, and called the product $J ■ ■ - S' a divisor-product
belonging to the set x[, •■■, x0^. X#t x(x) be a function for which
x($y) — x[%)x(y) f°r all values x, у satisfying a definite condition. Let
X(n) = ZxW, where d ranges over all divisors of n. Then
where on the left the summation extends over those sets of solutions x\,
• ■ ■ 1 xl which satisfy the condition mentioned, while on the right the summa-
summation extends over all the divisor-products belonging to these set» of solutions.
If we take x(x) = +1 or — 1, according as x ie a product of ал even or odd
number of primes (equal or distinct) and note that £x(d) = 4-l or 0,
according as n is a square or not, we obtain the theorem stated by liouville.157
Other special cases are obtained by taking x(x) to be the number фк(х) of
sets of k integers <x and prime to x, or p(x) of Vol. I, Ch. 19, and noting
)
( (
Several writers20* found two integers whose sum, difference and difference
of squares are all twelfth powers (square, cube and hiquadrate). Else-
Elsewhere101 was added the condition that the product of the nine roots of these
powers shall be a square, cube and biquadrate.
*■■ Nouv. Ann. Math., B)r 1,1S82, 455-7.
" Math. Quest, Educ. Tunes, 25, 1876, 76.
*»Nouv. Лил. Math., B), 18,187», 490»
'«I/intertn&fture d« matb., 22,1915, 101.
"SitxmigBber.Akfld, W«h. Wien (Math.), 95, II, 1887,606-9.
«•Amer. Matb. Monthly, 2, *885,128-9.
" Math* Quest. Educ. Tuusa, 60,1Ш, 37-38,

702 Нитовт of the Thrqby of Numbers. [Chap, xxiii
G. B. M. Zerr1" found six positive integers xx such that each diminished
by $(xi-\ 1-**)* becomes a fifth power.
Several*" found three numbers in arithmetical progression whose sum
is a sixth power.
E. Swift407 proved thatx=0, у=1250о* give the only integral solution of
{+у)(+1?)(+у)+у
A. Cunningham*8 discussed **тЧ-|/**Ч-**п='11-!-»|я-Ир'* (« = 1, 2) by
use of the identity <14+Ь*+(а+ЪL=*2(аг+аЬ+Ь*)\ Employ the usual
solution of i£+i?=v?t and set x=uf—v*—uv, y=2uv, z^x+y. Then
2C*=x*+yK+z*}
He208 expressed two special sextics and two octics ш the form Y^—
where Y, Z are functions of x, and g^ 17, 13, 19, 2.
A. Gerardin210 discussed the solution of sm+Ayp — /*, х™—Ау*=#ж
Thus 2xm=/24-^ 8° that ж is a sum of two squares.
G£rardmni treated the system z*—1=4угт ву111—1=^, by taking as
s, t the factors 2ут—1, 4у2л+2уп+1 in either order, or t = lt or, for #=2,
s = 2*-l or 22*+2*H-l where n=k-\.
E. N. Barisien2" noted that л13=Н+яэ—i8=u2-^-tt>2 for г=9з/*т
s=x4+9»j/a, ^Зх'уЧ-^у4, where u, v, w are sextic functions of x, y.
A. Cunningham213 noted that if NM=xm—ym, and mt n are primes both
of the fonn^il, we can set Nm=t^1nv^J N* = tl^mul, simultaneously.
He and R. F. Davis21* proved that we can express (х14-\-х7-\-1I(х*+х+1)
in the forms Л2+3£* and C*+7B*.
Cunnin^iam215 investigated N^<f>{z, у) = Ф(х', у*)- -- >t where
ф(х, у)=х*у*±х~уш
and x} у are relatively prime integers»
A. Gerardin*" gave solutions of the system
L. Aubry517 made P(x+y) +Qx and P{x+y) +Qy both nth powers.
. Matb, Monthly, 5, 1898, 114. ~ ""
8901489
"Т/Ш., 15, 1908, 110-1, Problem proposed by J. D. Williams in 1832.
ш Math. Queet Educ. Timta, B), 14, 1908,66-7 (reprinted, Mesa. Math., 38,1908-9,102-8).
"Ibid., B), 16,1909,105-6.
111 Амос, franc xv. ее., 37,1908,15-17.
m SfthinxOedipe, б, 1911, 141-2.
s» L'interm£di&ire dee math., 19,1912,194. Cf. G&udln" of Ol XXI.
™ Math. Queet. Educ. Times, B), 23,1913, 2Ь22.
Л41Ш., B), 23, 1913, 86-8,
я. M&th.t 44,1914^5, 37-17.
term&Uaire des nuth.T 21,1914,143H; 24,1917> 111-2.
3,1916, 33-4. Cf. Sphim-Oedipe, 10,1915» 26-27.

. ххш] Systems of Equations of Degree n>4. 703
A. Gerardin21* noted cases in which g*—x, &—yt s*~z are squares, where
B=x+y+z; s"~z, ---, sn—t are all cubes, where s=*xj-y+z-\-tt for
He noted (pp. 197-8) cases when P9~+Qxm+Rym+ - - ■, Ps*+Qzm+Rzm+
- - -y - - - are all pth powers, where $=x+y+- - -*
Gerardin11* gave the general solution of hk problem to make *±xf
8±y, • - *, &±a all pth powers, where s=x+yA \-a.
R. Goormfti^tigh gave solutions of x+y+z=*7&^&=sA*f &^if=B*9
8*—г*^С9, where p is 2 or any odd integer. Hem stated that, for
A <1000000T A ^1+*+ ■ - Ч-^щ-1Ч-уЧ- ■ ■ Н-»щ holds only for
8191 =
in addition to evident solutions if x or у is negative.
Despujols1*2 took ф\+ф\ = (а2-!-^)* in the identity
to obtain a congruent number*1 h of order ». He stated that every con-
congruent number of order n ifl of the form 20*A/i, where (PQf+p1) =x*> and
conversely,
Ы1х\=П (i=lf 2>3)seep. 174, p. 186.
Pafbbs кот available гол вжровт.
P. L*cteibauer, LebreuUe und Aofgabea Qber Qleichbateo ale Beitrag жиг
beetimmetn Aiulyaie, Progr. Mflnnecrtadt, 1834.
G. Ал Longoni, Sui problemi <li onaltai indefcerminata, Мовла, 1840.
С F. Meyert Bid diophantidcbe Problem, Progr. Potedam, 1867.
Poeechko, AuflOwingemethode unbeetinunter GLV Progr. St, PoJten, 1869.
X Slavik, Solution of Indeter. equations (Czeoh), Progr- K5niggratJ, 1877.
F. M. Coeta Lobo, Resolution dee equations indetermiiieea, СсппЬпц 1885.
С. AUun&t Element! della teoriA generate delte equ&zioni . . . e delle equasloni
naBte, Napoli, 1891.
А. ЗДшд, L'analim diofantea, Trapajiip 1900»
H. Ziwchlag, Diophantiecb« Gleich., Berlin, 1908.
J, Edaljii, Note on indetermiDAte equations, Jour. lodiftn Math. Soc., 3,1911, 115.
H. VerhageD, Ад equation in three unknowns, Nieuw Hjdechrift voor Wiskunde, 3,
307-H.
ae ., 23, 1916, 100-170.
»r 207-8»
24, 1917,23-24.
, 88 (p. 153, correction).
261919 14-15-

CHAPTER XXIV.
SETS OF INTEGERS WITH EQUAL SUMS OF LIKE POWERS.
If i—$(a+&4-c), at b, с and t-~a, t—Ъ, t—с have the same sum and same
sum of squares; this double property shall be denoted by
A) a, bt c=t-a, t-Ъ, t-c, ЫЦа+Ъ+с).
The separation of two sets of numbers by the symbol = shall denote that
they have the same sum of Ath powers for jfc=l, ---,n<
Chr. Goldhach1 noted that
a+0+6t л+Г+«, 0+у+В, *=«+*, fi+l, y+t, а+0+у + Ь
L, Euler2 remarked that a, b, с, а+Ъ+с^а+Ъ, a+c, Ъ+с* This is the
case 5=0 of Goldbach's result, but it implies the latter since (Frolov7) each
number may be increased by any constant 5.
IP* N be chosen so that N, N—ait * • -, N~at have the same sum as
я, n+au • ■ ♦, «4-a{, then the sum of the squares of the former numbers
equals that of the latter.
E. Prouhet9 noted that 1, - *, 27 can be separated into three sets,
two of which are 1, 6, 8, 12, 14, 16, 20, 22, 27 and 2, 4, 9, 10, 15, 17, 21, 23,
25, such that the sum and sum of squares of the numbers ш any set are
the same as for the other sets. As a generalization, it is stated that there
are nm numbers separable into n sets each of n"~l terms such that the sum
of the Ath powers of the terms is the same for all the sets when k<m>
F. Pollock4 noted the fact, equivalent to A), that
p, p+a, ^
P. Proth5 noted that
and the numbers derived by interchanging Ъ and с have the ваше sum and
вит of squares.
E- Cesaro* proved that if a, • * •, h form a rearrangement of 1, ■ • ■, 9
and
a, b, c, d^d, ej7 g=g, Л, A, a,
then a=% 6=4, c=9, d=6, €=l,/=6, tf=S, A^=3, A?^7. Note that the
three sets of four numbers each may be placed on the sides of a triangle,
with g, d, g at the vertices*
1 Correep. Math. Phye. (ed, Fuas), 1, 1843, 526, letter to Euler, Juiy 18,1750.
1 Лг£, 649, letter to Goldbadi, Sept 4, 1751. Special caee by Niobolflui^ of СЫ ХШ,
*• New Senee of Mith, Hepoeitory (ed., T. Leyboom), 3,1814,1, 76-77.
* Compb» Bendua Parid, 33,1851, 225.
4 PhiL Тпшя. Boy. Soc. London, 151, 1861, 414.
1 Nouv. Gorreep, Math., 4, 1878, 377-8.
•7b.p 293^5. Question by P. Ph)th.
705

706 History or the Theory of Numbers. [Chap, xxiv
M. Frolov7 noted that 2а*-=ЪЪк, ZaNstf, k=l, ••',», imply
For n=2 there must be at least 3 terms a; for л = 3, at least 4» For n —3,
the least terms are stated incorrectly7* to be 1, 5,S, 12 and 2,3, 10, 11. For
я=3, there are examples when the a's and b's together give 1, 2, -.., 2m.
J. W. Nicholson* noted the identities
b, a+Zb,
5a+10b, 4а+Ш, 3a+5b, 2a+8fr, 3a+36, 2a+6b, a, b
-5а+Ш, 4а+6Ь, За+Ш, За+85, а+5Ь, 2а+ЗЬ, 2а+&,
there being one more term on the left than on the right. But for n = l,
• • -, 5, the sum of the nth powers of the ten numbers a±32} odb24, агЫ8,
0, а±4 equals the sum of the nth powers of the ten а±30, а±28,
A. Maitin9 noted the special case of A):
Also,
—с, а—^4-с, -о-|-Ь4-с=2а, 26, 2c.
IL W. D, Christie10 noted that, if t**e+f+g+ht
s+e, s-j-/, &+gt s?±hf 8—t=s—e, $—f, s—g, s—h, s+L
[Since we may reduce each term by s, we obtain an evident identity J
A. Cunningham11 noted that x+y, Ъ, с^х, у, b-\-c if ху^Ъс. Next, if
д, Ь, c=z, y, zf then
a, b, c4-b, ^c=a:, ^ z+kef кг.
Similarly a solution in two sets of n numbers yields one in two sets
numbers. J. H. Taylor noted that if ai-j-ua-f-
then
ai+1, a2po»+l^ai> ■", aJrioi, a,+lp a»,
И ^.j.. .-+6,r^2r(n-r)-rf then
H. M. Taylor noted the generalization of A):
R. W. D. Christie noted that ab+cd, be, ad^bc+ad, ab, cd, and
n—1, n—2, n+3, n—4, Л4-5, n+6, «—7
i 2, n-3,
Bull. Soc, Math. France, 17, 1888^-9, 69-83; 20, 1892, 69-84. The second waa reprinted
ш Sphinx-Oedipe, 4, 1909, 81-89.
On the proof-eheete Eecott noted that 5, 1, 4, 8 J= 2t 2, 7, 7 has smaller terme. It is de-
derived from 3,-1, 2, 6 — 0,0, 5, 5 of Eeoott« by increasing each term by 2»
Amer. Math, Monthly, 1, 1894, 187»
Math, Magazine, 2,1898, 212-3, 220,
Math. Quest, Educ, Timea. B), 2,1902, 40. Hie condition *=a+6+c-H i* Uimeceeeary-
/Wrf., B), 4,1^03, 9&10D

Chap, xxiv] Integebs with Equal Sums of Like Powehs. 707
A. Gerardin1* noted that ^Н-^+г3^*+1K+(У—2)*+0+1K is equiva-
equivalent to 4х+Дж=(£-1J, where Ax=>x(x+l)/2. He took Дх=1, 3, 6, 10, 15,
- - - in turn and found the possible z's^lOO by use of a table of triangular
numbers. He found 13 solutions like
The sum of the squares of 1,15, 12 exceeds that of 2, 10,16 by 10» Consider
two of our 13 solutions for which the ratio of the excesses mentioned is a
square тг\ multiply the numbers of the first solution by 7?i and add to the
second solution; in this way we get
2, 4, 20, 22, 33 = 1, 6, 16, 26, 32;
1,4, 12, 13,20-2,3, 10, 16, 19;
3, 4, 15, 20, 23, 26=2, 5, 17, 18, 22, 27;
2, 6, 30, 46, 53, 73=3, 4, 34, 44, 51, 74;
2, 6, 44, 58, 63, 91 = 1, 8, 40, 60, 65, 90.
Others follow by adding two of these. From х+у-\-г=х+2+у~4+2+2,
he got
1, 19, 23, 24, 32, 48-3, 15, 20, 25, 40, 44.
Gerardin13 noted that 14, 23, 25, 138 = 7, 26, 30, 137,
2, 12, 15, 35, 38, 48=3, 8, 20, 30, 42, 47,
while x+h, y+p, z^xf y, z+h+p *s impossible. [The last fact is a case of
Bastien's4* evident theorem J
H. B. Mathieu" noted that
I, l—rn—an, 1+(а—1)т—п=*1—т~п, I—an, l+{a—l)m.
U. Bmi15 gave a+&, c, d=c+dt a, b if ab=cd. [Cunningham.!1J
E. B. Escott" showed how to find all solutions of
B) 1>. = 2у,, ±4 = 2$,
for n = 3. Set Xi=Xi+S, yi-Yi+S, where ZS^Xi+Xz+x** But, if 2$,
is not divisible by 3, take S=Zxit 3^ = X<+£, Zy^ Y<+S. Thus
XXi^Q^XYi.
Using these to eliminate Хг and F, from 1>XxXi=?>YiY2> we get
C) JCj+XA+Xi^+^^+H.
Hence the problem reduces to solving C). To find all its solutions, let
N be any number all of whose prime factors are of the form 6rc+l or 3,
besides square factors common to X\t Xif Ki, F2. Then represent N in all
ways in the form z2-\-xy+y2,
"Sphinx-Oedipe, 19О&-7, 120-4.
Ш., 1907-8, 27, &4^5, Alao, a case of A),
* I/interm&Jiaire dcs math., 14, 1907, 20b All the eolation*, ibid.t 50r 200-3P by the other
writers &ге вресЫ cases of A).
u Ibid*, 227. His other solution is equivalent to A)>
"Ibid.t \5t 1008, 109-Ш.

708 History of the Theory of Numbers. [Сялр. xxiv
A. Gerardm17 noted that
1, m+3, 2w-2, 4m+2, 5m-3, 6m-l
=2, TO-1, 2m+3, 4w-3,
x, з+З, s+5, x+6, s+9, x+10, x+12,
also the result due to G. Tarry:
c, a+3&, 2a-b-c, 4a+56-3c,
=&+c, a—b,
Ge*rardinie noted tliat 624-ab-a2p a24-2a5-4b2p 46l and 4b2 have the
same sum and sum of cubes as Q2+ab—Ъ2, 4b*+2ab—a2, ¥ and ¥.
G, Tarry" gave
c, 3a-46-c, ia-b-Qc,
-26-19с, 11а-2Ц 12а-10Ь-13с, 12а-8Ь-15с,
-7&-20с
-564-2с, 2а-ЗЬ, За-|-2
-Ис7 ба-106-с,
Welsch*0 stated that the general solution of B) is
with iti, yt- (г = 1, - • •, n—2) arbitrary, where
and \, }x are of the same parity as a—X, a—Y* E. B. Escott (pp» 213-4)
noted that one can proceed as he;< had done for n—3.
H. B. Mathieu21 asked if the general solution is
2su—uu+st, tt+tv, su—2uv-\-tv-^t—uvf 2su—2uv+st+tv, su+tv.
Numerical solutions not of this type were cited in reply-22
A» Ge'rardin*3 noted three cases of A) in which c=2a+2b=t pvTartin*],
and that 4p*—3mp, 3m*+4mp—4^ have the same sum and sum of cubes
as Gvf-Zmp, 2j?+4mp-9npt Зт2-2р*.
U. BiniM set y4^x4+r9 m B)^ whence Zrt=0. By the latter, rm is
eluninated from the quadratic equation, which is then treated as a quadratic
for fj- Next, let
D) sn4-yn+2"=u*4-f*-r-t0n (л-1, 2, 4),
irSphinx-Oedipe, 190S-9, 96; errata, 144,
»/M&, 4,1909,44.
«• L'mteim&iiaire dee *oath., 16,190&, 89^90» Forn-3. ibid., 15, 1908, 2МЫ.
n /6tip 16, 1900, 219-220-
«Ли., 17,1910,72,165.
" Авэос. Ьащ. av. ac., 38,1909, 143-5.
" MheiB, C), 9,1900, 113-8; same method in Periodico dl Mat., 25,1910,119-128.

Chap» XXIV] INTEGERS WITH EQUAL SxJMS OF LlKE POWERS. 709
where x, y, z are not a permutation of u, v, w. Then z+y+z=0 and the
equation given by «=4 is a consequence of the others. Replacing z by
—x—y and to by — u~vt we get
Let z\t у\, щу t?i be one solution; the general solution is
where xt, уг, щу V} are arbitrary. Various special solutions of D) are given,
A- Gerardin* noted that
He* gave 2d+3x, 4d+2z, d=d+2xt 4d+Zx, 2d.
Welsch27 stated that the general solution of B) is
with Xi, yi (г=1, •-•, n—3) arbitrary [false if n>3, since in Zx*=SyJ
only the terms free of the s's and #'s cancel].
E- N, Barisien^ gave the relations involving 1, - • •, 32:
J, 8, 10, 15, 20, 21, 27, 30=4, 5, 11, 14, 17, 24, 26, 31
-2, 7, 9, 16, 19, 22, 28, 29=3, 6, 12, 13, 18, 23, 25, 32.
C. Bisman* gave six relations like the last, a numerical example of
*^2bk (k=l, •••, n) for each n^9, and three identities of the type
а-Ъу a-2c, а+Ъ+Ct а+2Ъ~с=а+2Ъ, а+с, а-Ъ-с} а+Ъ~2с.
L. Aubry« treated Ъх<= Xuu Z*J = Zii! (г-1,2, 3) by setting xt^ l+y%n,
l hence 2yv^2^ The cubic equation holds if
E- B, Escott11 applied hisie method to the last problem.
A. de Farkas» noted that, if Ss, 2x\ Zx* and zt+3z4+ ••• + (
equal the analogous sum» involving y's, then xi+a, zt+a+d, - • ^
+(m—l)d have the same sum and sum of cubes as yi4-a, -•- [falsej.
G. Tarry» stated tliat the first 2*Ba-f 1) integers can be separated into
two sets each of 2n~1{2a4-l) integers having the same sum of fth powers for
t^l, --,n- Fora=l,n = 3, the first set is 1,3, 7,8,9, 11, 14, 16, 17, 18,
22,24.
5▼.«>,, 39,1,1910,44; врЪшх-Oedipe, 5,1910,182.
" Sphin*45edipe, 6,1910,177.
« L'lnterm^diaire dee math., 18, 1911, 60 (for n=3), 205.
» Mfttbesie, D), 1,1911, 69.
»JW*, 205-3,264.
L'i^iiie dee math., 19,1912, 156-7. E. Miot (p. 3) gave two вшпепса! воЫОош,
/fctrf^ 182. Hieremari; on p. 131 ie the саде n-2 of Frolov'fi» fir^t reeult,
/W*200

710 History of тш: Theory of Numbers. [Chap, xxiv
Tarry" gave A) and noted that, for x arbitrary,
imply
By use of this lemma he found
Оа-ЗЪ-Sc, 5a-9c, 4a-4b-3c, 2a+2b-~5c, a-2b+c, Ь
=6a-2b-9e, 5a-4b-5c, 4a+b~8c, 2a-3b,
H. B. Mathieu** gave as the general solution of B), for n-d,
l±(ab+ac), l(l-hd)+qab=Fac, l(cd+l)^ab-qac.
L. Aubry (p- 234) noted that x+y+z^u+v+w implies xyz — uvw.
0. Birckw noted that, if з+у+г=0,
{ix~ky)n+(iy-te)n+{iz-kx;y = (iy—kx)n+(iz—ky)*+(ix-kz)n,
n-0,1,2,4.
*' V. G. Tarfete 'w noted that
e=2p 4.
Further such cases were given by E. B. EJscott and A» Gerardin.3*
E. Miot^9 stated that any 2nBa-fl) numbers in arithmetical progression
can be separated into two equal sets having the same sum of fth powers
for t=l, •■., n, if a>0, n>l; while f = l, * >-, n —1 if a = 0. Hence, if in
Tarry's" example we replace x by a+(x—l)r, we get
a, a+2r, a4-6>, - •-, a-|-23r=a4-r, a4-3rp --pa+22r.
Tarry40 noted that the number of teims in each member of the equations
deduced in his" lemma is 2k—d, if к is the number of terms in each member
of the given equations, while x is expressible in d ways as a difference of
two numbers belonging to the same member. Given
1, 5, 10, 16, 27, 28, 38, 39=2, 3, 13, 14, 25, 31, 36, 40,
take
x-11 ^16-5=27-16=38-27=39-28= 13-2= 14-3 = 25-14 = 36-25.
Thusd=8,
1, 5, 10, 24, 28, 42, 47, 51-2, 3, 12, 21, 31, 40, 49, 50.
E. Miot {p. 85) noted that
1+n, 2+n, 104-n, 12+ft, 20+«, 21+n
=«, 5+rt, 6+n, 16+n, 174-n, 224-«-
* L'interm6diaitt! dee math., 19,1912, 219-221. Cf. Tarry «
«1Ш., 225. \
" Ibid., IS, 1912, 252-5. Cf. Bini*11 of Ch. XXII. i
Ш.. 129; & 201, 250.
»1Ш. f2l9 19Hr 126-9.
»iftidL, 20, 1913, 64-5. GeDtinfiiation of Tarty.» ,]
« ibd., 68-70. i

Chap, xxivj Integers with Equal Sums of Like Powebs. 711
0. Birck (p. 182) took x+y+z =0 and
£=&c—ky, it^iy—кг, {=гг—кх, х=гу—кх> к=гг—ку, p^ix^kz.
Then
П+£ ft-fc П+V, n-ij, Л+f, П — Г = П4-Т, 71-Т, tt+*, П~«, Jl+p, 7l~f>.
0. Birck*1 noted that
for
subject to the condition №—I'+fft— ii)*J—|iy*—^ From one solution
(г, i7 д;, v) of the latter.he derived two or more new solutions.
A. Gerardin42 noted that
), р*+2р(а+Ъ)+2аЪ, p(a+b)+2ab
=ap, Ър, ^4-^
Е. В. Escott43 noted that D), for n=2, 4, has the solutions
y= 2m2-4wn—л2, z= Zm2-2nl)
v = -m2+4mn4-2?iz, №= -2^2+3n2,
where ?n, n are odd, and gave two analogous solutions. Gerardin gave
(ibid.) a process to obtain solutions.
Crussol4* treated the last problem with the restriction y+z=v+w.
The equations can be written in the form
(*+рт)Ч*+|да}*+(*-ря)*-(*-рт)*+^--ря)*+(в+ря|)*, *-2f 4,
where яг, n are relatively prime. Thus
Set e^3a?-^(n2-4?n2). Then the solution is
р=3л24-)82(л2-4ш2), г4-у=л54-2о^(п2
Crussol46 noted that the system
*-(*+e)*+(*-a)*+tf+b)
fc-2,4,6,
is equivalent to
Set ^=og—pp, У^«р4-^д, *=«ff+ft?, ^=ap—% Thus
The discriminant of this quadratic in у, В must be a square. The first of
« L'interm£dutre dea m*th., 20, 1913, 273-7.
c Sphim-Oedipe, 8, 1913, 134; oortwtion, 157.
/№1413 a. papera20^7<rf СЯь XXU,

712 Ншговт of the Theory of Numbers.
three epecial solutions is 2, 16, 21, 25; 5, 14, 23, 24, given by p = l=Z,
$=2, t-5.
G. Tarry4* republished his14 results and noted that
Au ^yAk=Bh^tBk, А<+А^~2Н=В<+В^(г=1, ->fk)
imply Ai, ***,Ak^lBb >-, B^ as shown by subtracting A from every term
of the given equations. A. Aubry concluded that
Л, B, C, -A, -B, -C±A', B\ C\ -A', -B\ -C
if
А =аЬ+ар+Ъ<х-&хр, В = -аЬ+а04-а&+За0, С
Takea=l, a=2t Ъ=Ъ, /5=4 and add 32 to
every term; thus
1, 12, 21, 43, 52, 63=3, 7, 28, 36, 57, 61-
Aubry noted that Ai+x, BL+y=Ah Bh x+y if Atx+Biy=2%. Hence set
Ai=ab, B^cd, x=ca, y=b*t a=a+d. Thus, if A, B=(, ij, f, then
A=ab+bd+cd, B=ab+ac+cd, whence
But at+ad+d1 has besides 3 only prime factors of the form 6&4-1. If
A*-AB+B* is divisible by 3, A+S = 3A ^d A, B=A-h, B-h, 2k.
Невсе А, В=%, ъ fis solvable if and only if A2—AB+.B2 is a multiple of
3 or has at least two prime factors 67:4-1-
Crussol47 solved а, Ь, с, d =ai, hi, ch dx. After adding a suitable constant
to each term we have a+b-fc-|-d=0. Set
А=а+Ъ= — с—rf, Ai
2B=a-b, гВ^ад-bi,
Then
The general solution of the latter is A=\px, B+C=jiqy, B—C=vnf
Ai^^rr, Bi+&^vpyy Bi—Ci=bqz. Then the former condition becomes
ez7=fyz+gz*f where e-^V-X2?2, f~№-&?9 g = vh*-\*f. From the
evident solutions (x, у, г) = (t>, X, д) and (g, r, p), we get the general solution
L. Bastienw proved the impossibility of Xi, — ',3n=yi, • ^^
^e do not form a permutation of the #'s. For, the elementary symmetric
functions of the x's equal those of tike y% so that the z's are the roots of
the same equation of degree n as the у'в.
- Sphinx-Oedipe, пцгоёго врёсЫ, June, 1913,1S-23; Penfleignement math., Щ 1914,
(prepared for pram by Aubry after Tany'e death).
« Spfaiux-Oedipe, 8,1913,156-7; special слйе А»л-г-1, рЛ34,
«ЛЛ 1712

Сяар.ххгу] Integers with Equal Sums of Like Powers, 713
E. X. Barisien" noted that 1, 5, 9, 11, 15, 16 and 3, 4, $, 10, 14, 18 and
2, 6, 7, 10, 14, 18 and 1, 5, 9, 12, 13, 17 have the same sum and sum of
squares; also that
3, 4, 8, 11, 15, 16-2, 6, 7, 12, 13, 17.
A. Aubry50 gave known and new solutions of 2Ja —Za, Za*=2<*2, and
proved the impossibility of x, у £ tf u, v*
X. Agronomof" noted the case a+c+3=2b of A).
A, Gerardin5* gave a solution of XA = 2Jf, 2Л1=
N* Agronomof*3 gave an 8 parameter solution of
i>*-g^ № = 1,2,3).
For any solution of this system, we have
(*=if 2, з, 4),
i«l i—1 »—1 »=1
being arbitrary» Proceeding similarly^ we can solve
By specializing the solution first cited, he obtained solutions of
!>{ = Sfft (ft = 1,2,3: js^I or 2 or 3).
A. Filippov5111 stated that the specialized solutions just mentioned are
trivial since they reduce to xi=yi or у>=*0.
A. Gerardin" noted that 2ж*=2а, 2z*
. Goormaghtigh66 solved the same system by setting
C=Pk+Qp, b
Then the equation obtained by eliminating z between the proposed equations
determines P/Q as follows:
»Matbelis, D), 3, 1913,69.
«•Annaea Sc. Acad. Polyt- do Porto, 0,1914, 141-15L
*l SuppL al Penodioo di Mat., 19,1915, 20.
«Nouv. Алл, Math,, D), 15, 1915» бв4; l>mtemi&fiaire dee m*tfa.r 22, 1915, 130-2 (cor*
rectionforA-2); 23, 1916» 107-10. Cf. papers 130, 302, 43&4D, 442 of Ch. XXI.
" T6hoku Msth. Jour,, 10, 1916» 207-14.
ttfl/6id., 15,1919,143.
■« L'mUrmediftJts dee math,, 24, 1917, 65 (correction, p. 153}.
*IbL 25,1918,20-21.

714 History of the Theory of Numbebs. [Chap, xxiv
An Equivalent Problem in the Theory of Logarithms.
The system of equations Za* = ztf (fc = l, - -, n) which we have been
considering is equivalent to the system 2a1=2biy 2alu2 = 2bibi, •••,
2aiaj - - * Ori = S6ibx • • • Ъп. Consider the equation having the roots au att
- - - and that having the roots bt, b2, • ■ •. Thus our problem is equivalent
to the following: Find two equations of the same degree each having all
its roots integral and the first n coefficients of the one equal to the corre-
corresponding coefficients in the other.
The lattsr problem occurs in the investigation of rapidly converging
series convenient for the computation of logarithms. In the familiar series
take, for example, m^s2, n = (x-l)(z+l). Then log (x-J-1) differs from
21ogx-log (x-1) by a series in k = lfBx2—1). In general, we desire
that m and n shall be polynomials in x whose roots are all integers such
that к becomes a fraction whose numerator is a constant. We may remove
the second terms of the polynomials by a linear substitution.
J, B. J. Delambre* took m^xt+-px-\-q1 n=x*+px—q, and assumed
that m=Q has the roots a, bt —a^b, and л-O the roots -a, —b, a+b,
whence p— —а2—аЬ—Ъ*, q=a*b-\-ab*. For a=b = lf we have the formulas
m, n=s*-3x±2, ascribed to Borda.
J. E. T. Lavernede*7 gave an extensive treatment of such polynomials,
chiefly of degrees 3 and 4, and noted the examples
S. F. Lacroix68 quoted the preceding results and the following, attributed
toHaros:
m=x4x-5)(x+5), n = (x-3)(x+3)(z-4)(x+4)=m+144.
John Muller55 had made only the following contribution to our subject:
log ((S+lJ-log d+log
log (d+3)>=log (d-bl)'-Hog (d+A)-]ogd-bgq,
The latter is applied when d=^14 to find log 17, knowing log 15, log 18 and
log 14. Then $=2025/2023. Taking a=2024, x-1, we have q={a+x)l
(a~x), a series for the logarithm of which is found by subtracting the
** J. С de Borda'e Tables trigonometriquee d^cicnales ou Tables dee logarithiuea . . . revues,
augmenteea et publi6es par Delambre, Paris, on IX A8(XM}. Introduction.
» Notice dee travaux de l'Acad. du Gard, 1807,179-192; Anaalee de Math» (ed.r Geigoone)r
1, Ш0-11, 18^1,78-100. SeeAllman^
*• Traite du Caicul Diff, . . - Int., ed. 2, I, 1810, 4&-S2.
** Traite analytique dee sections coniquee, fluxjone et fluentes . T ., Рапя, 1760, 112. Thk
topic does not occur in the earlier English edition, A Math, Treatise: containing ш
System of Conic Section*; with the Doctrine of Fluxions and Fluents . . lf London, 1736.

Chap. XXIV] PROBLEM Iff THE TkBORY ОГ LOGARITHMS. 715
series for log A—s/a) from that for log A-bx/a). [If in the second formula
we take d=x—2, we obtain Borda's" result. If in the first we take
d^x—1, we obtain the example m=x*, n=x*—l given before the report
on Delambre.5*]
W. Allmanw gave the result quoted under Delambre* and the first two
results cited under LavernMe.
T. Knight61 started with x=(x-\-n) {xf(x+n)\, changed x into z+n'
in the fraction and multiplied by such a fraction as will restore equality:
In the final fraction change x into x-}-n" and restore equality by annexing
the new factor
х{х+п+п*){х+п+п")(х+п'+п")
(х+п)(х+п')(х+п")(х+п+п'+п"У
The expanded numerator has its first three terms the same as the corre-
corresponding terms of the expanded denominator, and also the fourth terms
alike if n" = n+nf. The rest of the paper is on the case n^nf—n"= * ■ •
= — 1, and gived the general factor explicitly.
Secretan" noted that
E. B. Escott83 spoke of a&*+ai$*~l-\ and ад?*-} as having
exactly their first r terms alike if aQ=ae, •■ -,a^x^<C-i, aT^a'T' He readily
proved theorem (I): If / and g are two polynomials in x having exactly
their first r terms alike, then f(x)*g{x+d) and g{x) -f(x-\-d) have exactly
their first r+1 terms alike. Starting with f^x—a^ g=x, and taking
d=— Ъ> we see that (x—c)(x—Ъ) and x(x—a—b) have two terms alike»
Taking the latter as / and g, and d = — c, we see that (Knight)
(x—a)(x—Ъ)(х-с)(х—a—b—c), z(x~a—b)(x—a—c)(x—b—c)
have three terms alike. Proceeding similarly, we obtain theorem (II):
If we form the equation whose roots are the sums of au -*-,a» taken 1, 3,
5, - • - at a time, and that whose roots are the sums of the a'g taken 2, 4, 6,
■ ■ * at a time, we obtain two functions of degree 2*'1 having exactly their
first n terms alike» For special a's common factors occur and may be
removed. Thus, if n=4 and if the a's are a, b, a-}-b, a+26, four of the eight
roots will be соштоп and the remaining ones are 0, a-\-3b, 2a+6, 3a+4b,
and a, 6, 2a+4b, 3a+3k If in (I) we take g=P(x)=x(x+d)(x+2d)< ••
\x-\~(n — \)d\ &ndf—P+c> and remove the common factor Pjx, we obtain
two functions (P+c)(x+nd) and (x-\-nd)P+cx of degree n-\-l with exactly
their first n-|-l terms alike. Again, taking g=P(x)*P(x+a) and f=g-\-c
in (I)> and removing the common factor gf[x{x+a)), we get
(x+nd)(x+a+nd)(g+c\ {x+nd)(x+a-\-nd)g+cz(x+a),
Т, Roy» Iriuh Acad., 6, 1797, 391^34.
л Phil. Trans. Roy. Soc. London, Ш7Г 217-33»
» Comptea Rendae Paris, 44, 1S57. 1276-9.
« Quar. Jour. Math,, 4lr 1910, Ш-167.

716 History of the Theory op Numbers. [Chap, xxrv
having all terms alike except the last two in each» Taking n—2 о* 3
and making suitable assumptions, we find that these functions have two
common linear factors (pp. 148-50, with changed notations)* Besides
employing roots in three or more arithmetical progressions, leading to a
solution of degree 7 (p. 152), various special methods are used.
Escott, after reading the proof-sheets of this chapter, pointed out its
relation to the derivation of formulas for the computation of x:
tan1 -?- -i-tan^1 ~-J- — = tan'1 Цг,
where X is a real polynomial in x whose degree equals the number of frac-
fractions in the left member. Since
taxr1--—log*-1—.,
у 2г y—ai'
it suffices to have (z+ot-\-ai)(x-\-P+bi)' >>=X+pi. Of the polynomials
m, n in the above problem on logarithms, we may employ here those con-
containing only odd powers of x and a constant term. If in Delambre's* ex-
example we replace a by — ai and b by — Ыу we have
—tan^—^^tan / . , . ч ■
By the former we have a product of factors like xJ-}-al expressed as a sum
of two squares (cf. note 13, p. S82 of VoL I of this History). Escott noted
that hisw general results include as special cases Goldbach's1 and Euler's*
formulas, the first identity by Nicholson8, the two formulas by J. H. Tay-
Taylor,11 as well as the following (after reducing each term by such a constant
that the sum of the terms in either member becomes zero"): GeVardin's11
2, —, 47, Gerardin,17- * Tarry,17 Miot,« and Aubry."
In Sphmx-Oedipe, 10, 1915, 30, occur two examples of two sets of five
numbers having equal sums of feth powers for k = l, * • ■, 4, the numbers
being functions of six parameters.

CHAPTER XXV.
WARING'S PROBLEM AND RELATED RESULTS.»
Waking's problem.
E. Waring1 stated that every integer is a sum of at most 9 [positive
integral] cubes, *lso a sum of at most 19 biqu&drates, etc. Every integer N
of thepn)performisaBumofafinitenmnberoftermfl£^aa;w4-bx*-h«r+- ■ •
(N being a multiple of 3 if t=3x*+fa?+24). Cf. MailleL"
J. A. Euler1 stated that, to express every positive integer as a sum of
positive «th powers, at least T=p+2n~2 terms are necessary, where ?
is the largest integer <C/2)». For n=2, 3, 4, 5, 6, 7, 8, T=4, 9, 19, 37,
73, 143, 279 [cf. VaccaM].
A. K* Zornow,1 at the suggestion of C. G. J. Jacobi, constructed a table
of the least number of positive cubes composing each number ^3000*
The number of cubes was stated to be ^8 except for 23, ^7 for numbers
>454, ^6 for numbers >2183. The final statement and the second for
239 (which requires 9 cubes) are erroneous. Corrections were made by Z.
Dase, who computed a table extending to 12000 and communicated it to
Jacobin The largest Dumber within the limits for which 7 cubes are
required is 8042; for 8 cubes, 454. Jacohi considered the problem to mid
all the decompositions of a given number into the least number of cubes. He
tabulated the numbers < 12000 which are sums of two cubes and those
which are sums of three cubes.
C. A. Bretschneider* constructed at Jacobi's suggestion, a table giving
all the decompositions of numbers ^4100 into a sum of biquadnttes, and ft
companion table showing the numbers wbich equal the sum of a given
number of biquadrates but not fewer. For 79, 159, 239, 319, 399, 379 and
559, it is necessary to use 19 biquadrates; for the remaining numbers, at
most 18* As far as 4096^4*, he verified that 37 fifth powers are needed,
and 73 sixth powers. He repeated Euler's2 statement.
J. liouville* was the first to prove that every positive integer is the sum
of a fixed number Ni of biquadrates, in fact, of at most 53. He first proved
that the product of any square by 6 is a sum of 12 biquadrates, in view of
*A. J, Kempner read critically the reports ш Шй chapter and compared them with the
original papers except for 2, 6, 38b, 44<z, 54? 60-62» 64, 69, 72, which were not accessible
to bim. The statements concerning incorrect results in papers 6a, 13 and 17 are made
on hie authority.
1 Meditation» algebraicae, Cambridge, 1770, 204-5; fed. 3,1782, 349-350.
1L, Eider's Opera poriuma, 1, 1862, 203-4 (about 1772).
1 Jour, fur Math., 14, 1835, 270-280.
* Jour, fur MatL, 42, 1851, 41-69; Jacobi, Werke, VI, 322-354, and 429-431 for correction*
of the Journal article.
9 Jour, fur Math., 46,1853, 1-28,
9In hie lectures at the College de France; printed in V, A. Lebeflgue's Exerolcefl d'AoaJyae
Num&ique, Parts, 1859, П2^5. а. Е. Mullet, Bull. Soc. Math. France, 23, 1895,
bottom of p. 45.
717

718 Histoby of the Theoby of Numbers. [Chap. XXV
But any number is of the form 6p+r, r=0, — -, 5, while p is a sum
nJ-j \~n\ of four squares. By the earlier remark, 6p is a sum of 48
biquadrates. Hence tf4^48+5.
E. Lucas*1 gave the identity
A) 6(z;+^-hxJ-h^ = 2Cxv+^-hS(^-Xy)* (i,j =1, -,4; i<j).
Clt becomes Liouville's6 identity for Xi=x+y, Xi = x—y, xt = z+t7xA-=-z—Q.
Lucas also gave the incorrect identity
Assuming that every integer is a sum of nine cubes, he stated incorrectly
that it follows that every integer is a sum of at most 26 sixth powers.
Lucas7 noted the identities
the second being erroneous ["Fleck13! sbice the left member exceeds the
right by GOta^+sW+a^V+y2*2**)-
S. R6aUs* proved that 47 biquadrates are sufficient by using the result
that any integer is a sum of 4 squares, one of which is arbitrary (under
certain restrictions) and hence may be chosen a biquadrate.
E. Lucas9 reduced the number to 45 as follows. Let fc=6p+r. If
p=Sh+j (j = l, 2, 3, 5 or 6), p is a SI, and, by A), к a sum of 3 12+5
biquadrates. If p=Sh or 8fe+4, p-27 is a SI; then
so that at most 3 12+2+5 biquadrates are needed. Finally, if p=Sh+7,
p—14is a ED, во that
*=6n;+6n;+6tt!+3<+3+r, N<±Z-12+4+5.
Lucas10 obtained the lower value tf4^4L Since 8ft+j (j=l, 2, 3, 5,
or 6) is a SI, 48А+Ф is a sum of 36 biquadrates. By subtracting at most
five of the biquadrates I4, 2*, 34 from any given number, we obtain one of
these numbers 4&k+t (*=6, 12, 18, 30, 36). By the tables our theorem is
true for numbers ^5>34.
E. Maillet11 proved that every positive integer is a sum of 21 or fewer
cubes ^0, five or more of which are 0 or 1. He employed the identity
to conclude that &*(c?+m) is a sum of at most six positive cubes if O^m^a2
and if то is a sum of three squares, i. e., if m#4*(8?i+7). Under the similar
conditions on tn\ 6Л = 6л(«2+то) +6V(«'*+*»') is a sum of at most twelve
«■ Nouv. Correep. Math,, 2, 1876,101.
• Jour, de math. 616m, et »p&., 1, IS77, 126-7, Probe. 33, 39. Quoted byCA.
RecueU de probl&mea de math., dg&bre, 1895, 125,
• Nouv. Corrap. Math., 4, 1878, 209-210.
• Ibid., 323-5.
" Nouv. Ann. Math., B), 17,1878, 536-7.
" Амос, footy. av. ec., 24, II, 1895, 212-7.

Chap.xxvj Waking's Problem. 719
positive cubes. For a and a' odd and relatively prime and for every
^'suchthat a<a'<<*78l8aa'^lA/^l<*y3, it is shown that there exist positive
integers m and mf satisfying the earlier conditions and also сит-\~а'т'=А'„
Hence every integral multiple 6Л of 6, for which
wither, a'odd and relatively prime, is a sum of at most twelve positive cubes*
Taking a —7—2t <x'=y, we see that the intervals obtained by varying 7
overlap if у exceeds a finite limit and is odd. Hence every multiple of б
exceeding a certain finite limit is a sum of at most twelve positive cubes,
whence Ni^X2+5 (at least five cubes being 0 or 1).
G. Oltramare17 proved that any positive cube is the sum of 9 smaller
cubes ^0. Any number N is the sum a^-i-b2-}-^1-}-^ °f f°ur squares. Then
8x*+SzN is the sum s of the cubes of z±a, x±6, s±c, x±d. For N odd,
N—2x+lt we have JV* = l3-}-a. For Nl=2kN, where N is odd, we multiply
the last formula by 2***
G. B. Math ewe1' argued that there is a considerable probability that
all sufficiently large integers are expressible as sums of p-}-l pth powers,
at least for some positive integers p. According to Kempner4*, this is not
true when p is 6 or any power of 2.
E. Maillet" proved that if ф(х)=ах*-\-а1Х4-\-' - --ha* equals a positive
integer for every integer x ^д, then every integer n exceeding a certain func-
function of a, - - -, afi is the sum of a limited number N of positive numbers ф(х)
and a limited number of units, where N is at most 6, 12, 96, 192 when ф is
of degree 2, 3, 4, 5> respectively. For each function Ф(х), the number of
representations of n obtained increases indefinitsly with n.
E. Lemoine1* stated that every integer equals p-M, where s is a cube or
a sum of distinct cubes, while pis one of the 24 numbers 0—6,8 —17,27—33.
L. Ripert1* proved this statement.
R. D. von Stemeck17 gave a table showing the number of cubes needed for
the representation of all numbers ^40000. From 8042 on, six cubes suffice.
He stated incorrectly [Tleck*] that 3&* is not the sum of three cubes un-
unless they are equal. He conjectured incorrectly [Kempner**^ that always
about ten of any thousand consecutive numbers are sums of two cubes.
G. Vacca,1* after citing Euler's statement, noted that 2n<v — 1 is the
sum of v—1 numbers each 2n and 2n—1 unite. [Thus, for n=2, 7 is the
sum of 4, 1, 1,1, but not a sum of fewer than 4 squares; for n=3,23 is the
sum of 8, 8 and seven units, but not a sum of fewer than 9 positive cubes;
for n=4, 79 is the sum of 16,16, 16,16 and 15 unite, but not a sum of fewer
than 19 hiquadratesO
2,1895,30.
рт v 25, 1895-6, W.
u Jour, de Mrfh., E), % 1896, 363-380; BuIL Soe. 1Ш. Fnra», 23» 1805, 40-49. Cf.
papers 68, 72, 73, 117,181-2 of Cb. L
onv. Ann. Math., C), 17, 1898,19&
, C), 10, 1900,335^6.
br АЫ. WmL Wim №th.)v 112, Пж, 1908, ОД7-в6.
iid^ II, 1904, 2Ю4.

720 History of the Theory op Numbers. [Crap. XXV
E. Malllet19 erroneously concluded that there is an infinitude of integers
not a sum of fewer than 128 eighth powers >0.
A. Fleck40 noted that in the proof by Lucas10 it suffices to subtract at
most three biquadrates unless the given number is 48m-M, J=10, 11, 26,
27, 42, 43. For £ = 10, subtract 14+34; we get 6tf, where N=4Bm-3) is
a GE unless 2m-3=7 (mod 8), i. e., m=l-\-4fi. In the latter case,
is a sum of 36 biquadrates since 4(8^—27) is a EC. Treating similarly the
remaining />s, he concluded that #4^39. He found that Nt^13 by
employing Maillet's11 result and the formula, following from r*=r (mod 6),
E. Landau21 proved that every definite integral rational function of x
of degree n with rational coefficients is a sum of 8 squares of integral
rational functions with rational coefficients, and gave references to related
problems.
A. Fleck22 proved that the square (cube) of every definite integral
rational function of x with rational coefficients is a sum of a finite deter-
ininable maximum number, independent of the degree and coefficients of
ihe function, of fourth powers {sixth powers) of integral rational functions
of degree ^1 with rational coefficients, L е., linear functions and constants.
Fleck" remarked that Maillet's14 limit 192 for Ns can easily be reduced
by about 36, but that the new limit is still far above the ideal limit 37
suggested by tables» To show that N* is finite, he used the identity
Hence 60n* is a sum of 184 sixth powers. Thus if m is any integer, 60m
is the sum of at most 184JVs sixth powers. Since any integer is of the form
бОти+7-, r=0, 1, - - -, 59, we have tf6^184iV*+59.
E. Landau24 lowered the limit for Na to 38. Setting xA=Xi in A), we
see that 6n2 is a sum of 11 biquadrates if л is representable in the form
х\^-х\-^2х\, which is true if n is any odd number m. Hence 6m2 and
6-16 m2 are sums of 11 biquadrates. As above, &k+j (j=lf 2, 3, 5 or 6)
is a sum of three squares at least one of which is odd. Hence 6 times such
a number is a sum of 114-12-J-12 biquadrates* By arguments of the type
used by Fleck,20 we get iV4^38. Except for numbers 48п-И, *=Н, 27, 43,
he proved that 37 biquadrates suffice. For these cases, A. Wieferich*
showed that 37 suffice. Hence iV4^3
"Aniiali di Mat., C), 12,1905, 173, note. Error admitted in l'intermediuire de» math., 20,
1913, 202.
"SiUmigBber. Berlin Math. Geeell., 5,1906, 2^9.
« Mmth, Annalen, 62,1906, 272-281.
*JWa, «4,1907,567-572.
■ Ibid., 561-6. To ^=192 must be Added the number of tmita.
"RendieDirti Orcob Mat Palermo, 23,1907,
* Mrib AniulfiD, 68,1909,106~&

Сна*. XXV] WaRENG'S PROBLEM. 721
EL Maillet" proposed the following generalisation of Waring's problem:
Can к be taken sufficiently large that there shall be integral solutions of
where *i, • ■ -, nm have given value?, and Nlf - -, N* any values satisfying
suitable conditions? For a»2, n(=2l 74 = 1, fc^4, there is always a
solution (Cauchy, Ch. VUI, p. 284) if Ng is odd and N1 ie odd and
E* Maillet17 proved Waring's theorem for eighth powers, but gave do
explicit limit for AV He proved ш an elementary way that there is an
infinitude of numbers each not a sum of n or fewer nth powers,
A, Hurwita* proved that every integer is the sum of at most
37F-4+«Ы2+48+6'8)+5039=ЗШ9
8th powers, in view of ЛГ4^37 and the identity
In general, if there exists «n identity (in a, b, c, d)
where pf pi, *•*, Pi are positive integers and cti, • ♦ ■, ^r are integers, then
( )
so that Л, would be finite if N* is. He proved by use of the gamma
function that there is an infinitude of positive integers each not the sum of
n or fewer nth powers.
X Schur" found tha identity which prove* iV» finite:
A, Wieferich* proved that ЛГ»^9 Cexcept for a limited set of integers
arising from a case*1 overlooked]. The proof consists m showing that any
positive integer IB the sum of three cubes together with k=6a?+6amt where
0<A andт**х\+х\+х]<А\ For, by Maillet,11 к is then a sum of 0
positive cubes.
* ЪЪйеппбйш da matb,, 15,1908,196, tnd МдШвС"
" Ban. 80c. Mtfb de France, ЗЬ, 1908, 09-77; Comptes Hendue Ftom, 145,1907,1399.
■Mb*L Ана^оц d&t 1908, 404-7*
» Matb. Anmlen,«, Ifloo, 105 (m д i*per pubfiehed ty Undaa,)
* Math* Авшк, ве, lfloo, tf-10L
tt The оме'-4 is 1064S< @.4N^-'. AUaitioo wuG»Uedto this gup in the proof by P.
ВЫ, Ffledere Z*hlentboo»ie, 2,1010,344, who indicated in hii ZoaKLM*> pp. 477-^
ldftilJidbl^dii Thl
gg>
meofpocstod £л tbo nrwiiycBBfaJ ftttarapi by E. Lejneek (Math, Aim., 70v 1911,
454-*) to fill tbo «ар. ТЬ» «ар in Wleferich'* proof wu filled by Kempnc?**

722 History of the Theort of Numbers. cCha
E. Landau32 proved that every integer г exceeding a fixed value is the
sum of at most 8 positive cubes. He proved that there exists a prime p
not dividing z such that 8рэ Шг < 12jP-&ud such that p*(p — 1) is not divisible
by 3. Hence fP=z (mod p3) has positive integral solutions 0<p*. In
M Then
By the paper of Wieferich,80 ws can find an integer 7, 0^7<96, such that
Afi—-у^бго, where 77i=xf-h^+Xs. For * sufficiently large, m<pt, so
that 0
A. Wieferich" proved that ЛГ5^59, ЛГ7^3806. Не gave a table showing
the least number of fifth powers required to represent each number 1,* - -,
3011.
D. Hilbert34 proved Waring's assertion that every positive integer z
is the sum of at most Nm positive wth powers, where Nm is a finite number,
not determined, depending upon m but not upon z. He first proved, by
use of a five-fold integral (a 25-fold integral in the first paper,) the lemma
(stated by Hurwitz,2* who was unable to prove it) that there exists for every
m (and r = 5) an identity in the re's
where the ал are integers and the pA are positive rational numbers. It is a
simple step to prove Waring's theorem for powers whose exponents are
2*, к ^ 2. The case of any exponent is derived from this by an elemen-
elementary, but long, discussion (not using calculus).
F. HausdorfP* proved Hilbert's lemma by use- of integrals involving
exponentials, the method being more suitable for computing the a's and
p's.
E. Stridsberg38 proved easily that Waring's theorem for /cth powers
would follow if it were shown that, if В is any real number, every positive
integer ^B can be written as SpxP;, where the P's are integers ^0
and рл is a positive rational number depending only on p. He noted
that Hausdorff's elegant modification of Hilbert's proof can be reduced to
an elementary study of binomial coefficients» Using symbolic powers of h,
let h* denote B/0 !/д! for all even integers 2^0, and h*+l=0 for all odd
integers 2^+1^1. A theorem of Hausdorff's becomes the simple one that,
if f(x) is any polynomial which is never negative for a real value of x, then
«Math. AnnaJen, 66; 1909, 102-5; Landau, Htmdbuch der Lehre von der Verteilung der
PrimwihleD, 1, 1909, 555-9. Cf.Lwidwi."
» Math» Annalen» 67, 1909, 61-75.
M Gottingen Nftchr», 1909, 17-36; Math. Ann., 67,1909, 281-300.
» Math. Annalen, 67, 1909, 301-5, Cf. Hurwitx.*4
x Arkivftr Mat., Astr., Fyaik, 6T ШО-11, No. 32, No. 39. French i&mn£ in Math.
72, 1912, 145-152,

Сна*. XXVI ЭТлВПЮ'я FfeOBLBK. 723
f{x+h) >0 for х real [cf. Hurwits"! since
being true forf(x-\-h)**h\ Hubert's lemma is proved by use of
whence follows HiuwiUs's theorem that Waringfs theorem is true for л=2»>
if true for n=m- finally, be simplified the second (elementary) part of
ffilbert's proof of Wiring's theorem-
A. Boutin*7 gave the identities
the exterior sign being the product of the n interior signs.
P, Bachmann* gave an exposition of several of the preceding papers.
A. Fleck"* and W. Worn4* proved that every definite quartic function
of x with ratioaal coefficients is a som of five squares of rational integral
functions with rational coefficients-
Б. Landau19 gave a new elementary proof that all numbers exceeding a
certain limit and prime to 10 (or to the product of any two primes of the
form 3m + 2) are sums of at most 8 positive cubes. He here avoided the
theory of the distribution of primes used in his11 former proof.
J. Kurschilc40 generalised lioavffleV identity A) to give
where on the left occur all possible combinations of signs and all sets of k+1
of the 3*4-1 variables a* • - ■> «i*. For m^3, there is no identity
A. Gerardm* noted that (a^-f-O/I is the sum of the cubes of as1, y*f
V» V, 3sV 3*у», &*>. Also (^4-%*)' ifl the sum of the cubes of s1,
Зу1, Szy*, 2x*y, sty- Ь. Boure remarked that the former is the sum of the
cubes of *\ Ыу> Ф* 3*У*> 6^Л-
A. J. Kempner*1 considered the number C(k, n) of the positive integers
^k which are sums of n or fewer positive nth. powers, and the superior
limit Sof C(h, n)fk for k=> «>. He proved that S<l/n\r where&s Hurwit2n
and MaiBet" bed proved merely th&t S<L It follows that there is an
infinitude of positive iiitegere of each of the forms 9^ 91+1, ■ -, 9^4-8, sudi
that each is not a sum of fewer than four positive cubes. There is an
infinitude of positive integers each not a sum of fewer than nine sixth
"ЬЪИвщ*1ш1Г*<1(»1щ»иц 17, lftlO, 123-3,238-7. Seepepea«И»below,
»• Archir M*th- РЪтв., C), Ш, 1900,2^38; (*), U, lftlO, 275-6,
^ViU^bbrift N*f O^dL Zuri^i, 56,1911,110-21
& Ш 24S^S2
pp, , ,19, KL
jjber d» W*ri^pdw РгоЫвщ nod
Extract m MsflL Airaaltfi, 7% 1912, 887.

724 History op the Theoky of Numbers. CChap. xxv
powers, and an infinitude each not a sum of fewer than 2e+: powers, with
the exponent 2*, for #>1. He lowered the known limit for Nt to 970 by
use of the identity
for с=d and for d=0, and the fact that every number is of one of the forms
a*+ №-\-h? for к ~ 1,2. For the determination of upper limits for iV« and Nu
from known limits for Nt and JVV, he gave identities expressing Zfa^-K^+c*)*
as a sum of Bn)th powers, for n=6 and 7, where I is a suitably chosen
integer»
K. Remak43 noted that Stridsberg** used integrals hi a single place and
applied the result proved by them only for the special case in which
/{аг)=0*(а). For this case Remak gave an elementary proof by use of the
fact that a quadratic form in n variables is definite if the determinant of the
part involving the first v variables (suitsbly chosen) is positive for v= 1, 2,
• - *, n. Hence the proof of Waring's theorem is reduced to algebraic pro-
processes.
A. Hurwitz44 gave a new elementary proof of the theorem, used by
Eausdorff jU Stridsberg** and Remak,41 that if the real polynomial
not identically zero, is ^0 for every real x, then
is positive for every real x-} likewise toTf(x)-\-f(z)-\--
L. Orlando44* amplified Hurwitz's*4 proof*
G. Frobenius* also gave an algebraic proof of Waring's theorem by
altering Stridsberg's proof at the point where he had used integrals,
E. Schmidt4* used Minkowski's convex point sets in space of q dimen-
dimensions to give a more luminous exposition of Hilbert's first lemma,
G. Loria47 remarked that if Waring's тпт1тппщ 19 for ДГ4 could be
lowered to 16 ([overlooking the facts noted by J. A. Filler]], one would hope
for a proof that every number is a sum of n2 exact nth powers.
E. Landau46 pointed out errors in the same journal on sums of cubes*
W. S. Baer41 proved that every integer ^23 • 1014 is a sum of 8 or fewer
positive cubes, likewise every odd number >175 396 368 704. and every
number =8 (mod 16). The following numbers are sums of 7 or fewer posi-
positive cubes: every number 2744s (s odd), all sufficiently large multiples of
16 or 27, all sufficiently large numbers =0,8,16,24,28,36,44,48, 56, 64
, Annafen, 72,1912,153-6.
«• /Ш., 73,1912,173-6. Cf. Orbmdo<* For д generalisation ж* G. P6Iyaf Jour, fur M&th.,
145. 1915, 233.
<" Atti della £L Accad. lined, Readioonti, 22,1,1913, 213-5.
• SiUungsber. Akad. Wva. Berlin, 1912, 00^70.
« MMh. Аопд1ел, 74,1913, 271-i.
« L'eneeignement math., 15, 1913, 200-1.
« L'intermeduire dee math,, 20,1913,177,179.
"Bi nun WAimgachen Probtem, EHsbl, Qottiiigen, 1913» 74 pp.

Chap. XXV] SUMS OF UnLTKB POWSB*. 725
(mod 72). He reduced the limit for JV« to 478, that for N6 to 58 and gave
a simpler proof that iV<^37. For £=2744, it is shown by elementary
methods that every number =k (mod 2k) is a sum of 7 or fewer positive
cubes; hence if Cj(x) denotes the number of positive integers Шх which are
decomposable into 7 or fewer positive cubes,
B) ^-<^^^1 for aU sufficiently larga*.
His transcendental methods enabled him to replace IfBk) by 13/72* He*0
later gave a direct elementary proof of the last result B) for jh=4096 by
noting that the integers ku, where и is positive and odd, can be decom-
decomposed into 7 positive cubes all of whose 7 bases exceed any assigned positive
number g for every v exceeding a limit depending upon g.
E. Stridaberg51 gave a brief elementary proof of Kurwitz's lemma £HiI-
bertMJ without the use of integrals (Remak,41 Frobenius4*) or the gamma
function. The proof is admitted to be otherwise essentially the same as
hie* former proof*
G. H. Hardy and 3. Ramanujan" proved that the logarithm of the
number of ways n is a sum of rth powers of positive integers (xearrangemente
of the ваше powers not being counted as distinct) is asymptotic to
j
where {* denotes the Riemann zeta function, and Г the gamma function*
Hardy and J. E. Uttlewood**' made use of the theory of analytic func-
functions (of, Ch* DP*1) to prove that every positive integer, which exceeds a
certain number depending on к Alone, is a sum of at most k-2h~1-\-X posi-
positive fcth powers; for example, a sum of at most 33 biquadratee. The
transcendental method leads not only to a proof of the existence of repre-
representations, but also to asymptotic formulas for their number* They since
communicated to the author the improved result that at moetfifc—2JM+5
positive fcth powers are necessary; this gives 9 cubes, 21 biquadratee, 53
fifth powers, 133 sixth powers, ete.
Numbers expressible as bums of unlike powers.
D. Andre** proved that every even integer is the sum of & cube =£0 and
three squares (since every 8п+3 is а БН). In general, if 9 is odd, every
even integer > 7'is the sum of an 5th power +0 and three squared each +0.
G. de Rocquigny64 noted that every integer except 1, 2, 3, 4, 6, 7, 8,
10, 11, 18 is a sum of three cubes and three squares. He* stated many
" Math. AimaleD, 74+1013, 511-i.
11 АШт ftr М*Ц Aetr,, Tymkt 11, 19X6-7, No. 25, pp. 36-9. H* m*wd pep» witb tltt
мяе title, <bid.t 13,1919, No. 25, deals at Uogth (pp. 31-70) witb definite azbd tomb
definite polynomials in x and incidentally with their occdrre&ee in the titeiatuz^ oo
Waring'i problem,
» Ptdc, London Matb. Бос., B), 16,1917,130.
^Qoar-1""- Mftth., 4£, 1919, 272 eeq.
WNouv. Ann. MaUl, B), Щ 1871,186-7,
MTr»v*ux8c,d«iTTaiy. Rennet 3,1904,42.
* L'infenridian* de* math., 10, 1003, 100, 212; 11, 1904, 31, ft, 81, 99, 140, 171, 214.

726 History of the Theory of Numbers. [Chap. XXV
theorems like the following: Every integer > 36 is a sum of four squares and
four biquadrates each Ф 0; every integer > 14 is a sum of four squares and
four cubes Ф0.
P. F. Teilhet" verified that every integer up to 600, except 23, is a sum
of two squares and two positive or zero cubes.
G. Lemaire57 noted that 3, 6, 7, 11, 15, 19, 22, 23 are not sums of any
number of powers of distinct numbers.
G. Rabinovitch" proved that every number >23 is expressible in one
of the forms a*+b*f am-\-b*+cp, ■«•, where a, b$ «• ■ are distinct, and m,
n,p} ■ • • exceed unity.
A. Gerardin** proved the theorems due to Andre.43
Every number a sum or three rational cubes.
S. Byleyw solved G=x3-hya+za by taking x = p+q, y = p—q, z—m—2pr
Then
will, for p = см^/б, equal the square of av—ai?(m—at^/6) if m*=2av(m—а
Let m=dv. Then v=Qad!D, where D=3<F+a*. Hence
_(H)H
Ileference is made to a less simple method in Leed's Correspondent, Quest.
211.
T. Strong*1 showed how to express any number a as sum of three or more
rational cubes. Take x, p—x, m—p, r, s, -«- ae the roots of the cubes*
Thus
The right member will be the square of Zp(p—2m)+2c if
p=<?lCa), с{2т-р)=т*-\-г*-\-&-\ .
Set с^mnt r^mr\ s=m', — ^. The second condition gives
Hence giving any rational values to n, r'f s't -"Де get rational values for
x=m—c/(Zp)t mt pt r, s, — ». Since we can in particular express 4 as a
sum of three positive cubes, we can divide unity into three positive parts
such that if each be increased by unity the sum is a cube fEvans,414 Davis,4*
and Tebay*28 of Ch. XXI].
Wm. Lenbart," to express Л as a sum of three cubes, selected any cube
r3 and from Ar* subtracted a cube s2 chosen by trial such that the difference
«I/intermediaire das math., 11,1904,
|T Ibid., 19, 1912, 218.
»/Wd., 20,1913,157.
»/bid., 22, 1915, 207,
» Ladies1 Diary, 1825, 35, Quest, 1420,
* Amer. Jour. Arte, Sc. fed, Sillimjm}, 31, 1837r 1Э6-&.
*» Math. Miet^llany, Fiuahing, N. Y., 1, 1836,122-8.

Chap, XXV] SUM OF ТНКЙВ RATIONAL ClJBKS. 727
is a number t found in his1" table (Ch. XXI) of numbers expressible as a
sum of two positive rational cubes. Or, let Лг|+«* = ^о*+Ь1, Then A
is the sum of the cubes of or, bxt ex if c=p—* and
Hence
а(т*А-**)
та та та
As an application, 2 and 4 are expressed as sums of three positive rational
cubes. The same table is used tentatively to express n-\-l orn-1 as a sum
of n cubes each >1 or each <1, with examples when n=4, 5, 6*
Several*1 expressed any number n as the sum of three rational cubes*
Let their roots be (l±*)/Bs), (ox*— l)fx. The sum of their cubes is л if
Assuming that **=1— 2аа?-\-$пх*> we get х~
EVZKY POSrnvX NUMBER A SUM OF FOUR POSITIVE RATIONAL CUBES, ETC*
G* Libri* noted that if щ nf r are solutions of йа?-\-Ьу*-\-с&=0, then
aX*+hY*-\-cZ*=d is solvable for d arbitrary* Set X=mp+qp Y=np+8,
Z=rp-\-L The new equation lacks p1 and will lack p3 and hence determine
p rationally in terms of ^ ^ if we take q^ — (bn28-\-cr*f)f(am*)r
If Л is a multiple of 24, it is a sum of four cubes [not neceesarily positive]:
ff±l
Kext, let A*=24z-\-b, 0<6<24, If Ъ is one of the numbers 1, 3, 5, 7, 8, 9,
11,13,15,16,17,10,21,23, V-6 is a multiple 24* of 24, when<5e Д~У+*,
where 8=2A(x—n) is a sum of four cubes, so that Л is a sum of five cubes.
If & is not one of the above numbers, 6ztl is one of them* Hence every
integer is a sum of eix cubes one of which is 0 or 1. If
we have the identity in r, 8, t,
A)
9
Every integer is the algebraic sum of 17 biquadrates, taken positively or
negatively. The proof, similar to the above for cubes, follows from
Mtth. Quest. Edtux Tfmee, 13,1870, 63-4.
М^тогш «оргя la teoti» d« ситегц Fbv»^ 1820r 17-23,

728 Нштовт of хвв Theoby о* Numbers. [Chap, xxv
Again, if P--1-B/480,
lbese two quartic forms repeat under multiplication.
Iibritt proved that any positive rational number m equals the sum of
four positive rational cubes* In the identity
B) m
B) „
we can reduce the right member to a eum of four positive cubes. In
take a = (m-[-(tf)IW)> b=m/(fitf). Then the sum of the first and third
terms in B) is a sum cf+fP of two positive cubes if (m-\-6<f)*>2m*f where
Now use C) for a^a, b=m/(Gtf). Them а1-{^/(бд2)}1 is a sum of two
cubes each positive if
which implies the preceding inequality and can be satisfied. Formula
A) is here repeated. It is stated that %x*-\-yA—zi—Zui represents all
rational numbers.
P. Tardy" gave the generalization ton factors of 4ab=(a-\-b)*— (а—Ъ)*
and
This formula had been given by С F. Gauss.87
E. Rebout" noted that, in this formula, also 24abc is a cube if a=3,
6=4, c=6.
V. A. Lebesgue* remarked that every positive rational number is a sum
of four positive rational cubes:
D) «
where m* is a rational cube lying between n/0 and л/12, while
-. ffir Math., 9, 183% 288-292; Mem. &&**&* pftre divoe Savants Acad. R, Sc
lOostttut de France (M*tb PhyB.)f 5,1838, 71-5. In Comptee Rendue Paria, 10,1840»
Ш, tibri stated be bad proved the theorem in Ыя book, «Memoir» de Math, et de
РЬув., Fbrence, 1829,152-168.
«Annali di Sc Mat. Ym.f 2,1851, 287; c£. Noa*. Ann. Math., 2,1843, 454. Cf. Boutin.»
nW«ke, 11,1863,387. Of. H. BrorawJ, Nouv. Соггеч>. Math, 4,1878,136-8.
••Nouv. Aim. Math., B), 16,1877, 272^
"Ешвгбяхт d aoAlyee iram&ique, Farii, 1859,147151.

Chap. XXV] StJM OP FoUB POSITIVE CUBBB. 729
E. Lucas70 remarked that Lebeague* appears not to have guessed
what eeems to have led Euler to obtain formula D), viz., the problem to
express a number as a sum of two cubes. Any positive rational number JV
is expressible in an infinitude of ways as a product or quotient of two sums
of two positive rational cubes. To prove the former (which corresponds
to Euler^B theorem), employ the identities
* + (SLM -
Divide their product (member by member) by (/^Н-ЗЛ/2)8. Hence
is expressed as a product of two sums of two cubes. Take L=Bl?f
М<=2^^ЪГ^Аа%. We get a decomposition of ЛГ~2*ЗГ4£\ and we can
choose a'/b* to make all the cubes positive. As a corollary, E, Fauquem-
bergue7*1 proved that the quadruple and square of 4pe+27$* are sums of
two cubes, a problem proposed by Lucas.
G. Oltramare71 noted that every integer is a sum of five integral cubes.
R« Xorrie7* gave the identity
He expressed (p. 58), 5, 17 and 41 as sums of five integral cubes, not all
positive. Other solutions had been given by A. Cunningham.7*
v. Ami. Math., B), 19, 1880, 80-91; Bull. Soc. M*th. France, 8,1879-80,180-2. No
reference Ы m&de to Euler'e writingp. The author of tbie History has found do formula
like B) or D) in Eulert papers or books. Nor did libri* or Lebeegue* imply that
eufih & formula is due to Euler. The fact (hat Lebeegue вроке Ы &) *в the trans-
transformation of EuJer may bare led Lucaff to infer too ЬаяШу that abo B) is due to Euler.
"•Nouv. Ann. Math., B), 19,1880, 430.
711/iDterm&liaire dee math., 1, 1894, 25. a. 165-6, 244; 2,1895, 325.
71 University of St. Andrews 600th Annivertary Mem. Vol., 1011, 68.
19 Math. Quest, Educ, Типов, B), 4, 1903, 49.

CHAPTER XXVI
FERMATS LAST THEOREM. axr+by*=cz*. AND THE CON-
CONGRUENCE
For proofs of the impassibility of xn-\-yn=z* for n=3, 4, see Chs.
XXI, XXIL
Leo Hebreus,1 or Lewi hen Gereon (J288-1344), proved that 3*±1 *2*
if m>2, by showing that 3*"±1 has an odd prime factor. The problem
had been proposed to him by Phffipp von Vitry in the following form: All
powers of 2 and 3 differ by more than unity except the pairs 1 and 2, 2 and 3,
3 and 4, 8 and 9.
Femmt,* commenting about 1637 on Diophantus II, 8 (to solve
о*), stated that "it is impossible to separate a cube into two cubes,
or a bicjuftdrate into two hlquadrates, or in general any power higher than
the second into two powers of like degree; I have discovered a truly remark-
remarkable proof which this margin is too small to con tarn/' This theorem is
known as Fermat'a last theorem.
Claude Jaquemet' A651-172$), in ft manuscript in the Bibliotheque
Nationale de Paris and first attributed to Nicolas Malebranche4 A638-1715),
attempted to prove Fermat's last theorem. In a*=x'-\-y* we may suppose
x, у relatively prime. The quotient of x'-\-y* by x-\-y is
Then x-\-y and Q have no common divisor d other than factors of z. For, it
would divide
Adding 2yx^*-\-2y*z^\ we get ЪуЬ*-* . БЬаПу, we get zy*~K But
у is not divisible by d since xfy are relatively prime; hence г is. Similarly,
x—у and (x*—y*)/(z—y) have no common divisor not a factor of z.
Suppose that a, x7 у are relatively prime integers for which a*=z£-\-y*t
z odd. As just proved, at moat one of the powers is divisible by z. First
let x£ and y9 be not div&ble by z. Let x*=p'q*, y*=r*&*t where r and s
are relatively prime, also p and <?. Then a-pq=r\ а—Г8 = р*> Thus the
divisor p—r of p*—r* divides pg— r*. Dividing the latter by p~rt we get
the remainder p$—p* or rq—rs> neither zero, and "Ъу continuing this
process to infinity, we get no new remainders, so that p~r is not a divisor
of pq—rs>" As pointed out by E. Lucas4» the last conclusion is wrong;
♦ H, 8, Vandiver iad стйааШу the proof-eheete of tbi» chapter and believes that the roports
ал *ccmte' Both be and the author compared the report* with the original papeia
wben available.
1 Cf. JT. Ckrkbaeb, Bin. Heufelbf-Eg, Berlin, 1900, 62-4.
1 Отпев, I, 291; Fraich tranaL, Ш, 241» Diophunti Alexandrini Arith. Iibri «ex, ed,r S.
Fermat, Tofosae, 1679, 61. Precis <ks Oeuvrea matL de P. FeJtQatr par £. Braesinde,
Mem. Aead. 3* ToutaMe, (*), 3,1S53, 53.
' а. Л. Mane> Bull. BAL Stmia Se. M*t. Fk.r 12, 1ЛГ9, S8«^8M.
*Cf. C. Heniy, AttL, Wak8.
*• IbuL, £68, Suice be omitted the factor p before g-«j t*ke к to b« a multiple of p.
731

732 History of thb Theort of Ntjmbxbs. CGhap. xxyi
take any integer к and set p(q—«)—Jfe(p—r). Then pq—гз=(р—r)(e+fc).
The second case in which a' and x* are not divisible by г differs from the
preceding only as to signs.
L. Eider's* theorems on the linear forms of the divisors of a"db6" are
cited under Euler1»1 of Ch. XVI of Vol. I of this History.
Lagrange's10 method for тщ—А$ш=дш is given in Ch. ХХШ.
A. J. Lexell1 considered a?-\-fr=<?. Set x+y=<f, x-y=V. ТЪеа
Since the factors are relatively prime, x=7?t s*~4z*=g*. Hence
N. Fuss F noted that, if 1±4х*=П is possible in rational numbers,
r*_l_p«^g» would be possible in integers. To reduce the former to integers,
set x=p$li*; then i*u±4p*q* = O, say the square of r*-\-2v, where v is
prime to r. Then Tkp*<T=p(r*-\-v), whence v=pn, rw+p—qn*
L. Euler8 multiplied ап-\-Ъ***сп by 4an and added &m. Thus
Eulex* noted that he had failed in attempts to prove xn-\-yn=zn im-
impossible if n>2.
C. F. Kausler10 proved that xl-|-yi=^i is impossible in integers. For,
if possible, set x ^тя, where m is a prime. Of the forty cases, all are imme-
immediately excluded except two:
я*+£у*+у*=т*п9 or mn*, *-tf~l or »A
For the second alternatives, eliminate z*. Then
and m is a factor of 3^*, If у is divisible by m, z is, and 2, у, г have a
common factor. There remains the case m=3; then z-\-yy z—y are 3*, 1
or ZAj 3 or 39, 3J, cases readily excluded. The first alternative is excluded
by the lemma: There are no integers y, z for which
Sophie Germain11 A776-1831) stated in her first fetter to Gauss, Nov.
21, 1804, that she could prove that х*-\-уш=г* is impossible if n=p—1,
• Сопшь Arith., I, 50-6, 269; П, 533-5.
< EolePfl Open poetama, 1,1862, 23Ь2 («boot 1768).
ТЛ»Л»241 (about 1778). C£ Ealer •
•/ЫА, 242 (flbaat 1782).
♦Ли., 687; k«t№ to Ltgranee, МагсЬ 23, 2773L Group. M*lb. Ffaya. («d^ P, H. Fie),
1,1843,618,623, kttan to GoHb&uh, Aug. 4,1753, May 17,1755. Novi Gomm. Acad.
Petrop., 8, 1760-U 105; Сошш. Aritb. OdL, I, 39&
!• Nov& Act* Aead. Se. Pfetrop^ 15,1906, td шк» 1799-1802,1«-1$б.
11 The firrt And third letter «ere роЫЫю<1 in Oeuvree phfloeophiquee de S. Germaio, Pare,
1879,298. CSnq ktCns de Sophie Gamin h С F. СЬшт, раЬКбея par В. Boitoompacni,
Beriin, 1806, 7A pp. Reproduced in Aidav Maiib Phji., 65, 1880, UtL Baktt 250
pp. 27-31; 66,1881, Iitt Bericht 361r pp. 3-ia Beviewed, with Gam,11 by S. Gftn-
tber, Zeitadir. Math. Phjm., 36,1881, ШЛ.-JjL AM., pp. UKS6; Mm traneL, ВиП.
BibL Storift Sc MftL e РТв., 15» 1882,1744.

, xxvn Fermat's Last Theorem. 733
where p is a prime 8*+7. In her" fourth letter, Feb. 20, 1807, she stated
that if the sum of the nth powers of any two numbers iB of the form кг-\-п/*,
the sum of these two numbers is of that form. Gauss" replied, April 30,
1807, that this is false, as shown by lSP+&**=K*-\-lipt whereas
15+8+±4-11^.
С F. Gauss" gave a sketch of a proof of the impossibility of tf+b* +с*=0
and noted that the method is not applicable to seventh powers.
P. Barlow" proved that if n is a prime and хщ—y*=z* is solvable in
integers prime in pairs, then one of the four gets of conditions
x-y=r*
x—z=sn
r»
f
must hold. For, (xn—yn)f(x—y) is not divisible by a factor Ф» of x—y,
and if divisible by n, the quotient is prime to x—у and to n. Hence zn is
divisible by z-y, and, if n is a factor of x—y, by n(x—y)t while the quotient
is prime to n and to л— у. In the first case, z—y=rn. In the second caser
, y
His attempt to prove хш~уш=г* impossible if n>2 involves the error
fcf. Smith,*8 Talbot*4) that a sum of fractions ш their lowest terms is not
an integer if the denominator of each fraction has a factor not dividing
all the remaining denominators.
X. H, Abel16 stated that, if rc is a prime >2, а" = Ъщ+с* is impossible
in integers when one or more of the numbers а, Ь, с, а-\-Ъ, а-\-с, b—c,
a1'", blf*, clfM are primes [cf. Talbot77, de Jonquieres117]. If the equation
is possible, then a, b, с have factors x, y, z> respectively, such that either
[cf. Barlow15]
or values derived from the second set by permuting af Ъ, and x, y, and
changing the signs of с and z; or values derived from the third set by
replacing a by b, b by — cf с by a, x by у, у by —z, and z by x* Thus 2a
must have one of the three forms listed, where x, y> z have no common
factor* Finally, 2a ^9Ii+5Ii+4f*; the least one of a, hf с cannot be less than
(9л-5*+4*)/2. The editor, L. Sylow, remarked p. 338 that these theorems
appear to contain some inaccuracies.
* Published by E. Sobering, Abb. Gwell. Wise. Gottmgen, 22,1877, 31-32.
» Letter* medita di C. F. Oauas » Sofia Germain, publicata da В. ВопеотвдЫ,
1879. К*фгск1игЫ ш Archiv Math. Phys.t 65, 1880, Litt, Bericbt 257, pp. 5-9.
w WeAe, П, 1863,39(W, poeth. peper.
-AppeiJdii to En^iishtrtiuri. of Euler^a Algebra Proof "completed" by BaHow in
Nat. РЬЯ. Cheon. and Arts (ed., Nicholson), 27, 1810, 193, and reproduced in Barlow's
Tbeoty ^ NtimbttH, London, 1811,100-0.
18», 264-5; мат. &L, 2; 1881, 354^5; fetter to Holmbo* Aug. ^ 1823-

734 HiexoET or тик Theory of Numbers. [Chap. ~XXyl
A. M. Legendre1* remarked that the French Academy of Sciences had
offered one of its prises for a proof of Fermat's last theorem, hut without
awarding the prue» He considered xn-\-y*-\-zn=O for n a prime >2 and
for relatively prime integers x, y9 г each + 0. He noted (§§ 3, 4) that
x+y+z is divisible by n, and its nth power by (х+у)(у+г)(г+х)> by *
proof criticized and completed by Catalan.*1 Let
be the quotient of y*+z~ by y+г. Then (§ 7) y+z and Ф have the g.e.d.
л or are relatively prime according as x is or is not divisible by n.
First, let x be divisible by n. Then (§§ 8,10)
y+z=-an, ф(у, z)=na*, x=-a*>
n
W z+x=h% *(*, s)~0*, y= -bpt
x+y**cn, ф(х, у) =yn, *=-cy,
where a is an integer divisible by n, and each prime factor of or, fi or у jg
of the form 2An-hl. Each prime factor of a is of the form 2fri1-|-l (§ 11)^
and x, assumed divisible by nt ш divisible by n2 (§ 13), both results being
credited to Sophie Germain in the foot-note to § 22.
Second, let no one of the numbers x, yt z be divisible by n. Methods
applicable only in the special савев n=3, 5, 7, 11, but not to n= 13, etc.,
are given in §§ 14-20. To Sophie Germain is credited the proof (§§ 21-22)
that, if n ш an odd prime <100,
B) x*+y»+9»=Q
has no integral solutions each prime to n. This proof is called " very
ingenious, quite simple, and of an almost absolute generality." As noted
above, y-\~z is prime to ф{у> z)t and their product equals (—х)ш; hence we
may Bet
у+г=а% ф(у7 z)=an, x=—aa,
C) z+x=b*, ♦(«,*)-p«, V~-hfi9
x+y=c*t *(x, y)=y*, z=-eyt
whence
D) 2г^Ь|>-Ьс--л% 2y=a*+c*-bnt 2г=а*+Ъл-сп.
Theorem. If there exists an odd prime p such thet
E) r+if+r-O (mod p)
has no set of integral sohitions £, i\y f, each not divisible by p, and euch that
n is not the residue of the nth power of any integer modulo p, then B)
has no integral solutions each prime to n»
For, if z, у9 z are integers satisfying B), they satisfy congruence E)
во that one of them, say x is divisible by p. ТЪеп, by D),
fr4-c"+(-a)*=0 (modp).
q j у p r b ИЬЬшёте do FerniftL
AcmL IL Sc de HivtHdi de France, в, шппбь 1823, Pmm, 1827, 1-60, 8*m9f
рЬ m to pagntg, ТЪёсжю drt цшиЬгея, ed. 2,1823, moond euppfeiwnl, 8ерЦ 182S.
140 (npsodnecd in Spbmx-Ocxfipe, 4,1908, 07-128; шЫл, 5,1910,112).

Chap. XXVlJ FerB^at's LAST TheOKEM. 735
Hence a, b, or с is divisible b;y v. But if b were divisible by py then, by C),
y= -Ь0 would be divisible b;y p> and hence by B) also z would be divisible
by V> whereas xt у, z have no common factor. Similarly, с is not divisible
by p. Hence
a~0, x^O, zm-y, ф(х§ y)=y*-*§ 4>{y>z)=ny^ (modp).
Thus, by C), yn-y"^\ <x* zzny»-1. Hence ny»=a« (mod я)- By the
final equation C), у is prime to p. Hence we can determine an integer yi
sucb that r^i^ 1 (modp). Tbusn^fariI1 (mod p), contrary to hypothesis.
The theorem applies if n^7y p^29, since the residues of the seventh
powers modulo 29 are ±1, =tl2, no two of which differ by unity, and no
one of whkh is congruent to 7. Smulaily, for eafth odd prime n<100, S*
Germain gave a p for which the theorem applies.
The condition that n shall not be a residue of an nth power requires
that p be of the form топ-Ы, where evidently m is even. Legendre proved
(§5 23-28} that m must be prime to 3 and that both conditions in the
theorem hold if p^mn+1 is a prime and m^2, 4, 8, Ш, 14, 16 (but over-
overlooked the exceptional character of n = 3 when m=14, 16; cf. Dickson1").
He concluded that (l)hasno solutions prime to n forn an odd prime <197.
He proved18 (§§ 3&-47) that х*+у*+г*=О has no integral solutions and
that if solutions of B) exist for n=7, 11, 13 or 17, they involve a great
number of digits (§§ 29-37).
Schopis19 argued that, if a^_ys=u^ wnere xyw ie prime to 5, then
x-y=u\
ana
+
is a fifth power, say (u*+z)b. Thus
Thus z ifl divisible by 5 and the second member by 25. Thus A is divisible
by 5, which is seen to be impossible.
G. I* Dirichlet30 proved that there are no relatively prime integars
x, у such that х6=ЬуБ=2-5||Лгя, m and n being positive integers, пф2,
and A not divisible by 2, 5 or a prime Ш+l. With the same restrictions
on A, the theorem holds also if n=0, m^O, and 2-Л^З, 4, 9, 12, 13, 16,
21, or 22 (mod 25). If n>0> пф2, and if Л is not divisible by 2, 5 or a
prime 10fc-|-l, there exist no relatively prime integers x9 у such that
х*±уБ=5МД The last sho^s thet x*±if=z* ш impossible in integers
(since one of the unknowns, Say Zy must be divisible by 5); the proof is
analogous in the two cases г even and z odd, whereas Legendre1* employed
two methods.
"This proof waa reproduced in Legends Tbeoiie <Ы гютЬля, ecL 3, Hp 1830, tibe. 651-
663/Pp.361-8F Goman tfintf, by H. Maser, 1893 r 2, pp. 3524?. If г «the unknown
divtsibJe by St the proof for tbe «saee * even is like Diridtfet**." while that for г odd Ь
Ъу я вресЫ mxafym.
» ЕЫвв fiatwj Unbent Analytic, Р^,^^ СшвЫппеп, 1825,12-15.
»Joor. fur Math^ 3,1828, 354-376; Werke I, 21-46, Head at the Pirie Acad- 8fc,, July 11
■ttd Nov. Hp 1826 w»d printed pri^tely, Wftfke, Ip J-30. Cf. Lebeegue."

736 Histoby of the Theory of Numbers.
A. M. Legendre* stated that the discussion of B), at least for special
exponents n, can be facilitated by a consideration of the cubic equation
whose roots are xty,z\ for integral roots, the discriminant must be a perfect
square. He was not entitled to conclude that x+y+z and xyz are divisible
by n*, as he had not proved that one of the unknowns is divirabls by n*
V* Bouniakowsky* argued that if xm+ym+zm=O, where m is A prime
and x,y,z&re integers with no common factor, and if N is chosen so that
m=+(N)—1 (which is possible for each prime tn<Zl, except *n=13),
then xyz{xy+zz+yz) is divisible by N. But he used Eider's theorem
s*<Jf)sl (modiV) which is valid only when x is prime to N.
Mrichlet" proved by descent thet B) is impossible in integers for n = 14,
also the impossibility of
G» Iibri" considered the number Nt of sets of positive solutions <n
of **+уЧ-1=0 (mod n), for a prime n=3p+L The equation for the
three periods of nth roots of unity is found in the form
Comparing this with the known cubic, we get Nt~n±a-2, where
pin1"} Since a is comprised between zero and r = Dn—27I/1I we have
Vi^n—r—2. Hence Nt increases indefinitely with ft, and from a certain
limit od, я'-Ь^-Ы^О (mod n) is always solvable with neither x nor у
divisible by n. Having N%, we can find the number of positive solutions <n
of s'-rY+^-blsO (mod n).
If n is a prime 8tfi+l, so that n=at+1662 in a single way, the same
method of proof shows thet the number of solutions of з?+у*+1=0 (mod n)
is гс±6а—3, which increases with n. It is stated thet one can prove
[Pellet,1** ** Dickeon199, Cornacchia,^ Mantel577] that a limit to the prime
p can be assigned such that, after passing it, the number of solutions of
zn+y*+l=zQ (mod p) will always increase. Hence it is futile to try to
prove u"+v*=zn impossible by trying to show thet one of the unknowns is
divisible by an infinitude of primes.
E, Б. Kummer* considered xn-hy2X=2axl where X is a prime, and x, у, ш
are relatively prime by pairs. We may take у even. The third of four
possible cases is
ThiflifltheonlypoeaibilityifX^Sn+^orif 2X+lisftprime. If the initial
equation is solvahle in integers, so is r2X-h«ax=2g2x. As auxiliary to tha
Dombiw, ecL 3, П, 1830, art. 451, pp. 120-2; Gtfman trtiwL, Mutf, П, pp.
» M6m. Acad. So. St. Р#*Ыюъгк (Math.)p F), lv 1831,150-2.
« Jotzr, № ДЫЬ., 9V1832,390-3; Werke, 1,180-194. E^produced by ОноВДт pp. 164-7.
■• Jour, flip M*th^ »p 1S32, 270-6.
« Jour. Шг Math., 17,1037, 203-9.

c&a».xxvij Fbbmat's Last Theorem. 737
proofs, it is ehown* that if
and a±ft have a common factor, it divides the last term ±n(o6)(m^1)rtl and
hence is the prime n if a and Ь are relatively prime. Since the coefficients
n, n(n—3)/2, • ■ ■ are divisible by n, the exponent of the highest power of rt
dividing ап±Ъп exceeds by unity that in a±k
F, Paulet" attempted to prove Fermat'e last theorem, but concluded
without proof thet a=fi in acr=fa, where
In hie second proof he equated corresponding mimmands of equal sums.
G* 1лтё* proved that xT+y7+zJ=O is impossible in relatively prime
integers. One of the unknowns, say x, Is divisible by 7 (Legendre17)*
It is shown that x+y+z=7AP, P=»pp, where p, v, p, 7 are relatively
prime integers such that
He made use of the lemma (pp. 197-8) that [bouniakoweky*4]
Thus A must be a square B*. Then
2a=27B*P, 2a*+2ab=BD, аЬс
EUminatmg а, Ь, с, we get an equation whose solution is shown to depend
upon tbe impoeeible equation
For simplifications of this proof, see Lebesgue" and GenoccbL*
A. Cauchy25 reported on Lame's preceding paper and stated that his
leinimi is obtained by tekmgn=7 in the generalization that (x-\-y)*—xn—y*
is algebraically divisible not only by nxy{x+y) but also (if n>3) by
q=x*+*y+y\ and if n=6*+l by g*.
V. A. Lebesgue" simplified Ьатё'8м proof by use of the lemma that
is impossible in odd integers p, q, r, relatively prime in pairs, r +0, if a is
a positive integer.
Lebesgue11 proved that if X*-\-Yn=Z* is impossible in integers! then
»*1 is impossible.
Abo in Nour. Ann. Math., 7,1848, 289, 307-8.
О Mih (fid., A. Qoeteki), 11,1839, 807-313.
iwParie, 9,1839,45-6; Jour. <fo Maih., 6,184OP195-211.
vante Acad. Sc de 1'Inrtfttrt de SVanoo, 8, 1843, 421-437.
Rendue Piidfl,^ 1830, 359-363; Jour, do Math., Бр Ш0, 211-6. O*uve* de
Cwicty, A), IV, 49Я-ЫН.
"Jour. deMjith.,6, l&W, 276^p 3484) A^яют^ of <AecurityfOptx)of<rf lemma).
»7W., 184^5,

738 History of the Theory of Numbers. [Chap, xxvi
X, Iiouville" noted that if un+v*=wn m impossible in integers not zero,
then г1*—у**=2хл is impossible.
Cauchy**expressed {x+y)* — xn—yMin terms of
for n odd ^13.
V,. Bouniakowsky** proved for m=2, 3, 4, 5, 6, 7 that
is impossible if R is rational and the radicals irrational. For m=7 set
C=(ABI17. We get ^-Л-^ = 7ЙС(^-СI, which implies the lemma
of tame21 (Cauchy*)* For, by Betting A = a7, В **b\R=*a+b, С **ab, we gpt
(а+Ъу-а7-Ъ7=7аЬ(а+Ъ)(а1+аЬ+ЪгI.
E. E. Kumrner3* eubmitted to Dirichlet about 1343 the manuscript
giving what he then believed to be a complete proof of Fermat's last theorem.
Dirichlet declared that the proof would be correct if it were shown not
only that every number an-\-aia-\ |-ax_iaA~l (where a is a primitive
Xth root of unity and the a's are ordinary integers) is always a product of
indecomposable numbers of that form, as shown by Kummer, but also
that this were possible in only one way, which is unfortunately apparently
not the case.
Frizon* announced a uniform process applicable to prime exponents
V* A. Lebesgue*7 supplemented Dirichlet's* results by proving that,
if A has no prime factor 10m-f-l and no factor which is a fifth power,
s*-|-y*=AB*u8 is impossible in integers if Л is a multiple of 5, or if Л^±2,
гЬЗ, гЬ4, ±6, =Ь8, =Ь9, ±11, or dbl2 (mod 25). A like treatment is
apparently not applicable to the remaining cases Л = ±1, ±7 (mod 25).
The equation 31*±yw=ilz* is impossible if A has no prime factor
As auxiliary propositions, а*=!>4-|-50Ь$са-|-125с4 is impossible, while
which can be reduced by descent to the case in which Ь and с are odd, is
impossible if c=5'2*-A*.
E. Catalan88 expressed his belief that хш^ул = 1 holds only for 3l—2* = 1.
S. M. Drach1* argued that хп+у*=г* is impossible in integers if
n=2*i+l>L For, by Eider's Algebra, 2, Ch. 12,
Take a=z,
" Jour, de Math., £, ,
»Eierciceedpun*lj»ef!tdephyB.mialLp2, 1841, 137444; Oeuviee, B), XIIP157-166,
u Mem. Aced. Sc St. P^tefBbomg (Miitb), F), 2, 1841, 471-192. Extract ш Bull. St.
Fften., Щ 1-2.
■ K. Hened, Ged£ebtnkrede auf E. E, Kummer, AbL GeedL Math. Wit*Lt 29, Ш0, 22.
CCf. the lees technical nddre» by Ветве!, К. £1 Ешплмг uod der groese Fernmteobe
SftU, Marimrser Akademtebe Ratal, l910r No. 23.3
^ Camptca Bendtn Pane, 16.1843, 501-2.
"Jour, de Math, 8» 1843,49-70.
" Jonr. fur Math., 27, 1844, 192. Nouv. Ann. Math,, lv 1842, 620; <2), 7, 1368, Ш (n-
pwted by К lionnet). For n-2p Lebeegue» of Ch, VL
w London, Edmbargh, ОаЫш РЪЛ. Mag., 27,1845, 286-9.

Chap. XXVI3 FsRMATf8 LAST THEOREM. 739
Then
Then Z\z~ and Yfym give
From the sum and difference of the resulting values of p *fz±q >£,
9 y
Developing the difference of the two members by the binomial theorem,
we get ft series in yjz with every coefficient negative if n>L Next, the
case n m is treated at length.
C. G- X Jacobi40 gave a table of the values of m* for which l+gm=g*
(mod p)f where p is a prime ££103,0^191^102, and q is a primitive root of p.
O. Terquem41 proved the theorem of Lebesgue11 and the corollary of
Liouville*1*
A. Vachette** noted that zm~yn=(xy)* is impossible in integers. For
p=mn, set z*=(xy)n and take n=wi. Thus zm~ym**zm is impossible if z
is a power of xy.
J. Mention41 proved the formula [cf. Kummer*J:
F)
У. A. Lebesgue44 obtained F) by applying Waring's formula to the
quadratic equation with roots а, Ь- Applying it to the cubic with the roots
ct, &: y, we get (of-hj5-h7)m. For n=7y the latter result is said to have been
employed [in papers 2S-30J to prove the impossibility of x7+y1=z7 by a
method simpler than that for exponents 3 and 5.
G. Lame45 claimed to have proved that, if л is an odd prime, £*+#•=*•
is not satisfied by complex integers
G) <h+alr+ ■ -H-ft-if-1,
where r is an imaginary nth root of unity and the a's are integers.
J. Liouville* pointed out the lacuna in Lame's proof that he had not
shown that a complex integer is decomposable into complex primes in a
single manner.
Lame (p. 352) admitted the lacuna and believed (on the basis of exten-
extensive tables of factorizations) that it could be filled; he affirmed (pp. 569-572}
that the ordinary laws for integers hold for complex integers when n=5>
« Jour, fikr Math,, 30,1846, 181-2; Werke, VI, 272-4.
-Nout. Ann, Math,, 5, 1846, 70-73.
ш Ibid., W-70.
«Nouv Ann. Math., 6, 1847, 3W(propoeed,iJ, 1843,327; 18,1859,172,249).
«/6&, 427-431.
« Comptee ReDA* Pbrta, 24,1847, 310-5,
•7Ы*, 315-6.

740 Hibtobt от the Theory or Ntjmbebs. CChap, xxvi
Lame stated (p. 888) that Fermat'a equation is impossible for a series of
exponents including я=5, 11, 13.
Lame47 presented hie arguments in two long memoirs.
Ov Ter^uem" suggested a subscription to Lame for bis* proof (!)
declaring it the greatest discovery of the century in the mathematical world.
E» E, Kummer" pointed out the falsity of Lame's4* assumption that
every complex integer can be decomposed into primes in a emgle way.
L. WantzeF proved thet Euclid's g.c.d, process holds for complex
integers a-b&V^l [а^^^У proved by С F, Gauss61] and for complex
integers formed from an imaginary cube root of unity, and stated that a
like result holds for complex integers G), with n arbitrary, since the norm
(or modulus) of G) is < 1 when a0, ■ ■ •, a*-i are between 0 and 1 [erroneous,
у}
A. Cauchy" showed that the final statement by Wantzel" is false for
n=7 and for any prime n=4m-|-lsl7. He pointed out lacuna in the
proposed proof by Lame44 of Fermat's last theorem. He denned the
factorial of G) to be its product by the complex members obtained from it
by replacing r by the remaining primitive nth roots of unity, and obtained
upper limits for such factorials [norms} He" proved that any common
factor of Мц=Аг*-\-В and Af* divides Mй if A and В are relatively prime.
Caucby*4 attempted to prove the false theorem that the norm of the
remainder obtained on dividing one complex number G) by another can
always be made less than the norm of the divisor. He concluded (falsely)
that a product of complex integers G) can be decomposed into complex
primes in a emgle manner, and that the other laws of divisibility of integers
hold for these complex integers.
Cauchy" noted (erroneous) conclusions which follow from the assump-
assumption that his preceding theorems hold for a given number n; in particular,
errors relating to the factors А+т*В of Ал+Вп. Не promised to discuss
later the objections which can be raised against proofs in his preceding
paper.
Cauchy* further developed the subject and admitted at the end of his
final paper that his" basal theorem is false, failing for я** 23.
Cauchy87 obtained results moat of which are included in Kummer's
general theory. In the fifth paper, p. 181 (Oeuvres, p. 364), he stated that
а*+Ъп+сп=О is impossible in relatively prime integers not divisible by
* Jour, dc Math,, 12,1847,137-171,172-184.
« №wv. Ann, Math., 6,1847,132-4.
» Oomptee Rendus Fbrfa, 24,1S47, «Ю-flOO; Jour, de Math., 12,1847, 136.
» Comptee Beadufl Fans, 24,1847, 4ЭСМ.
« Сотш. Soc. Sc. Gotting, Recentioree, 7,1832, &4fl; Werke, П, 1863,117, German traneL
by H. Maaer, Qanstf Unteraichungen fiber hChere Arfth., 1889, 656.
"Comptee BendiM Pans, 24,1847, 460-481; Oeuvree, <1), X, 240-254.
"Ibid.. 347-S; Oeuvn», A), X, 224-6.
NJbtf., 616-628; Oeuvwe, (l), X, 25±-26&
"Ibid., 578^584; Oeuvne, A), X, 268-275.
"Ibid.. 633-6, 661-6, 996-9,1023-30; Oeoyies, (l)r X, 276-286, 296-308.
"/W*, 2fi, 1847, 37, 46, ИЗ, 1Й2,177, 242, 285; Оеита, a), X, 334-351, 354^371.

Свар, xxvi] Format's Last Theorem, 741
tha odd prime л if , „^ , л*-* ■ _l
is not divisible by n £i* e., if the Bernoullian number £(^«/1 is not divisible
by пЗ, or if a certain number « (p. 359) is prime to n* Cf. Genocchi,"
КшшпегЛ*
E. E* Kummer58 proved that x*—y^^is impossible for theeenes*
of real primes X for which (A) the number of non-equivalent ideal complex
numbers formed from an imaginary Xtb root a of unity is not divisible by
X and (B) every complex unit B(a), which is congruent modulo X to a
rational integer, equals the Xtb power of another complex unit. These -two
conditions are satisfied if X=3, 5, 7, but probably not for X=37.
G. L. Dirichlet40 noted that Kummer's condition (A) relates to a theory
closely analogous to the fact that a number m for which X> is a quadratic
residue is not always represented by x* ~Dy*, but by one of several quadratic
formsl and similarly for the forms in X^l variables defined by norms of
complex integers based on <*.
Kummez*1 proved that, for the domain defined by an imaginary Xth
root a of unity, where X is an odd prime, the number of classes of ideals is
the product H=kxht of the two integers
P , D
where ц «* (X—1)/2, and P, Д Д are defined as follows. Let 0 be A primitive
root of 0*~* = 1, and g a primitive root of X. Then
where gi is the least positive residue of gi modulo X. Next,
is a complex unit (a divisor of 1). Then, if h denoted the real part of log x,
feC«)
d. Wim. Bcrim, 1847,
** "I jm>ve that it is impossible for an Infinitude of prime* X, but do not knew for just which
A'e the aasumptioDe hold." That tbeee X^e are infinite ic лш&Ьег тгая belfeved, but not
proved, by Kummer. He called the remaining primee exceptional (л& 37r etc.), Tbe
name etatemente were made m 1847 in letter to Кгопескег (Кипш№Ггм pp. 75, 84).
In bv Voriwaneen fiber Zablenthooife, 1, lfi01v 23, Xronecker stated that Kmnmer
proved the impoeedbflftу of *x+jrx-t*xforan infinit«deof primes X and at fiwt believed
that hie proof applied to nearly all Xra, bat later believed the oontrary. Кшптет* р,
32, ifl ebewbere quoted ae believing it probable tb*t there am шррюшалШу us many
regolar primes ae irregular (exceptional) pnmee. A. Wieferfcb* TfiudtestKicb fur
Mathematuer и. Fbywker, Leiptis> % 1911, lOfr 111, stated thftt Кшшпег proved
Fermat'a last theorem for an mfinite eedos iff «xponeotfl.
••BerichteAkAd.Wiea. Berlin, 1847, 13P-141, W«te, П, 254-^,
* Beridrte Abd W«l Bcrim, I847r 806-319. Seme in Jour, fur Matb., 40,1SS0, 03-138;
Jour, de Matlb, 16,1861, 44tt&

742 History of the Theory op Numbers.
Let ci(a), <»<, e»-i(a) be units such that products of powers of them
multiplied by ±аш give all the units. Then
| Ui(a)
A=
It is shown that Ai is divisible by X if and only if X divides the numerator
of one of the first (X—3)/2 Bernoulli&n numbers 2?i = l/6, 2**=l/30, ..-;
while if ht is divisible by X also hi is, but not conversely. He proved that
if X is not a divisor of H, condition (B) of Kummer" is satisfied. Hence if
X is an odd prime not dividing the numerator of any one of the first (X—3)/2
Bernoullian numbers, хл+ух=гл is impossible in integers.
The French Academy of Sciences*2 offered as a prize a gold medal of
value 3000 francs for a proof of Fermat'e last theorem. After several
postponements of the date fixed for the award, the prize was finally (C. R.,
44, 1857, 158) awarded to Kummer for his investigatione on complex
numbers, though he had not been a competitor.
Kummer48 proved by иве of prime ideals that, if X is an odd prime not
dividing the numerator of any one of the first (X—3)/2 Bernoullian numbers,
**x=0 has no solution in integers! nor in complex integers
where a is an imaginary X-th root of unity. Thus there is no solution for
X<100, except perhaps for X=37, 59, 67. This proof has been given In
modern form, by use of Dedekmd's ideals, by Hilbert.151
A. Genocchi" proved that, if n is an odd prime,
* (modn)
and noted that this, in connection with a statement by Cauchy,*7 shows that
х*-\-у*+гщ*=0 is impossible in integers not divisible by the odd prime n
when n is not a divisor of the numerator of the Bernoullian number B^n-tytf
the last one of the Bernoullian numbers in Kummer*e condition.
Kummer*6 noted that his assumption that Bn is not divisible by X for
n= (x—3)/2 (as well as for smaller n'&) corresponds to CauchyV* condition
+0 (modX).
If not both £(x-i>/t and £<x^»)/i are divisible by X, one of the solutions xy y,
of ix-hyx=«x must be divisible by X. Proof by Kummer,78 pp. 61-^5.
« Compt4» Rendus Ibrie 29, Ш9{ 23; 30, 18501 263-4; 35.1*52,9№Ю. There were
compctmg zDeznoin for the pn*e proposed for 1850 ana eleven for the postponed р
for 1853, btrt поле were deemed worthy of the prise* (X Near, Ann. Мш., 8, 1849,
ЗЙЗ-3 щи!, for ЫЫк«га1)Ьу, 363-4, 9r 1850, 386-7.
ttJ<HB4 fur M*LLp40, 1850, 1Э0-& (Щ; Joar. de M«tb.v 16P 1861,488-98. Reproduced bf
Gftmbioti,'71 pp. 169-176.
•* Annrnb di Sc MkL e F»., 3, 1852, 400-1, Summuy m Jour, fur M*tb.p 99, 1886, 31*
Thia ecmgmenc» ie a epetial сабе of one prored by Conchy, M6xl Acad, S& Рдпв, 17ц
1840,205; Oteraia, О), Ш, p. 17.
• Letter to L. Кпэвеско-, Jan. % 1852» Kunmtcr^p. 91.

xxvi j Febmat's Ьаят Theobbm* 743
H. Wronski" pretended that the impossibility of x*-\-y4=z*, n>2,
follows from hie67 results on**—Nv*^Mu*.
F* iAndry48 proved Legendre's17 statement for р*=тп+19 *n = 10 and
14, when n>3, noting-thet A47ztl)/A4ztl) are primes,
Landry49 employed two primes ф and в=21ф-\-1, and an integer с
belonging to the exponent Ф modulo $. The congruence
(mod &) can be reduced to 1-|-€±е'=0ип1еввз^^огя=|
By use of tbe subetitutions «=«Г1, «=«}'*, etc., we can reduce
to a similar congruence with z replaced by the integral residues modulo ф of
1 s-J. _1_ _*_
*' *' z' z ' 1-z9 z-V
Excluding 2=1 or 2, these eix expressions are incongruent modulo ф unless
ф is of the form 6J+1 and then they reduce to two for two special values
of z. If all three relations 1-H—«*=0, 1—«_|_С'=0, 1—«—1*=0 are im-
impossible for a single one of the above eix values, then 1-N— ^=0 is im-
impossible for all six.
For Landr/s third memoir (on primitive roots), see Vol. I, p. 119,
p. 190 of this History; for his fifth memoir (on continued fractions), see
bindry* of Ch* XX above.
iAndry™ recurred to the exception arising if 2*2= ±1 (mod B), where $
is a prime 2кфп+1, п a prime >2. For ^=5, 7, 11, 13, 17, 10, he found
all the cases in which 2*^1 has such a factor в. For exampls, if 0=11,
only when n=31, 9=683. Aside from these exceptions, l-h«±^=0 does
not hold for х = ф or z=0 when 4^19; nor for **=2, J, —1, orz=3, 1—3,
J, etc., except for a few special values of Л
Landry71 proved that, if $ is a prime 2кфп+1(п>3), l+«±«'=0
(mod &) are each impossible for ^=5, 7,11, 13,17,19, aside from the excep-
exceptions for 0 = 11, 13> 17 noted by Landry,70 and the new exceptions, aris~
ing for 0^19: 0^761, n=5, k~4; $^M77 n**17, k = l; 0=419, n*=ll,
H. E. Heineria considered Pm—DQ^^=1J where P, Q, D are polynomial
b« Calzolari12 noted that any given numbers x, у, г can be expressed in
the form x=*v-\-wy у=«-Ия, z=tt+tr-j-u? [since we may take u=z—z,
« Veritable science oauttque dee тдгёее, Рдгё, 1853,23. Quoted in I'intermediaire d«e oiath.r
23, 191», 231-4, and by GuimAraee^
«7 Rlfonoe du sayoir ЬидшЬ, 1847, 242, Sra p. 210 of Voh 1 o! this History.
•• Premier mdmoire eui fa. theorie dee Dombree. DemoneiAtion d4m prioripe de Legendre
rdatif an theorime de Fermat, Farisr Feb. 1853,10 pp.
" Веилёше memorie eur I& theorie dee oombiee. ТЬбог^ше de Fennatr Paris, July, 1853,
18 pp.
r° Qufttritme шбтопе виг la throne dee uombree. ТЬёотёше de Fermat, Paris, Feb. 1855,
27 pp.
*£ixi&me шётопе ww Li theorie dee nombree, ТЬёогёте de Fermat, У livre, Paris, Nov.
1856,24 pp.
«•Jour. fQr Math., 48,1854, 256-9,
* Tentativo per dimoetrare 3 teorema di Fermat . . . x*+y*-z*t Ferrara, 1856. Extract
by D. G*mbloli,^ 168-16L

744 HlBTOBY OF THE THEOBY OF NUMBEBS. [Сна*. XXVI
v=z~y, w=z+y— t}* *** *m4-j/*=**, and set х=г—щ y=z—v. Then
Hence u*+v* is divisible by z. Similarly, <*<=«*-f-(«—u)m is divisible bys,
and £—t?*-f-(u—t?)m by p. His argument that Fermat's equation is im-
poesible if n is odd and >3 is unsatisfactory. By Cotes' theorem,
where X = l, 3, 5, • • •, n—2. The Xth quadratic function has the factors
u+v±2tIuv cob Xx/Bn).
But ttw+»*bae the factor z=u-f-y-f-u>, whence
Similarly, since ais divisible by x ^u-\-(v—u)+v>sBXid 0 by y=v+(u—v)+v)t
t л г^——ч А х
2V( )
whereas the one is real and the other imaginary. He also claimed that the
first и is symmetrical in u> v, while the third w is not. He made also the
error of assuming that an even factor of a product of an odd by an even
number must divide the latter.
J. A. Grunert7* proved that, if n > 1, there are no positive integral values
satisfying x*-\-y*=z* unless x>n, y>n, simultaneously. Set z=x+u and
apply the binomial theorem; hence yn>nxn~iur
L, Calsolari74 considered a triangle whose sides are integral solutions of
x~+yn-z*,nodd >1. Thus z*=x*—oxy-\-yt=P2{vr a suitable value of a.
It is stated that the polynomial P*^z*-\-y* is divisible by P2, the polynomial
quotient P^t is divisible by Рг, etc., and finally the symmetric quotient
-Pi=3-h# equals z, which is impossible. If n=2m, P?^P», a=0, m^l,
G. C« Gerono74 considered the integers x, у for which ax—bv=l for
primes о, Ъ. If a>2, then 6^2, а^2ж-\-1, andx=l, y=n when n>l, with
also *=2, j/=3 when n=L If a=2, then b«2«-l, a;=n, y=L
E. E. Kunimer78 proved that for алу relatively prime integral solutions
of xx+i/x=2x, where X is any odd prime, and xyz is prime to X,
(8) Лоь-здР*, У) ^° С™0* х) fr^3- fii ' • •' x-2)>
where Б,- is the jth Bemoullian number and Pt-(^; ^) is the homogeneous
polynomial of degree i for which
He proved that Fermat's equation is impossible in integers for odd
prime exponents which satisfy the following three conditions:
»АгеЫт Math. ГЪун., 26,1856,119-120. Wrong inference by Land»*11 p. 54.
« Aimali di Sc, Mat. * Yis., 8» IS57, ЗЗ^чНб.
л Noav, Aim. Mitb, 16^ 1857, 3M-8.
«Abh. Alout W^B«Jm(MatlL.),fv 1857,1858,41-74. Extract in Mcmafeb. Ak*I. Wk*.
ВШ 1857, 275-83.

Chap. XXYIJ FORMAT'S Ъа&Г TheoBEM. 745
(i) The factor Ai of the class number H is divisible by X, but not by Х*„
(ii) For/*, 0, e(a) defined a» by Кшшпег61, and for the integer *»<(X—l)/2
such that Bw^0 (mod A), there exists an ideal with respect to which as
modulus the unit
jb not congruent to a Xth power, whence the second factor h* of H is not
divisible by X.
(Ш) The Bemoullian number BrX is not divisible by X2.
All three conditions are satisfied when X=37, 59, 67, the values <T100
for which he had not previously proved Format's theorem. P^ut Kummer
(pp. 46^50) repeatedly used an earlier*6* congruence involving logarithms
which is not true in all cases, as noted by F. Mertens-7" The remark that
this error vitiates also the present paper, and two farther criticisms
were made by H. S. Vandiver/*5 First, Kummer (p. 42, bottom) relied
on his paper in Jour, de Math., 16, 1851, 473, where he reduced fti modulo
X, but not modulo Xm, п>1, ав now needed. Second, Kummer (p, 53)
employed a decomposition of Фг(<х) which holds only when it contains only
ideals of the first degree. Although the theorem on p. 61 is really subject
to this restriction, it is applied (p. 67) to ideals Ot(a) which are not proved
to be of the first degree* Кшшпег,7*1 р. 120, had given the different de-
decomposition when there occur ideals not all of the first degree/)
H. F. Talbot77 proved (I) If n is odd >1, ащ=Ъ*+сп Js impossible m
integers if a is a prime ^AbelMJ; (П) If n is any integer >1, and if an=bm—cm
is possible when a is a prime, then b—c=l. For (I), F-hc)n>b*-hc*^afl,
Ъ+с>а\ Ь<а9 с<а9 Ъ+с<2а. Hence Ь+с is not divisible by the prime a,
contrary to the given equation. Similarly for (II). Generalizations are
given. If a is a prime and m<n, a**=h*+c* is impossible if n is odd, while
а~=Ъп—с* is impossible if &—c>I.
K. Thomas*8 attempted to prove Fermat's last theorem.
H. J* S, Smith79 gave numerous references on Fermat's last theorem,
noted that Barlow's1* proof was erroneous, and reproduced the proof by
Kummer" for regular primes.
A. Vachette80 proved F) and concluded that, if а, Ь are integers and n
is a prime >2, (a+&)m~am—Ъ* is divisible by яаЬ(о+о), and gave several
expressions for the quotient* Set
Then Aim+i is proved divisible by (a1—1)* [Cauchy*]. There are proofs
(pp. 264r-5) of F) by induction on n and by Waring's formula*
F* Paulct^ gave an erroneous proof of Fennat'g last theorem*
*• Ктпшег, Jour, for Math-, 44, 1852,134 (error, p. 133).
~SiUuii«sber. Akad. Wias. Wien (Matb.), 126,1W7, Ila, 1337-43.
**< Proc Nutioaal Ac»d. Sc., April, 1920.
* Trana. Hoy. Soc. Edinburgh, 21 1867, 403-6.
n Dad Prthagoriische Dreieck und die Ungende Zahl, Berlin, 1859, Ch. 10.
71 Report British Лаос for I860, 14&-152; СоП. Math. Р&регв, 1,131-7.
w Nouv. Ann. Math., 20,1861,160-6.
* Совами, 22, 1858, 3S5-9. Correction^ p. 407, by R, Badau.

746 History or the Theoby of Numbers. CChap. xxvi
L. Cabolari*1 attempted a proof, etarting as before."
P. G. Tait* stated that if хш=уш+гш has integral solutions when m is
an odd prime, then x^y^l, 2=0 (mod m).
H. F. Talbot84 noted that Barlow14 made the same error in his proof of the
impossibility of z*—yn—г* as for the case n^3 (p* 139), where he stated
that, if r, s, i are relatively prime in pairs,
since each fraction is in its lowest terms and each denominator has a factor
not common with the other denominators, and hence the algebraic sum of
the fractions is not an integer (by the f&lse Cor. 2, Art. 13) ♦ On the con-
contrary, we have
23^35^25 e
A, Genocchi86 abbreviated Lame's58 proof for n—7. Let z, у, г be roots
of !?—pt?-\-qv—pq+r=0. Then sr-H/'+*r=O is equivalent to
After excluding the trivial case p=0, we may change q to pfy r to pV,
and get 7r*—7r(l—^+^) = — 1. The radical in the expression for the root
г must be rational. Thus (l-g+g1I/4^1/7 м a square. Set 2q-l~*lt.
Then
Proof of the impossibihty of the latter is not given.
Gaudinw attempted to prove that7 if n is an odd prime, (x-\-h)*—z*=z*
is impossible in rational numbers. Treating x/h as a new variable, we are
led to the case h=h To avoid the author's complicated formulas, take
n = 5. Then
is of the form НМ-И. Since г5 is of that form, г = 10*-И and
1,
which is said never to equal the first expression. Hie remaining two argu-
arguments are trivial,
I* Todhunter87 proved Cauchy's29 theorem and that, if g
Ъ=ху(х+у),
<Tr ^(«-rl)(*r2)-(iii3r+l)
m+ q °
r
2m ~m+ Br)!
* AimaK di Mat., б, 1Ш, 280-6.
« Froc. Roy. Soc. Edinbuigh, 5,18634, 181.
■* Trane. Roy. Soc. Edinburgh, 23,1864, 45-^2.
«AudaH di Mat., 6,1864, 287-8.
"Comptee Hendrn Perie, 69,1864,1036-8.
" Theory of Equation^ lB6l, 173-6; ed. 2,1867,189; 1858,186, 18S-9.

Chap. XXVI} FbRMAT's LAST ThEOHEM. 747
summed for r — l9 2, • • •♦ The first formula had been given earber.*8
Housel** proved Catalan's8* empirical theorem that two consecutive
integers, other than 8 and 0, can not be exact powers [with exponents > 1].
E, dataian* stated this theorem and those given under Catalan.1Me
Catalan** set p^x+y+z, Р=ря^х*—уп~гя and proved that the
quotient Q of P by ($-\-у)(у-\-г)(г+х) is (for n odd >3)
If n is a prime the coefficients of P and Q are divisible by n. Also,
where ^ is д polynomial in ^ у, г with integral coefficients,
G. C. Gerono** proved that, if x or у is a prime, xm=yn+l holds in
positive integers >1 only when x=n=3, y=m*=2. See Carmiehael.*2*
A. Genoechiw stated that х*+Ьх*у*—у*17=г? is impossible in integers.
Hence zT-\-y7+z1=Q is not satisfied by values of xy yf z which are roots of a
cubic equation with rational coefficients, a generalization of Lame's28
theorem.
E. Laporte" would deduce Fennat1» last theorem from the fact that the
series of powers higher than the second are formed by the summation of
terms of arithmetical progressions preceded by extraneous terms.
Moret-BlancM proved that the only positive integral solutions of
are y=0; y—ly x—2\ y=2, x=Z. A, J. F. Meyl* showed that the only
positive integral solutions of (x+l)*^^^1^-! arez=0, з— y^l, x*=y=2.
u N. M- Ferrets and J. S» Jackaon, 8olutione of the Cambridge SeDate-Houw Problems /or
1848-ДО1, pp. 83^S5.
» Catalan's Melanges Math-, Liege, ed. 1, 1868, 42-48, 348-9.
**IUd.t 40-1; Revue de lJinetructioii publique en Belgique, 17, 1870, 137; Nouv. Corresp.
Math,, 3,1877, 434- P^wfe by Soona™
11 Melftn^e Math., ed. 1,1868, No. 47,196^202; Mem. Soc. Sc. liege, B), 12,1885,179-1S5,
403. (Cited In Bull, dee ее. math. «rti\, B), в, 1,1882, 224.)
" Nouv. Ann. Math., C2), % 1870, 469-471; 10,1871, 204-6.
* Comptctf Rendm Paris, 78, 1874t 435. Proof, 82,1876, 910н$.
M Petit e«ai but quelquee method*» probables de Format, Bordeaux, 1874. Reprinted in
Sphinx-Oedipe, 4,1909, 49-70.
* Nouv. Ann. Math., B), 15, 1876, 44-6*
"IbuL, 546-7.

748 History of the Theobt of Numbeks, [Chap, xxvi
F, Lukas* set y=x—a, z=x—b, a<bt in yn+z*=xn, n>2. Hence
-, wn be ite roots, all positive. Then
which are said to be impossible if n>2. This error was noted in Jahrbuch
Fortschritte der Math., 7, 1875, 100.
T. Pepinw proved that хг-\-у7-\-гг=0 is not satisfied by integers not
divisible by 7, by uae of the fact that u*=x4+7*yA has no integral solutions
with v+0 (proved by descent). He proved (pp. 743-7) that the first
equation has no solution in which one of the unknowns is divisible by 7.
J, W. L. Glaisher* expressed Cauehy's* theorem in a new form. Let
n be odd and set ж=с—Ъ, у=а—с. Then
Then En is algebraically divisible by Ег=*Ъху{х+у)> If n=67rt=fcl, En is
divisible by Еъ=2(х*+ху+у*). If n=fon+l, Ещ is divisible by Et=i£l.
Glaieher100 expressed (x+y)n—xm—ynt for n odd ^13, in terms of
Р=з?+ху+у* and 7— xy(z+y). [E&rh&r by Cauchy«wJ
T. MuirlW noted that x, y> -x—y Are the roote of to1—
Hence by Waring's formula for the sum of like powers of the roote,
(—2H—8) ,
P ^"^ 1-2-3 P ^
He gave a similar formula for (s+y)*m+^M+i^m* For three variables, set
0=?&+Хху, 7^2х^+2хуг, 6=жуг(х+у+2).
^, j/, z, —z— y— * are the roote of w4—0i08-h7«'~^=0. Thu^s
summed for all integral solutions ^0 of 2г+38+4*=*2т-|-1, Since
the sum has the factor y=}i(z+y+z)*—x*-tf—J}.
Glaieher1*1 noted that Newton's identities give a recursion formula
for x*+ • * * +oC* extended Cauchy^s theorem to negative exponente, and
gave recursion formulas for and factors of the sum of the pth powers of all
the quantities ±а1=*:*-*±а» in which r of the signs are negative,
» AnAIr Meth, Ицт^, 58,187Й, КЮ-112.
« Compte» Benduu Peiv, S271876, 676-9.
w Qaar. Joor. Math,, 16, 1878, 365-6.
'« Мзд»едвег Math., 8,1878-9, 47, 63.
« Qaar. Joor. Math., 16,1879, ft14
^/Wd8&^8

. XXVI] FeRMAT's LAST THEOREM- 749
A. Desboves1" noted that aXm+bYm=cZ* has integral solutions if
and only if с id of the form ахш-\-Ъу*\ we can find a function с of а, Ь and
as many parameters aa one pleases such that integral solutions exist.
Next, let n=tn. Then we can find a, bt с so that there are two seta of
solutions and these determine a : Ь : с. There exists such an equation
with three sete of solutions if and only if
P^+QT+RT^U^+V+T-, PQR^UVT
have integral solutions +0* We can solve X4m—a*Y*~=:2х if
viz., by X«
A. E. Pellet104 considered, for p a prime, the congruence
Atm+Bu«+C=0 (mod p), ABC±0 (mod p).
Let J be the g,C.d. of m,p~l; ^L that of n, p—I. Set*s{". Then x must
satisfy the two congruences
Conversely, to each of the p common roots of the latter two congruences
correspond dd\ sets of sohitions of the proposed congruence, which therefore
has pddi sets of solutions. For m—n=2} the two congruences have at
least one common root, since the second is not x{r~1}f*-\-l=Ot being of
higher degree. Hence AP+Bu*+C=0 (mod p) is solvable (Lagrange,9
etc., of Ch, VIII).
JL Iiouvillc1* claimed that X"+Y*=Z* is impossible if n> 1 and X, У,
Z are polynomials in a variable L Set <x=XfZ< Then
is a polynomial in Tll—<xn=*YfZ. Since dU{dl is the argument of the
second integral,
„()—*(
must be the product of Г by a polynomial A. Hence
Thus X^1 divides Z4(Y/Z). Call the quotient В and set P=* Y/Z, Then
But in the latter, the left member is infinite for a root of P" = 1, while the
v. Ann. Math., B), 18> 187», 481-9.
"Compfas Bendus Pfirie, 88, 187&, 417-8,
>« Compt^ Bendiu Paris, 89, 1879,1106-ia

750 Histoet or the Theory op Numbers. [Сяа*. xxvi
right member remains finite. This argument was called insufficient by
E. Netto.1"
A. Korkine1*7 modified the last proof. Let Z be a polynomial in t
whose degree m is not less than the degrees of X and Y. Then one of the
latter is of degree m, say Y. Let m—X (X^O) be the degree of X, Differ-
Differentiate (Y/X)~+(Z/X)"+l=0 with respect to t Then, since K, Z have
no common factor,
XY'-YXf ZX'-XZ'
is an integral function. As the degrees of the numerators
and that of the denominators is m(n—1), we have
A. Lefebure1" separated into two classes the primes p=2kn+l. Into
the first class, put the p's such that the algebraic sum of any three residues
of nth powers modulo p cannot be & multiple of p. Into the second class,
put the p's for which die algebraic sum of three residues is a multiple of p.
It is claimed that all the p's in the first class are divisors of one of the integers
satisfying zn+y**=znf so that every p is a divisor of x, у or z, or is in the
second class. Hence if the first class is infinite, the equation is impossible.
But the first class is not finite when the second is infinite [correction by
Pepin1*].
T, Pepin10* noted that Iibri" long ago pronounced judgment on an
attempted proof like Lefebure's,1* To prove Iibri's assertion on
s4-jf+l-0 (modp=3A+l),
Pepin showed (by use of Gauss, Disq, Arith., art. 338, on the equation for
the three periods of roots of unity) that the number of seta of solutions of
the congruence in positive integers <p is p-f-L—8, where L is determined
by U^IM1-^ and Z/^l (mod3). Hence 7 and 13 are the only primes
ЗЛ+1 which cannot divide a sum of three cubes without dividing one of
them.
0. Schier110 claimed to prove x*+yn =zn impossible in relatively prime
integers if n is an odd prime. We have x+y=z (mod n). Expand by the
binomial theorem
cancel xn+yn with zn} and divide by the factor n. Thus
Hence also the left member must be divisible by n* It is stated that this
divisibility depends on that of the factors xy and x-\-y occurring in every
™ Jahibuch Forttthriti* Math., 11, 1879, 138.
^Comptes ReoduePari^ 90,1880,3084 (Math. Soo^Moecow, 10, 18821 54-6),
"•JW4 90,1880,1409-7,
»*1Ш.Р 91,1880, 366-8. Reprinted, SphinxOedipe, 4, l«»r 30-32.
"•Sitmmgeber. Akad. Wise. Wlen (Math.), 81, H, 1880, 392-3,

Format's Last Theorem* 751
term* Hence n divides x or y* For, if %-\-y and hence z is divisible by n,
set z=*+rtu~y in the initial equation; the result is said to bold only if у
is a multiple of n.
F. Fabrem proposed the question of the divisibility of (х+у)я—x*~y%
by x*+xy+y* and M. Dupuy proved {ibid,, 1881, 282-3) that n must be
of the form 6a±L
If111 (ZaI*+I=ra**+1 is true for n**l it is true for any n, since
A. E. Pellet stated and Moret^Blanc11* proved that
(mod 7) is solvable if ABC is prime to 7.
E. CesAro1" proved that if $(n) is the number of sets of positive integral
solutions of Ах*+Ву*=п, where A and В are positive integers,
The ratio of *(n) to the number of solutions of s*-fV~n is A-,
in mean» Hence, for а=0=1, ^(та)=п/D.В), in mean. For a=ft*=2,
ф(п)=т{D^АВ), in mean. The mean Ы the sum of the pth powers of
all the positive integral values which x can take in xh+yt=n is found
(p. 229).
С. М* FiumaU6 noted that, if no one of the coefficiente А, В, С is divisible
by the prime m=pg+l, then Axp-\-Byq-\-C^Q (mod m) has integral
solutions if and only if Az-\-Bzi+C=Q (mod m) has solutions for which
z=x>, Zi^y* are solvable for xy y, i. e,, if
z(z*-l)=O, ^!(^-l)=0 (mod m)
are solvable. Thus the initist congruence has solutions if and only if
P=0 (mod m), where P is the resultant of the equations corresponding to
the last two and Az+Bz-i+C^0, so that P is a product of (p+l)(<?-M)
linear factors.
For g=2, there are solutions if C+A or C—A is divisible by mr or if
any one of the products — ВС, —В(С+А), ~Б(С—A) is a quadratic
residue of m. In particular, J^+Ify^-f-C^O (mod 7) Is solvable if no
one of the coefficiente is divisible by 7, Cf. Pellet.111
E. Catalan, P. Mansion and de Tilly11' gave adverse reports on two
manuscripte submitted for the prize offered for 1883 by the Belgian Academy
(p. 101) for a proof of Fennat's last theorem.
E. de Jonquieres117 proved that in ап-\-Ъп=ся> п>1, the greater of a, b
is composite. Set e=a+kt b>a. Then, by the binomial theorem,
ш Joar. de math, el&neutaire de Longchampa et de Bouiget, 1380, No. 273, p. 528.
ш Math. Quest. Educ. Times, 3d, №b 105-
^Nouv. Ann. Math.. C), 1,1882, 335, 475-6.
"*M6in. 8ocRSc.de UbgB, B\ 10,1883, No. 6,1&5-7, 224.
"■ AnnaK 61 MaL, B), 11, 1882-3, 237-245,
"' Bull. At*d. R. Bel^que, C), 6 aniiee 52,1883, S14-9, 820-3, 825-32.
m Atti Acoad. Pont. Nuovi Iincei, 37y 1883-4,146-9. Reprinted in Sphinx-Oedipe, 5t 1910,
2ВД2. Proof by 8. Roberts, Math. Quest. Educ. Times, 47,1887, 50-^58; H. W, Curjel,
71, W9,100.

752 History of the Thxobt of Numbebb. CCka*. xxvi
Ъя=(а+к)ш—a* is divisible by k. But if kmb, спШ{а+Ъ)п>а*+Ъ\
Hence Ъп is divisible by an integer A:, k>l, k<b. Similarly, if a is a prime
<b, then c—b—1. He118 stated tha,t if a*-\-b*=c* and a or Ь is a prime,
the least of the two is a prime and the greater is composite and differs
from с by unity.
G. Heppel11'proved that, if nis a prime >3, (x+yL—x*—y*\& divisible
by nxy(x+y)(x2-\-xy-\-yx) and found the coefficients of the general term of
the quotient.
P. A. MacMahon120 employed his generalization of Waring's formula in
Proc. Lond. Math. Soc., 15, 1883-4, p. 20, to prove the identity
summed for the integral solutions of 2a+3b =n, where
)= {х-у){х+2у){2х+у)
He gave a similar identity for three variables. The right member of the
initial identity becomes tey(£-\-y)(&+xy+y*O if n=7 [cf. Cauchy*}
К Catalan141 stated that if p is an odd prime,
where P is a polynomial with integral coefficients, holds only if p=7 and
P~x2+xy+y\ Hei*2 proved this by taking*=p = l. Thus2^~l=p^
where N is an integer. Set t=(p—1)/2. Since 21—1 and 2'+l are rela-
relatively prime, having the difference 2, one of them is a square. The first
is of the form 4n-f-3 and is not a square. Hence 2'-H =M*. Thus the
factors If+1, M~ 1 of 2'are powers of 2 and their difference is 2. Hence
Af-1=2, so that p = 7, N=S or p-3, ЛГ=1.
Catalan182* stated the empirical theorems: (I) (x+l)*~i*=i is im-
impossible m integers except for x^O or 1. (II) j^—j^^lis impossible ex-
except for s*=l, y=Q or z=3, y*=2. (Ill) z*—l=P, where p and P are
primes, is satisfied only by x=% p=3, P=7. (IV) xn—1**P* is impos-
Bible if P is a prime. (V) :г*—k=pm, for p a prime, is satisfied only by
3—3, p=2, *л=3; л=2, p=3, m=l. (VI) ж'—^*=1, where p and gare
primes, is impossible except when x=y-3t p=q=2. (VII) x8+yI=p1,
where p is a prime, is impossible except when x=2, 0 = 1, p=3, (VIII)
^={B*-s-l)*+l}/2*-2 is impossible except when n=3, z^l. Cf.
Gegenbauer.1"
G. B, Mathews1* proved for special primes p that xy+^ =z» is Impos-
Impossible if no one of x, у, г is a multiple of p. The method was suggested by
"Compttt Bendu» Farie, 98* 1884, 863^4. Extract in Oetmes de Fermat IVt 154-6,
*» M*tb. Quart, Educ. Times, 40,1884,134.
"Mmatgw MatfL, 14,188^5, 8^11.
ш Nour. Ann. MaUl, C), 3,1884, 351 (Jour, de math., ярёс., 1883, 240).
~ Ш&, тш 4» 1885, 520-L
»- M6m. Soc. E. 80. liege, B), 12,1885, 42-3 (earlier 1д
>» M<m«tf М&Оц 15,1885^ 68^74.

Chap. XXVI] Fermat's LAST Tet^OREM. 753
Gauss'remarksforp^3 (Werke, 2,1863,387-391). Since z=x-\-y (mod p)t
D^ix+yy-x'-yv^Qimodp*), П=рху(х+у)ф(х, у).
The equivalent congruence хугф{ху у)=0 (mod p) is proved iasolvable for
p=3, 5, 11,17 unless at least one of the three unknowns is divisible by p.
The method leaves in doubt the case p=Zn+l since the factor х*+ху+у*
of Ф has real roots.
E. Catalan121 stated 16 theorems on a*+b*=c*t n a prime >3. If a is
я prime, then a=l (mod n); a*=l (mod rib); every prime factor of c~
divides a—1; a+b and c~a are relatively prime; also 2a—1 and b
a and 6 exceed ra; a"—1 is divisible by пЬ(Ъ+1)(Ь2-\-Ъ+1)> Next, no one
of a+b, c~a, с—Ь is a prime. If a+b=d[, c—a^&I, c^b=a*, then с is
divisible by и. The 4> [of Mathewsm] is
\
tbe sign being plus if A; is even.
Catalan"* stated the same theorems. Also, if a*+hn=c%, where а, Ъ, с
are relatively prime in pairs, and a-f-6 is divisible by n, it is divisible by
nw"; if а+Ъ is divisible by a prime рфп, it is divisible by p*\ if a+b is
divisible by a power >n* of щ it is divisible by n2*'1; if a+b is divisible
by a power >p% of a prime p Фл, it is divisible by j?n.
L. Gegenbauer"8 proved that 17> 29 and 41 are the only primes p=4/i+l
not dividing a sum of three biquadrates prime to p, Cf* Euler™ of Clu
ХХШ.
C« de FolignacI2r proved that a*—2*^±1 is impossible unless a=3,
n=l or 2.
A, E. Pellet"8 found by use of inequalities in the theory of roots of
unity that x*+y*=z* (mod p), where p is a prime g«-f-l, has solutions
x, yf z each not divisible by p for every « exceeding a certain limit (not
specified) for which ды-f-l is a prime pibri"}
P. Mansion1* considered z"+yn=zn, where x, у, г are relatively prime,
x<y<zf n an odd prime. By de Jonquieree,117 у is composite. It is proved
here that z is composite. The proof that x is composite is erroneous, as
later admitted.
M. Martone180 attempted to prove Fermat's last theorem.
"• BulLAeuL Roy. Sc.Bd^que, C), I2,1886> 498-500. Eeprodoced m Oeuvree de Ferm^
IV, 166-7.
M& S IL 6c. lifige, B), 13, 1858, 387-3ffT («Mflwig» MatL, 2, 1887, 887^397).
f some of these theorems by Iind,*> pp. 30-31,4143, and by 8. Koberta, M&tb.
Q Juc. Times, 47t 1887, 56-8.
"Stauopber. AkuL Ww. Wten (MatEi.), 95, П, Х887, 842,
"" Meth, Quert. Educ Timee, 4C, 1887, Ю9-П0.
» BuD. Soc. Math, de France, 15,1800-7, 80-43.
mBulL Ac*L Roy. Sc. Belgique, C), 13,1887,16-17 (correction, P- 225).
^ Dimoetrft&oiie di un odebre teomna del Ferm&t, Catanamo, 1887, 21 pp. Nupoli, 188S.
Nota ad una dimostr Napoti, 1888 (attempt to complete tbe proof m tbe former
paper).

754 History or the Theory of Numbers» IChaf. xxvi
F. Borletti proved that, if n is a prime >2, xn+yn^z" has no positive
integral solutions if г is a prime, and x2"—у*»=г** has no integral solution if
one of the unknowns is a prime; £"dby"^2el4s impossible if n> 1, and x, у
are odd and relatively prime»
E, Lucas1" proved Cauchy's* result Set q^a2+ab+b2,
r=ab(a+b)t 8л^(а+Ъ)*+(-а)«+(-Ъ)\
Then Sv+t^qS^x+rSn- Hence, by Wiring's formula, Sn is divisible by
<fr if n=6m+l; by q> and not by r, if ti = 6w&+2; by r, and not by $, if
n = 6m+3; by q2, and not by r, if n=6w+4; by qr if п=Ът+5; by neither
q nor r if n ~ 6w, As a generalization, if p is a prime,
is divisible by Q=l+x-{ \-x^1 if n is odd and prime to p, ;md by <? if
n = 2p+l. For p arbitrary, let ф(х) = 0 be the equation for the primitive
pth roots of unity* Then without details it is stated that
is divisible by ф(х) for n odd and prime to p. [Apparently the term x*n
should be added, and X taken to be the degree of ф(х), which degree is the
number of integers <p and prime to p.7}
L. Gegenbauer1*3 proved that, if a is a positive integer with at least one
odd factor >1, and q is a prime, x*+y'^qn has positive integral solutions
only when q^2y n = aa+l, x=y=2% ora^q^Z, n = 2+3a7x=2*3*,y = Z*>
Hence 33 is the only power of an odd prime representable as a sum of the
<xth powers of two relatively prime integers. A special case of this gives
the seventh empirical theorem of Catalan.!2** It is proved that if q is a
prime, x***1—q* = l is possible only for x=2t n=l, a+1 a prime, or x = Z,
a = l, q=2, 7i = 3. Hence a prime other than 2n-l is not followed by a
power, while 32 is the only power followed by a power of a prime. These
imply the third, fourth, fifth and sixth empirical theorems of Catalan,
A. Rieke1*4 attempted to prove xp+yp—zv impossible if p is an odd
prime >3, «He proved and used F). From an equation of degree
t = (p—l)/2 in a quantity m admitted to be doubtless irrational, he drew
(p. 241) the meaningless conclusion " that ml has the factor p, and m the
factor pUi, and indeed for all values of m"
D. Varisco136 failed to prove Fermat's last theorem since he concluded
(p. 375) that there is a unique set of solutions <ri = 0, etc., of
whereas the four equations are linearly dependent and have further sets
of solutions. The fault seemed irreparable to 0, Landsberg.1*
M Rttde let. Lotnb&nfo, Rendiconti, B), 20,1887, 222-4. ~~
u* Aseoc. fran?. w. вс., 1888, II, 29-31; Theorie dee nombres, 1891, 276,
in Sitsungpber. Abd. Woe. Wien (Math.), 97, U\ 1888, 271-6.
»*Zeitodmft Math, Fhys., 3i, 1889, 238-248- Errors noted by a "reader," 37, 1892, $7.
d Rtbhl146
"»Gioraale & Mat., 27, 1889, 371-380.
"*/W*28l89052

Chap, XXVI} FeJUIAT'S LAST THEOREM. 755
A. Rieke137 again attempted to prove z*+yp=z* impossible, but again
confused (pp. 251-2) algebraic and arithmetical divisibility, even for
E* Lucas1* proved (p. 267, p. 275) the theorem of Cauchy,** and (p. 370-
1) the formulas A), C), D) of Legendre17, with the aim to show that, when
zy у, z are relatively prune in pairs, no one of them is a prune or a power of a
prime £cf. Markoffli7j, He proved (p. 341) the first result due to Jaquemet.'
D. MirimanofP3* found in terms of the units a necessary and sufficient
condition that the second factor QKummer41] of the class number be divisible
by A. He treated in detail the case X=37.
J. Rothholz140 used the theorem of Kummer*5 on the divisors of a*±bn
to show (?) that z2*±y**=zu has no integral solutions if n is a prime 4&+3
or if one of the numbers z, y,zis a prime and л is an odd prune; xn+yn=zn
is impossible if x, у or z is a power of a prime, the prime not being = 1
(mod n), while n is an odd prime; zn+yn = Bp)n is impossible if n and p
Are odd primes; znzbyn=z* is impossible if а:, у or г has one of the values
1, • ", 202. The history of the theorem is discussed at length. On p. 29
are pointed out two errors in the proof by Rieke*1M
* W. L. A. Tafelmacher141 proved Abel's formulas and congruencial
corollaries from them. In the second paper he proved that Fermat's
equation is impossible for n=3, 5, II, 17, 23, 20 and, in case z+y~z=0
(mod n*) for n=7, 13, 10, 31 [but with proofs valid only when no one of
x, y} z is divisible by n, since the argument pp. 273-8 does not suffice to
exclude the case in wblch one of these numbers is divisible by и].
H. Teege142 proved tbat x^+^^l has no rational solutions by setting
l fz, (qfp)b=9. Then
Since z is rational, Ds+lJ-4(s-l)Ds+l)^w2, Set m=5». Then
5/A Let v=bfa, where a and b are relatively prime. Thus
Hence a2 divides 5^> The impossibility of this equation is proved by
considering the cases a divisible or not divisible by 5.
H. W. Curjel14* proved that if x*—yi=l and x7 у are primes, then г is a
prime, t is a power of 2, and x or у equals 2.
Several1" proved by use of cube roots of unity the known result that,
if n is odd and not a multiple of 3, (x+y)n— z*—yn is divisible by х*+ху+у*.
S. Levanen1" discussed xi+^^2mzlk} for x} y7 z without common factor,
r M»th. Phy*, 36,1891» 24^54. Error indicated in 37, Ш2, 57, 64.
IM Theorie dee пощЪгев, 1891. Inferences ш Introduction, p. хлж, where it ifl stated faleely
that Кшшдег prov«d Ferm&t's theorem for all even exponents.
"• Jour» fur Math., 109p 18S2, 8^-88.
ш Beitr^ge eudo Fermat^chen Lebrs&ts. Dies. (Gieesen), Berlin, 1802.
"Analea de la Univenridad de Chfle, Santiago, 82, 1892, 271-300^ 415-37. Report from
Lind,*i p, 50.
"»Zeitochr. Math. Naturw. Uuterricht, 24, X893, 272-3.
"»M*th. Quest. Educ. Timee, 58,1893, 26 (quest, by J. J. Sylvester).
**1Ш» 112,
*• Ofy^TEigt af Finska Vetenskap*-Soc. F№handlingar^ Hekingfore, 35, 1892-3,60-78.

756
Hibtohy of thk Theory of Numbers.
[Chap» XXVI
and wi not divisible by 5 (since x*+y*^z\ is impossible by Legendre18).
By the residues of s*, x^+y6 modulo 25, we see that m is not in the set
2, 4, 7, 9, 12, - - >, 2n+[(n~l)/2J For z divisible by 5, we have z=5tr7
x+y=2№. Proceeding as did Legendre, we find that the equation is
impossible.
D.Mirimanoff1* proved by use of ideaJsthatitS7+|/S7+2a7=O is impossible
in integers.
H. Dutordoir expressed his belief that ап+Ъп=с* is impossible in
integers if n is a rational number other than 1 and 2. The fact that it is
impossible when n—1/2 and one of а, Ь, с is not a perfect square is a case of
the impossibility of
when e is different from a and h, and one of the four numbers a, ■ -7 dis
not a square (Euclid, Elements, X, 42).
A. S. Bang148 pointed out errors in various elementary proofs of special
cases of Fermat's last theorem.
G. Komeck"* claimed to prove Fermat's last theorem by means of the
Lemma: If n and к are relatively prime (n odd) and divisible by no square
>1, then in every solution in integers of nzi+ky2=z% x is divisible by n.
E* Picard and H. Poincare1" pointed out the falsity of this Lemma by
citing the examples n«=3, k~l, x=y=z=4, and n^5, k=3, x = l, y=3,
z=2. The Jahrbuch Fortechritte der Math., 25, 1893, 296, pointed out
that § 3 of Konieck's paper shows a lack of knowledge of the nature of
algebraic numbers.
Malvy1* noted that, if a is a primitive root of a prime p=2*+l, and if
ina^41so4 (modp) we give to /1 the values 1,2, - -,2я, we obtain for
Л as many even as odd values. If in а**+*+_1=ан we give to ^ the values
1, ♦ ♦ -, 2"^, we obtain a even and 0 odd values for kf while if p = 17, a=3
orp=257, a^5, we have a=/5.
E. WendtIK proved that if nandp^mn+1 are odd primes,
rn+sn+t*=O (modp)
has only solutions in which r, s or t is divisible by p if and only if p is not a
divisor 6?
(?) (")(")
■
M Jour, fur MAtb., Ш, 1803, 26-30.
«Ann. 80c Яс, bmxdk», 17,1,1893,81, a. Mafllet«
v* Nyt Tid^faift for MAtb, 4t 1893,105-7.
™ArcfaiYMAth.Pliy9.,C2),13]lH>*(l*ft5); l-9> He ш>Ы, рр, 263-7, tbt the Lemma tola
l3lldtffthi^bflitf^+<s
^ СоафЫя Rendufl Pane, 118, ISM, 34b
ш I/lntermeVlIaira dm math., 1, ISM, 162; 7,1900,193 (repeated).
«Jour. fQr M&tb., 113, im, 83Ы47.

ChaJ». XXV1J FORMAT'S LAST THEOREM- 757
which is the resultant of x"^l, (x+l)m=l. For, if we multiply the con-
congruence by w~, where ut^l, we obtain a congruence of the form z+l=*y
(mod p), where x and у are nth powers, so that their wth powers are coa-
gruent to unity.
He proved Legendre's17 result concerning the cases m=2, 4, 8, 16. If
m=2?nb can be chosen so that mn+\ is a prime not dividing Dm, where r
is not divisible by the prime nt then a*=bn+cn (n>2) is not solvable in
integers all prime to n. If mn-\-1 ia a prime dividing neither D* nor nm— I,
the same conclusion holds. [This result differs only in form from that
by Sophie Germain17}
D. Hilbert1" gave a simplification of KummerV* proof of Fermat's
theorem for regular prime exponents, and a proof that *4+fP='y* is im-
impossible in complex integers a+bi.
G- B, Mathews1*4 noted that, if p is an odd prime, and x, у, г are solutions
of x*+y*+zp^Ot it is possible to choose к in an infinitude of ways such
that kp+l=q is a prime not a factor of $, yt zy от y*—zp7 etc., and such
that k is not divisible by 3. Then, since xp> y*t грате distinct roots of J*^l
(mod q), their sum is divisible by q. Let r=etwif* and р*=П(гв+г*+г7),
where the product extends over all triples of roots r", r* rT of xk = 1. Then
Рк = ±г£, where ttfcis a positive integer. Thus u*^0 (mod q) if and only
if three roots of x*=l (mod q) hftve a sum divisible by q. Hence if it
could be proved that for a given p there is an infinitude of primes kp+l
for which «i=0 (mod q) Ы not satisfied, Fermat's theorem would follow
pibri24]
E. de Jonquieres165 noted that, if n>2, it is not possible to express
с and Ь as algebraic functions of p, q such that ся—Ъ* becomes (pq)n
identically, and stated that this does not imply the impossibility of integral
solutions.
G. Speckmann"* discussed T*-DU*=m\
V- Markofi™ noted that Lucas'1» proof of Abel's14 theorem that
ая=Ъ*+с* (ft an odd prime) is impossible when a, & or с is a prime is in-
incomplete as the case а=Ъ+1 is not treated. He asked if (х+1)*=х"+ущ
is impossible.
P, Worms de Romilly168 ptated tbat ор+Ь»=с*, p a prime >2, impHes
p and q odd and relatively prime, q > 1, and u, v, B, pt a integers ^0. [Since
c—b=y is a power of p, Fennat's equation is impossible by AbeTs" result.^
roberibt d. Deutechen Math.-Vereirngmig, At 1894^5, 517-25. French tnnd.,
Aimafea Fac Sc, ToulotBeJC), 1, l909ilC)p 2,1910, «8; <3), 3, lflll, for errata, table
of contents, and notee by Tn, Got on the Hteinttire concerning Fentt&t'a but theorem.
ш Meesenger Math., 24,1894-5, 97-90. Reprinted, Oeuvre* de Fermat, IV, 16ei
ш ComptM Ifendue F№s, 120,1895, 1139-43 (minor error, 1286).
"* Ueber unbeetimmte Gleichungen, 1895.
i" L'intermediafre dc* nutth.v 2, 1805, 23; repeated, 8, 1901, 305-6,
/Wd 2P 1$9SP 281-2; repeated, 11, 1904, 186-6.

758 History or the Theokt of Numbers. ПСнар. xxvt
If то is a prime 6fc+l, (a+l)*^1^!, am~i = l (mod m2) do not hold simul-
simultaneously. If m is a prime, the integers ut not divisible by m, which satisfy
are of the form и~am—1.
P. F. Teilhet"* found A for which x*-Ayn = l by taking *=#*+!, or,
when n is even, x=y4—1* H. Brocard (pp. 116-7) found special solutions
when «=3, n = 5. T. Pepin (pp. 281-3) noted that we may apply to
xn—Ayn the method of Lagrange in his Additions to Euler'B Algebra to
mid the minima of any homogeneous polynomial in x7 y.
W. L. A. Tafelmacher1*0 treated x*+y*=z* and proved x*+y*=z* to be
impossible.
H. Tarry161 mentioned a mechanical device of double-entry tables for
solving indeterminate equations, in particular, xvt-\-ym=zti.
F. Lucas1*2 used Cauchy's5* theorem to prove that, if x, у are relatively
prime and m is an odd prime, when x+y is prime to m it is prime to
but when x+y is divisible by m, m(x+y) is prime to Qfm. From this he
deduced Legendre's formulas A) and C),
Axel Thue183 noted that, if L, M, N are functions of x such that
Ln—M»=Nn for all values of x, where n>2, then aL = bM=cN, where
а, Ъ, с are constants. If А*-ВЯ = С*, then
If p"-g«=rn, thenxt-if-Jipqr)" for
E. Maillet1" considered, for at Ъ} ct x, y, z integers not divisible by the
odd prime X, the equation
A necessary condition for solutions is that the congruence
have a solution щ such that 0<^<X, а+^фО (mod X), where <xc~at
&c=b (mod X). This is applied to show that xx-\-yx=zx is impossible for
X = 197, hence extending Legendre's limit to X<223, By the method of
Kummer it is shown that, if X is a prime > 3,
is impossible in complex integers, formed from a Xth root of unity, relatively
prime by twos and prime to X, if X* is the highest power of X dividing the
u* L'iDtermediaire dee math,, 3P 1896, 116.
"■ Analee de la Universidad de Chile, 97, 1897, 63-80.
ж Aseoc. frans* av, ac*, 26,1897,1, 177 (five lines).
"" Bull. Soc. Math. France, 25, 1897, 33-35» Extract in Sphmx-Oedipe, 4, 1909, 190.
'» Archiv for Math* og Natur., Kristiania, 19, 1897, No. 4, pp. 9-15.
'u Алое, fran?, av, ec,, 26,1897» U, 15&-168.

Сол*, xxvi] Гешслт'з Last Theorem. 769
number of classes of these complex integers, and hence for a value of t
exceeding a certain limit depending on X. He"* later proposed the problem
that the last theorem be proved without the restriction that x, y, z are
prime to X.
L F. Gram's1* paper was not available lor report.
EL Maillet157 applied Kummer's methods to zx+y*=cz*, where X is a
regular prime. The equation is impossible in integers if e=X. It is im-
impossible in real relatively prime integers not divisible by X if c^A\%
в^к\+0^17 0=0 or 1, when A —I orr*> ■ -ri', where ra> • • -, r» are distinct
primes +X, belonging to exponents/i, ---, /* modulo X such that
/i +/^X-l'
in particular, if А=г}1/г1ф1 (modX). For r a prime and Ъ<\, the equation
with c=r* is impossible in real integers if r*= — 1+JX (mod X2), where
fhae at least one of the values 1, - -,X—1; or if \~5,7,17,r*=4(modXI);
or if Х^11,г*-5ог47(т(н1111);ог if X = 13, r*=l7 (mod 13*). Finally,
х7+у7^сг7 is impossible in real integers for с a prime of one of the forms
49Jt=fc3, ±4, *5, 6, -8, ±9, ±Щ -15, ±16, -22, ±23 or ±24,
H. J. Woodalllte noted that zm+ym—1 is divisible by xy if y—zm—1
(m even) or if x=2t y^2m^ 1 (m odd).
T. ft. Bendz18* stated that х*+уя=гп has integral solutions if and only
if c^=4fin-i-l has rational solutions EEuIer8^, as follows from
He proved Abel's1* formulas, also x^-y=z (mod 3) and (p. 30)
(x+y)*—x*—yn=0 (mod n3),
when no one of x, y, z is divisible by n.
F. lindemftnn1* attempted to prove that xn=y*+z» is impossible if
n is an odd prime. He later (p. 495) recognized the error in the computa-
computation, but stated that Ыв work gives the first proof of AbeTs" statement
that if x, y, z are Ф0 and relatively prime in pairs
2x=p4+q*+r", 2y=p*+q4—r*, 2z=p*—q++rn
if no one of x, уy z is divisible by n, while, if z is divisible by nt
2x, 2y=p"+qn±n«-4-*, 2z=pn-qn+nn-hr~,
If x+y+z is divisible by n\ then, in B), «=0=т^1 (mod n*),
D. Gambioli171 proved de Jonqui&res'117 theorems, and the fact that in
xn+yn=zn (n>l), z is composite if n has an odd factor, or if x and у are
ш Congree intertiAt. dee math., 1900, Paris, 1902, 426-7.
«• F5rhandliBg*r SkandinAviaka NMurforekare, Gotheborg, 1896, 182.
" Comptea R«ndue Pane, 129, 1899, 198-4. Proofa ш Ada МлШ., 24, 1901, 247-266.
*• Mfttb. Queet- Educ. Times, 73,1900, 67.
m Ofver dioph^ntieka ekvationen ^+^=^\ Diee., Upeala, 1901, 34 pp.
^ SitBungsber. Akad. Wiea, Muncben (Math.), 31, 1901, 1S5-202.
iri Periodioo di Mat., 16, 1901, 145-192.

760 History of the Theoby of Numbers. [Chap, xxvi
composite; but erred in his proof that the least unknown is composite.
He gave abstracts of the papers by Cahsolari,7* Duichlet,* Kummer," and
Legendre,17 a list A91-2) of references on Bernoullian numbers and ideal
complex numbers, and A89-191) a short proof of the impossibility of
x*+y*=2*. In an appendix (гШ., 17, 1902, 48-50) he quoted Kummer*
and liouville4* on the insufficiency of the proofs by Lam6,* and Cauchy/4"*
Soons17* proved theorems stated by Catalan,90
P. Stackel1* proved Abel's theorem as given by Lmdemann.1*0
G. Candido174 proved a theorem of Catalan.111
* D. Gambioli's17* paper was not available for report.
P. Whitworth1" noted that if 21/x=0, Zs-1, then 2z*^x*+y»+z*
equals a series in xyz.
P. V. Velmine177 (W. P. Wehnin) proved that, if m, n, к are integers > 1,
there exist rational integral functions ut vy w of a variable which satisfy
um+vn*=wk only for the cases иш±1р=у>*> u*+tf=v?7 iu'+^t^ (when
the solution is easy), and u*+t^ = w*, the complicated formulas for whose
solution are not proved to give all solutions. Cf. Korselt.182
IX Mirimanoff1* studied P(z)^(z+1)}—xl-l where I is a prime >3.
Since it is unaltered when x is replaced by -1-х, a root a of P(x) = Q
implies the roots
(9) a, I/a, -l-«, -l/(l + «), -1-V«, -ar/a + e),
all of which are distinct unless a=0 or —1 or a*+a+l = 0. Now P has
the factors x(x+l) and zl+x+l. Set
r(, P(*)
where < = 1 if ?#1 (rood 3), « = 2 if 1=1 (mod 3). Then Я(я)=0 has only
distinct imaginary roots which fall into sets of six. Thus E{x) =Пе,(х),
where each e^z) ie of the form х*+1+Ъ(х*+х)+Цх*+х*) + B1-$)х*,
where t is real. If E{x) has a factor which is irreducible in the domain of
rational numbers, the factor is a product of certain of the e,(x).
A. 8, Werebrusowm denoted uf+uv—t? by (u} v). Then x5-; у* = А&
becomes
(tf+tfXrf-aW+lft x*-2zy+y*) -Az6.
This decomposes into two equations, one being the second factor equated
to Ajzf, the other being x+y=A&l, where AoAi = A, z&i=z, and Zi is a
product of primes 5n+l. Multiplying (u, v) by 1=9*—5 >42 and its powers,
, (), 2,1002, 109.
^Acta Math., 27, 1903» 125^8.
174 La formula di Waring e sue notevoli application], Lecoe, 1903, 20.
m И Pftagor*, 10, 1903^4, 11-13, 41-43.
m Mfttb. Queat. Educ Timee, B), 4,1903, 43.
in Mat. вЬогшк (Math. Soc. Moscow}, 24, 1903-1, 633-61, in answer to problem proposed
by V. P. Ennakov, 20p 1898, 293-8. a, Jahrbuch Fortacbritte Math,, 29, 1898, 139;
36,1904, 217.
"■ Nouv. Ann. Math., D), 3,1903, 385-97.
"Lintexibediure dee math., 11 1904,95-96; Math. Soc. Moscow (Mat. Sbomik), 25,1905,
466ph|73 (RQMim>). a. Jabrbach Fortschhtie Math., 36,1905,277-8.

C*ap. XXVIJ F£H&tATrS LaHT THEOREM. 761
we conclude that for each power we get six representations of a prime by
(Uy v); but only three representations of 5. A composite number has 2V
representations if p is the number of its distinct prime factors 5n±l.
Take Zi = {a, h). We get щ psuch that г\=(и, v) by using
(a, &)(*, т) = (шт+Ът, hr+ат+Ът).
Then
A0) (x~y)*~vs+(u+v)t, (x+yy=(iu-v)s+(v-Zu)t.
The product of the square root of the last вит by {*, t) gives Azlf so that
we have the general form of A. Taking x+y arbitrary, we get x—y and
then 8, t by A0):
Mirimanoff180 considered
A1) x4/+*x=0
for the case in which no one of the integral solutions z, yf z is divisible
by the odd prime X. By use of Kummer's congruences (8), he proved
that A1) is impossible in integers prime to X if at least one of the Berooul-
lian numbers' B^i, B^.if Д_*, B^-* is not divisible by X, where
also, for every X<257* In terms of КшптегЪ P,(J) = P,A, I), he defined
the polynomials
A2) ^
modulo X. Thus Hummer's criterion (S) is equivalent to the following.
If A1) has solutions prime to X, each of the six ratios t=z/y, * -', z/x satisfies
the congruences
A3) br-t(tl~0, Вь-опФМ^О (mod X) ({-3, 5, ■ ■ -, X-2).
An equivalent criterion not involving ВешоиШап numbers is that each
of the six ratios satisfies the congruences
A4) 0x-i«=O, ^,@^@^0 {mod X) (г-2,3, ■•-,,).
E. Maillet^1 proved by КиттегЪ methods that хл+^в=а^в (а>2) has
no real integral solutions +0ifais divisible by4; or if a is even and divisible
by a prime 4n+3; or if 2<a^lOO, a+37, 59, 67, 74; or if a has no prime
factor >17, Likewise for x«+ya=baz* if a is divisible by 4 and b is not;
or if a is of the form 4n4-2 and has a prime factor Х=4Л+3 such that Ь is
not divisible by X*; orifa=p*, b<p, p being a prime ^5 not exceptional in
tha sense of Kummer; or if а=3'} Ь^2 or 4, г^2. Probably the second
equation is impossible in integers +0if b^l or 2, a>2 or a>3, respectively,
E. Sauer proved th^t xn=yn+zny n>2, does not hold if x or у or z
is a power of a prime.
IT. Bmi18a noted that, if x+y+z=0 and k=2m+l, s^xk-\-yk+zk is
divisible by zyz. If lfz+lfy+l/z=O and jt=3A+2, s is divisible by
^ Jour, fur Math., 128» 190*, 45-68. ~
•If Br—i or By-t is not divisible by Л, the conclusion was drawn by Kummer.1*
" Annali di mM, C), 12, 1006, 145-178. Abstract* in Comptee Reudue Pftrie, 140, 1005,
12Э&; M6m. Acad. Sc. Inecr. Toulouse, A0), 5, 1905,132^3.
i" Eine polynomische Verall^memenuig dee FerroAtecbeu Satzes, Di*s^ Gk**n, 1905.
«• Periodioo di Mat., 22,190&-7,ЙЬЗ

762 History of thi Theory of Numbebs. CChap. xxvi
.> хщу*-+г*гщ+упгп is divisible by (xyz)* if n^5. Proofs184
have been given of the first result and the fact that, if x+y+z=0, a is ft
function of xyz and xy+xz+yz.
* G. ComacchiaJM treated the congruence x"+y"^zn (mod p).
P. A, MacMahon1» noted that the integral solutions of xn—aym-z
may be obtained by the development of alf* into a continued fraction.
F* Undemann1*' again1™ proved Abel's formulas and, after treating at
great length each of the three cases, concluded that Fermat's equation is
impossible in integers. A. Fleck18* pointed out a serious error and various
minor errors. I. I. Iwanov189 noted errors, also in IindemannJsno first
proof, where in F7) the modulus n* should be nb.
A* Bottari1*0 proved that if xf у, г are positive integers in arithmetical
progression such that x*+y*^z*, then either n=l and x=yj2=zlZ or
n=2 and я/3=^/4=з/5. If xt y} z, t are positive integers in arithmetical
progression such that xn+yn+zn=tn, then n=-3, x!Z^y!4=z/5=t/b. Cf.
Cattaneo.1"
J. Sommer191 omitted the restriction that n is a regular prime in stating
that Kummer proved that xn+yn=z"t forn>2, is not solvable in complex
integers based on an nth root of unity. He gave the proof forn=3 and
*=4.
P. Cattaneo192 gave a brief proof of the results of Bottari,190 but included
the false solution n =^ 1, x - yj2=zjZ=//4.
A. S* Werebrusowiw failed in his proof of Fermat's last theorem, the
error being indicated by L. E. Dickson and others (гШ., pp. 174-7).
Werebrusow1M stated that (х+у+г)п—хч—у4—г* has, for n odd, the
factor n{x+y)(x+z)(y+z). While this is true for n an odd prime, it fails
forw=9, x=y=z = l (ibid*, 16,1909,79-80).
L. E. I>ickson1M noted that, if a is a common root of the congruences
A5) z~~h (*+l)m^l (modp)
of Wendt, the numbers (9) are common roots and are distinct if 2M^ 1
is not divisible by p. They are the roots of a sextic in г which is unaltered
when z is replaced by ljz or by — 1— i* The sextic must divide zm— 1
modulo p. Set x=z+lJ, m = 2fi. The sextic becomes
From zT-l/z^^O we get/(x2)=0, where /(w) is of degree^-lor (ji-l)/2
»•* LTiDtmn6diaire dea matb,, 13, 1906r 142; 14P 1907, 23-23, M-&i~t~93H>b~2S8.
"■ Sulla Congruenza х»+ул=г* (mod p)t Tempio (Toitu), 1907, 18 pp.
Ш1¥ос. London Math. Soc., B), 5, 1907, 45^5S- For « = ±1, G. Comacchia, Rivieta di
fieica, m&t. ее. nat., Pavia, 8, II, 1907. 221-230.
'" Sitningeber. Akad. Wiea. Munchen (Math.)» 37,1907, 287-352.
111 Arcbiv Math. Phys,, C), 15, 1909,106-Ш.
Hi Kagane Bote, 1910, No. 507, 69-70.
»•■ Periodioo di Mat., 22, 1907,156-168-
w Vorleeucgen uber Zahlentbeorie, 1907, 184. Revised French ed. by A. L6vy, 1911, 192.
ш Periodic*) di Mat., 23,1908, 219-20.
in L'inteimediaire dee math., 15, 1908, 79-81.
m Ш<*., p. 125. Cwen-3, iDl'educAtionmftth., 18S9, рЛб.
»" Meewnger of Math., B), 38,1908> 14^32.

Chap. XXVI] FERMAT's LAST THEOREM. 763
according as p ш even or odd. Thus f(z*) must be divisible by
Hence ii^l. Forju^?,/^1)^^—бсг^+бх1—1 must be congruent to £(*),
whence p=2. For p=8>/(x8)=j^—oV+IOx*—4, whence p=17, contrary
to n>l. The cases я = 10, П, 13 are readily treated* The conclusion is
that, if n and p = mn+l are odd primes, m being prime to 3 and m—26,
the congruence ^ж+^я+Г"=0 (mod p) has no integral solutions each
prime to p, except for л=3, m^lO, 14, 20, 22, 26; n=5, »i=26; «^31,
in—22. A direct examination of A5) was made for m=2S, 32, 40, 56,.64,
By use of these results and the theorem of S. Germain,17 it was shown that
Fermat's equation is impossible in integers prime to n for every odd prime
exponent w<1700.
Dickson1* proved the last theorem for n<7000 by extending the range
of the m'& to include all values <74, as well as 76 and 128.
Dickson1** factored certain numbers mm — 1 for use in the last paper.
Dicksoniw discussed the following problem: Given an odd prime n, to
find the odd prime moduli p for which хп+у*+гя = 0 (mod p) has no solu-
solutions each prime to p. We may take p=mn+l, where m is not divisible
by 3, since otherwise such solutions are evident. The general results are
applied to the cases n=3, 5, 7. Forn^3> the only values of p are 7 and 13
[cf. Pepin1»]. For «=5, p = ll, 41, 71, 101 [verified up to 1000 by
Legendre1*]. For n^7, p=29, 71, 113, 491.
Dicksonlw proved, by use of Jacobi's functions of roots of unity, that
if e and p are odd primes such that
then я*+#*+2'=0 (mod p) has integral solutions z, yt z, each prime to p.
In particular this establishes the conjecture by Libri.24 Also, x*+y*s=**
(mod p) has solutions prime to p for every prime p=4f-tl exceeding 17
[and different** from 41J
P. WolfskehP01 bequeathed to the K. Gesellschaft der Wissenschaften
zu Gottingen one hundred thousand marks to be offered as a prize for
a complete proof of Fermat's last theorem. It may be noted that
WolfskehP02 was the author of a paper on the related subject of the class
number for complex numbers formed of eleventh or thirteenth roots of unity.
i» Quar. Joup. Math., 40,1908, 27-15, The omitted value n-6857 wsa later shown in MS»
to be excluded»
i" Amer. Math. Monthly, 15,1908, 217-222. See p. 370 of Vol. I of ibis History; deo, A.
Cunningham, Messenger of Math., 45,1915, 49-76.
in Jour, fur Math., 135, 1909, 134-141.
xUIbid., 135,1900, 181-8, Cf. Pellet,1"-** HurwitB,*" СогшюсЫа,лт and Schur.**
100 On p. 188, line II, it u stated that for/even and -^14, p—4/-H ie a prime only when
/=4}Р=ш171 thus о*ег1оокшБ/=10, р=-41. The/act that ^-Hy*^l (pod 41) hafl no
solutions each prime to 41 was communicated to the author by A. L. Dixon.
'"Cftttuigen Nachrichten, 1908, Geechaftliche Mitt., 103. Of. Jahrabericht d. Deutechen
Math,-Verieiniguiig, 17, 1908, Mitteflungen u. NacbrichUin, 111-3. Fermat'e Oeuvxes,
IV, 166. Math. Annalen, 66, 1909, 143.
«Jour, fur Math,, 99, 1886, 173-8.

764 History of the Theory of Numberb: [Chap, xxvi
No mention will be made here of numerous305 recent false proofs** of
Fennat's last theorem, published mostly as pamphlets. Errors in some
of these have been noted by A. Fleck,** B. LindM1 (p. 48), J. Neuberg,2* and
D. MirimanofL*07
E» Schonbaum208 gave a historical introduction to and exposition of the
elements of the theory of algebraic numbers; also Kumraer's proof, in
simplified form, of Fermat's last theorem for the case of regular primes.
* A. Turtschanmov200 proved and slightly generalized Abel's18 theorem
* К Ferrari*1* discussed the infinitude of solutions of each of
1Г\ ±y
A, Thue211 stated that there are no [not an infinite number of] integral solu-
solutions of any of the equations, with n>2)h and k given positive integers,
() кк (Y (ky
These results are consequences of the theorem (pp. 27-30) that, if r>2
and a, &, с are any positive integers, с Ф 0, there is not an infinitude of pairs
of positive integral solutions p, q of Ър*—(uf=с
A, Hurwitzm proved that, if m and n are positive integers not both
even, xw%y*+ymz*+zmxll = 0 has integral solutions Ф0 if and only if
u'+y'+ir1—0 has such solutions, where f—m*—mn+n*. (X Bounm-
kowsky,"' Cb XXIIL
Hurwitz, after citing Dickson's1M proof by cyclotomy, gave an ele-
elementary, but long, proof that, if a, b, с are integers Ф0 and e is an odd prime,
ax9+by9-\-cz9=0 (mod p)
has A sets of solutions x, yt z each not divisible by the prime p, where
-A>p+l-(e-l)<*~2)^-* (,-0,1 «гЗ).
Hence A >0 when p exceeds a limit depending on e>
A Wifrih214 d tht if v++p0 i
ce A >0 when p exceeds a limit depending on e>
A. Wieferich214 proved that if xv+yr+zp=0 is possible in integers
t h i dd rime, then 2P^1 -1 is divisible by p\ He
conditions A3) derived by Mirimanoff1*
. Wieferich proved that if
prime to p, where p is an odd prime, then 2P1 -1 is divisible by p\ He
deduced this criterion from the conditions A3) derived by Mirimanoff1*
** According to W. IietEmann, Der Pythagoreische Lehraatz, mlt einezn Aueblick auf due
FermAtecbe Problem, Lcip«igr 1912,63, more thAn a thousand false proofs were published
during the firet three yeaia after the announcement of the large prize.
'* Titles in Jahrbuch Fortechritte Math.t 39t 1906, 261-2; 40,1909,268-Ш; 41,1910» 248-
250; 42r 1911t 237-^9; 43, 1912, 254, 274r-7; 44,1913, 24&-50.
'«Arehiv. Math. Phye., C), 14, 1909, 284-6, 370-3; 15, 1909, IOS-Ш; 16, 1910, 105-9,
372-5; 17,1911,108-9, 370-4; 18,1911,105-9, 204-6; 25, 1916-7, 267-8»
*- MatheeiH, C), 8r 1908, 343.
M7CompteeBendusPariflt 157,1913,491; enwofE, Fabry, 156,1913,1814r^. I/enaeignfr.
nrent math., llt 1909t 126-9.
"■ Caeopie, Ptag, 37,1908, 384-505 (Bohemian).
*» Spacrinati Bole, 1908, No. 454,194-200 (Russian).
«"SuppL *1 Periodico di Mat., 11.1908, 4<Ъ2.
«^Skrifter Videnekabe^Sebkabet Ghrietiania (Math.). 1908r No. 3r p. 33.
*»Matb. Annalen, 65,1908, 42*30. Case m«2, n-1 by Eul«* and Vandiver,*" Ch. XXI.
*" Jour, fur Math., 136,1909, 272-202.
ш/Ш.,293-302. For outbe of proof, вее Шквоп"» 18^3.

» xxvi] Frrhat's Last Theorem. 765
from Kummer's criterion» Shorter proofs have since been given by
Mirimanoff5** aod Frobenius,5*8
P. Mulder216 noted that if n is an odd prime and an +&* is divisible by nt
it is divisible by n*. Proof as by Кшшпег.25
Chi\ Hies** argued that а**-\-Ъ**=с** (п>1) is impossible in integers
by considering the two factors of c2n whose difference is 2b", but assumed
that every prime factor of 2bn divides b.
G. Cornacchia217 employed the theory of roots of unity to investigate
the number of sets of solutions of ff*+j/n^l (mod p)t where p is a prime
of the form nk+h There are proper solutions for n^3 if рф7, 13; for
n^4,ifp#5, 13,17,41; for л=6, if p+7, 13, 19, 43, 61, 97, 157, 277; for
n=8,if , +17,41,113; fornany odd prime if p>(n-2)*n(n-l)+2(n+3).
For p a prime nfc+1, х*-\-у"-\-г*~О (mod p) has proper solutions for n=4
if p+5, 17, 29, 41 [Gegenbauer1*]; for n = G, if p+13, 61, 97, 157, 277,
31, 223, 7, 67, 79, 139; for «=8 if p=M7, 41, 113, 89, 233, 137, 761. He
proved a theorem like that of DicksoD,1M but with a limit
which is larger than Diekson'a if e> 3.
A. Flechsenhaar21* considered, for n a prime >3,
A6) zn+y»^2« = Q (modri2)
for xt yt г prime to n. We may set x<n, y<n7 x-\-y=z. Multiply A6)
by pfand pi in turn, where p&^l, p#^l (mod ft). Hence the solvability
of A6) implies that of
A7) 1+6*- (Ь+1)я=0, €*+l~(c+l)»s=0 (mod и2),
where Ъ^р&, с^рф, whence bc=l (mod n). These conditions continue to
hold after b is replaced by 6—n, and с by с -п. We get
Since these have the form of A7), it is stated that (n—b-l)(n~c-l) = l,
whence 6-hc+1^0 (mod ft), by a false analogy, as no proof had been given
that, for every pair of solutions b, с of A7), we have hc=l.
Admitting 6-hc+1^0, bc^l, Ьфс, we have п = 6т+1. Solutions b, с
then exist and are tabulated for n ft prime ^307. But (p. 274) for n a
prime 6ro—1, A6) has no solutions prime to ra.
J> Nemeth2" noted that x*+yk^zb, xl-\-yl=zl have no common sets of
positive solutions if k,l are distinct positive integers»
J. Kleiber220 stated that if л is an odd prime, x, y, z are relatively prime,
and j/, z not divisible by n, хя+у*=гп implies that
which readily give у=0. But he had assumed that the laws of factorization
of integers hold for numbers involving ^ had not specified the kind of
** Wiskiitkdige Opgaven, Amsterdam, 10,1909, 273-4.
*• Math. Naturw. Blatter, 6,1909, 61-3.
111 Granule di m*t., 47, 1909, 219-268. See Coraacchk^ and the rtjferencaj und«r Libri^
« Zeitechr. Matb Natutw. Untenicht, 40, J9(», 2^-275.
«• Math, ее FI^b. Lapok, Budapttt» 18f 1900, 229-250 (Hungarian),
"•Zetocb. Math. Naturw, Uatemcht, 40,1909,45^7,

766 History of thb Theoby of Numbers, [Chap.xxvi
quantity whose nth power is x-\-tyt and in giving the quantity the notation
p-\-cq had not specified the nature of p and q.
Welsch2" repeated a proof due to Catalan.111
IX Mirimanoff*** considered the relation of F=xl+yl+zl=Q to cubic
congruences. Let x, у, г be the roots of £—*i!4-*rf—*8=0. Thus
F= ф(*1, &i, S3), where Ф is a polynomial of degree I with integral coefficients.
We have *i*=0 (mod J). Let x} ytz be prime to L By Legendre,17 si— ¥
indivisible by l(z+y)(z+z)(y+z) = l(&18*~Si); call the quotient P(si,s2,si).
Since SxSi—st is prime to I, and since s\ is divisible by l\ F—Q gives
P@t sb 8*)=Q (mod J). Hence if J^O has solutions prime to I,
P-\-8tt-st=Q (modi),
subject to P=0, has three roots. For J=3, then P=l and F=0 is im-
impossible in integers prime to Z=3. For J=5, P= — ss; but if *i=0, the
discriminant of the cubic congruence is ~27s£, a quadratic non-residue of J,
so that it does not have three roots. The same argument applies to 1= 11.
For 1=17, the discriminant is a residue and there are three roots or no root;
the first case is excluded by the fourth criterion of Cailler (£&, 10, 1908,
486; see p. 255 of Vol. I of this History) for cubic congruences. The method
fails for (=3m+l, since we may now have «j=0.
Mirimanoff4** employed Euler's expression for 1—2*~*+3^* ±y*~*
as a polynomial in у to obtain a short proof of the final congruence used
by Wiefcrich to prove his criterion that 2^=1 (mod p1).
B. Iind224 proved that x*+tf=z* is impossible in integers. If xn+yn=zn
is impossible, so are Z2"—X*=4Y* and sB»+l)=P\ The last equation
implies *=£*, 2s+l =^*, tjk-tt whence £*— 1-2(ф", a case of Iiouville's21
equation. For a simpler proof, see Kempner.**1
J. Westlund225 noted that, if n is an odd prime,
is divisible by n2 if by n. Hence хщ+уп=пг* is impossible if z is prime to n*
R. IX CarmichaeF* proved that, if p and q are primes, pm—q*=l
ODly for m-1, q=2t p=2*-H; wi=tf=2, л=^р=3; я=1, p-2, g=2"-L
A. Fleck227 distinguished eases Л and i? according as none or one (say x)
of the integral solutions ^Q of x*+y*+z*=Q is divisible by the odd prime p.
Set 8 =x+y+z. Then
(A) y+z=a*, z+x=b>, x-\-y—t^y s=—abcj?GM,
(B) |/+z ^plp^Ia', z+x = hp, x-\-y=cvJ 8 =
He considered the six quantities
y3+yz+z2^GJ, x*~yz=GJiJ
z*+zx+x*=GKf у*-гх = GKh
GL, zi—xy=GLit
** L'intermediaire dee math., 10, 1909,
~™L'erurignemeiitm&th., 11, 1ЙО9, 49-5L
»/Ьй, llr 1909, 455-9. Summary by Dicbon^ p. 183.
*" Atchiv Math. Phye., C), 15, 1909, 368-9.
>» Amer. Math. Monthly, 16,190», 3-4.
«M, 38-9, Special casee by G. B. M, Zeir, 15,1908, 337. See Gerono.»
n» SitauD^ber. Berlin M^th. Geeell-, 8,1909,133-148, with Archiv Math. Fhys.t 15,1909.

Chap. XXVIJ FebMAT's LaBT THEOREM. 767
and proved that (i) 9 has no factor other than a divisor of £ in common
with one of these six expressions; (ii) any two of the six have no common
factor other than ft divisor of G, so that Jt - <■ •, L\ are relatively prime in
pairs; (iii) J, •—, L\ are products of primes of the form 6^p+l; (iv)
г«»=у'»«2*» (mod GJKLAKM*
G> Frobenius2*8 gave a simple proof of the criterion of Wieferich,
using MirimanoffJa1№ formulation of Hummer's criterion to show that
Г,4ш9
is congruent modulo X, for every £фО, ±1, to both
wheoce csO (mod X), so that 2^~l=l (mod X2).
A, Gerardin22* gave a brief history and extensive bibliography of the
subject* Ho conjectured that Fermat's last theorem could be proved by
showing that the difference or the sum of two nth powers (n >2) is always
comprised between two consecutive nth powers.
P. Bachmann28* gave an account of results obtained by elementary
methods, chiefly those by Abel," Legendre/* Wendt,1*2 and Dickson>JW~*
The remark (p. 461) that all primes <100 are regular was corrected on
p. 480.
H. Stockhaus*11 gave a- lengthy exposition- of known methods for expo-
exponents 3, 5, 7, with suggestions of doubtful value on the general case.
* K. Rychlik2" gave a proof for exponents 3, 4, 5.
* Ed. Barbette25* proved some inequalities*
R Bernstein23* proved Fermat's theorem under assumptions milder
than those of Hummer.™ The second case (that in which one of the three
numbers is divisible by the prime exponent Г) is proved by means of the
assumption that the class number of the field k{Z) of the Fth roots of unity
is divisible by I, but not by P; and again by means of the assumption that
h(Z) contains no class belonging to the exponent P, while the class number of
£(f+f~l) Ь prime to l} where ff=l. The first case (that in which the
three numbers are prime to t) is proved from the assumptions (i) that the
second factor A2 of the class number of k(t) is divisible by I, and (ii) if f is
the highest power of I dividing hit then in the " TeilklasseflLkorper" of the
Pth degree every ideal of k(t), whose fth power is a principal ideal in k(£),
is itself a principal ideal. [See Vandiver's*1 eriticisms.]
*"Sitiunpb*r. AbuL Wim, Berlin, 1909, 1222-4, Reprinted iD Jour, fur Math., 137, 1910,
314^
mHietorique du dernier tbeoreme de Fermat, Totibuee» 1910, 12 pp. Extract in
frang, ftv. ec., 39? lt 1910, 55-6. All of hia references are found in the present chapter.
^NiedereZahlentheorie, 2, 1010, 458-476.
m Beitrag nun Beweiti dee Fermatdchen Satxes^ Leipzig, 1910, 90 pp.
*» Caeopie, Pra«7 39, 1910, 65-80, 185-195, 305-317 (Bohemian).
Ш1е dernier tbeoremedeFermat, Paris, 1910, 19 pp.
M Gdttingen Nachricbten, 1910, 482-488, 507-016.

768 History of the Tkeoby or Numbers. CChap. xxvi
Ph. Furtwangler335 proved, in extension of Rummer's™ work, that if
«I+0I+7I=O, where at p, у are numbers, prime to L=(£— 1), of the field
А(Г), Г'—l, and if a=a, 0=&, 7=c (mod L)t where а, Ь, с are rational, and
if k(£) contains no ideal belonging* to the exponent 2j-\-l modulo L, then,
if xt у are any two of a, b, c,
By Mirimanoff,1* this congruence can not hold when j^ 1, 2, 3 or 4. Hence
if k($) does not contain ideals belonging to each of the exponents 3, 5, 7, 11,
Fennat's equation is impossible in numbers prime to I in k(£). The same
conclusion holds if the class number H is at most divisible by P.
E* Hecke2* proved that x*+yl-\-zl=0 is impossible in integers x, у, г,
each not divisible by the odd prime {, if the first factor hi of the class number
H of the field defined by an Jth root of unity is divisible by I, but not by P,
D. Mirimanoff,*** making use of hisiw criterion, proved that if
xp+y*-\-z*=0 has solutions prime to pt each of the six ratios xjy, - • - is a
root Jof
where «i, ^><хя%^1 are the roots +lof ** = 1. For m=2 or 3, at least two
of the six ratios are incongruent, so that our congruence, being of degree
<2, is an identity; taking t= —1 and applying
we get q(m)^0. Besides Wieferich's qB)=0, we have gC)=0. Thus tha
initial equation is impossible in integers prime to p for all prime exponents
p such that either gB) or gC) is not divisible by p; in particular, for all
prime exponents of the form 2*3*zhl or rfc2e±3*.
G. Frobenius118 proved the last two criteria and deduced A3) from (8)
more simply than had Mirimanoff.1» Set Ъ**=(~1)*~1В1Ч Ь^+1^0,
b1 = —}, во that the Bernoullian numbers are given symbolically by
) Set
) > у),
mF(x)=F(xt mb)-
•An ideal Q, prime to L-tf-l), as said to belong to the exponent ft modulo L if Q* i» %
principal ideal (я) such that »-rt (mod L*), while there existe no unit ij in the field Jb(f)
jrach that цж**Ъ (mod Lm+*)t where rt and r± are rational aumbera,
« Gtttingen Nachnchten, 1910, 554-562.
"Лй4ам
*" Comptee Rendus Fane, ISO, 1910, 2О4-& Reproduced.-
ш SitmngAer, Akad Win. Berlin, 1910,200-8.

Chap. XXVI} FeRMAT's LAST THEOREM. 769
Then
from which the results of the paper follow. The six ratios of the three
solutions prime to p of Fermat's equation satisfy the congruence Gm(x)=Q
(mod p) of degree m—2, Hence, if m=2 or 3, Gm vanishes identically*
A. Fleck2** proved, as an extension of his*3* theorem (iii), that the
prime factors of JIf Kt, Xiare of the form 6?p2+L Hence/, >> >, Lx are all
of the form 6^p*+l. For any prime factor j of the form 6/ip+l of J,
(ty)M=(k)**^l (mod j), where t=l in case A, t=p in case B. A like
result is said to hold for the prime factors of К or L.
E. Dubouis*40 defined, in honor of Sophie Germain, a " sophien " of a
prime я to be a prime 0, necessarily of the form kn+1, for which x*=yn-\-\
(mod B) is impossible in integers prime to $* He stated that Pepin10*
proved that the sophiens of n are finite in number, whereas Pepin proved
this only for n = 3. If the resultant of a*=l, (a+l)k=l is not divisible
by 0, then в is a sophien of n [WendtlM].
B. Iind241 gave an exposition of various papers dealing with Fermat's
last theorem without the use of complex integers or ideals, but unfortunately
interpolated careless remarks of his own» Of the results claimed by Lind
to be novel, equations A9)-B6) are correct, but long known, while B7)
is not proved, viz., that x+y—s^O (mod 9) if x*+yn^zn, it being proved
only for modulus 3. This error gave rise to later errors in bis inequalities
(p. 32) and his equations (95), A06b), His attempt (pp. 61-5) to prove by
use of congruences Fermat's last theorem contains several serious error»
besides the dependence on B7). The bibliography is quite extensive*
J. Joffroy24* noted that, if F^xrr-^y!l't-^7=O for integers x<y<z, then
FoTt^7-x=Pm, P=2 3 5 7 1319 37; so that
1\ Hay&shi248 proved that if, for n an odd prime, xn+p~^nz", or if
x*+yn=zn for z divisible by л, then bo-\-bi-\ \-Ъ,^0 (mod ti2), where
$ = (n~ij/2, and the Ъ'а are the coefficients of the polynomial Y satisfying
the identity
where
wtSitiungaber. Beriin Math. G«eeIlM 9, 1910, 60-3 (with Archiv Math. РЬув., 16,1910).
»"L'tatemi6diaire dee math., 17, 1910, 103-4.
h. Geechichte Math* Wiae., 28, П, 1910, 23-65. Reviewed adveively by A. Fleck,
Mv Math. Phye., C), Дбг 1910,107-9; 18,1911,107-8.
Nouv. Ann. Math,, D), 11, 1911, 282^3. Reproduced» Oeuvree de Pcnn*i, IV, 165-6.
Jour. Indian Math. Sac. Madraa, 3, 19llf 16-22; Ш4 Same in Science Report of
Tdboku Univeraity, 1,1913, 43-50, 51-54.

770 History of the Theory of Numbers. [Chap.xxvi
while
are such that ij2—(-l)^(xO2 has as divisors only 2 and numbers of the
formr2—(-l)'n*2. The initial equations are both impossible if n = 5or 13.
A. E. Pellet*" considered for a prime p—hn-hlj having g as a primitive
root, the Dumber hN* of times that
By use of the equation for the n periods of the pth roots of unity it is shown
that pNi has the limits A2d= V<p — AI, whence terror34*] the inferior limit
is positive if h>n^n. Hence in that case, xn+yn+l=0 (mod p) has
solutions prime to p. Cf. Libri.24
D. Mirimanoff246 reproduced his237 paper and used his first formula to
obtain results concerning ^E) and qG). Also he proved that ^p_i(/) is
divisible by p Dot only when / is one of the six ratios r=xjy, • * >, but also
for t= -г and t= -г2. Finally, he proved Sylvester's formula for q(m)
[VoL I, Ch. IV of this History}
A. Thue^7 proved that, if n is a prime >3, and € is an imaginary nth
root of unity, and each Bi is an integer numerically ^
| tan
|R ,R , ,R ж_, ^ tan тг!Bп)
if not every £>=0, Next, for R an integer, let PQ=Rn} where
% \\ \\
Then for a suitably chosen к and integers Д g^ such that
we have P/R = - B/A, where A = 2/V, Б = z^,-л It is stated that applica-
application can be made to Fermat's equation
If а*-\-Ъп=сп for relatively prime integers (p. 15), we can find positive
integers pt qt rt each < V3c, such that pa+qb = rc. Hence
whence 5"—rR is divisible by a.
Thue248 proved that if yn=x«+l, n>3, the most general solution of
where Л, В and each с are integral functions of x, is
/ч-(/*);=(Л)-
where / is an arbitral integral function of z.
w UiDterm^diaire dee math>r 18, 1911, 81-2»
** Thia deduction fails if n -5, h=2O.
«• Jour, flip Math., 139, Ш1, 309-324.
147 Sbifter VidenakapBeelflkapet I Kristiania (Math.), 1, 1911, No, 4.
"• Ibxd.} 2, 1911, No. 12, 13 pp, For Ыя paper, ibid., No. 20, eee1™ Ch. XXIIL

Fermat's Last Theorem. 771
* D, N. Ranucci wrote a pamphlet, Kisoluzione dell'equazione
con una nuova dimostrazione dell' ultimo teorema di Fermat, Roma,
1911, 23 pp.
F. Mercier2*** noted that we may take x<y<z if n>l, whence
njx<(xfy)*~1<l, n<x. This lemma, instead of helping him to prove Fer-
mati's last theorem, led him to commit the error of saying that Z*+yn=zn
is solvable when n is any integer > 1 because it is solvable when n *= 2.
Ph. Fuitwangter2** proved by use of Eisenstein's law of reciprocity for
residues of Zth powers, where I is an odd prime, that every integral divisor
r of Xi satisfies
A8) fHsl (mod*2)
if xj, x2, xz are relatively prime solutions Ф0 of ${+х1г+х*3=*0 and ^ is
prime to L Since one of the z's is divisible by 2, we have the criterion
of Wieferich. Next, every factor r of x<±xk satisfies A8) if z.-f-ff* and
Xi—Xk are prime to L Since one of the x's is divisible by 3 unless all three
are congruent modulo 3, it follows from the two theorems thatr if the x'n
are all prime to I, (IS) holds for r^3, which is the criterion of Mirimanoflf-
S. Bohnicek250 proved that integral numbers of the domain of the 24h
roots of unity do not satisfy Fermat's equation with the exponent 2n~\
2
H. Berliner251 considered x* = yv+zp for x, у, г not divisible by the prime
p>2* In Abel's formulas 2x = a7>+b7l-\-cpt •■-, we may take a>b>c*
Then a = b+c±2kepf where 2fc is the highest power of 2 dividing abc, while
ep is an odd multiple of 3. For every p, a<3(b+c): for p^5, a<3b\ for
Sl dlf*(b) for p^37, а<&*Ь. If pm5, 6>3p; if ?i=37,
L. Carlini262 proved that x"-\-y*=zn (n> 2) is not satisfied by three binary
forms in u, v, identically in the variables u, i\ Hence a like result holds
for polynomials in one or more variables.
J. Plemelj253 proved x5+y5+^5 = 0 impossible in R( Vs) more simply than
had Dirichtet20
*B. Bernstein254 gave some prop>erties of numbers satisfying хп+у*=ъп*
The latter is proved impossible under certain assumptions on x, y, 2.
R. IX Carmichael2" proved that^ if z'+yv+z11 ^0 has integral solutions
each not divisible by the odd prime p, there exists a positive integer
(-l}/2 such that
yf (mod tf).
«^ М^ш. Soc, Nat- Sc. Nat. et Math, de Cherbourg, 38, 1911-12, 720^7
"• SiteungB. АШ. Wise. Wkn <МаШ<), 121r He, 1^12, 589-552.
■и/бИ., 727-742.
«"Archiv Math. Phyu,, C), i9p 1912, 60-3.
**P6riodico di Mat., 27> 1912r 83-8.
"" Monataheft« Math, Phye., 23, i&12r 305-8.
ш Math. Unterr<f 1912, No. 3, 111-5; No. 4, 150-1
ш BuU. Amti. Math. 8oc,t 19, 19123 2336

772 History on the Theory or NtmsEna [Cha*.xxvi
We may (pp. 402-3) replace this condition by the simpler one2*4
as noted by G. D. Birkhoff. The test fails for p = 6n+l since the con-
congruence has a root. He2"" stated that ^±у*ф П.
N, Altiston noted that <x?±yr=zm has integral solutions if r, m are
relatively prime positive integers. R. Nome (pp. 33-4) treated the same
problem.
R. Niewiadomski^ considered dn=zn— x*—y~. If dm = 0 fornan odd
prime, then d*w+i is divisible by (аг+у)(«—x)(z—y). He gave linear rela-
relations between d*+h dn> d^ and expressions for dn when di=0 {mod n*)
and when i*=0. G. Metrod (pp. 215-6) treated the latter case.
E< Landau2*8 noted that the assumptions
xn=yr-i=i (mod?2), x+y=mp,
where p is an odd number > 1 not dividing то, lead to a contradiction. In
fact,
l=x^1^(mp-y)^1= ~(p-l)mpyr-*+l (modp4)
requires that p divide (p— 1)туг^2 and hence also m.
E. Miot** gave a false expression for the g.c.d. of 2X-1, 3X-L
H. Kapferer2 proved Fermat's theorem for the exponents 6 and 10 by
showing by descent that &=(г*±у*у—(уг)г is impossible.
H. Q Pocklington*1 noted that v*+-\-tf~=z* is impossible for all values
of n for which хщ+уп=*гп is impossible* For, if the former has solutions,
it has solutions with x prime to у and with у even. Thus х*=иг—&>
уп=2ш». Hence u+v=-a*f u—v=&n and u, v equal 2"~V\ *к ш some
order. Thusanifc^ = B7)\
J. R Rowe2" proved that if х*-\-уж=гп) where a;, ^nareodd, thenz+y
is divisible by 2* [evident since the quotient of x*-\-yn by x+y is composed
of n terms and hence is odd]. From this main theorem IT we obtain hie
theorem I' by changing the sign of y.
Ph. Maennchen*3 reported on the history of the theorem. Several
(p. 294) proved that 2*+l is an exact power only for 28+l =32,
W> Meissner3" proved that x»+yp=zv is impossible in integers not
divisible by the odd prime p if there exists no integer v<p for which
(H-l)p-t>>=l (mod p»), я>ф1 (mod p)
let Carmichael^jalsoif p=3k2-=fcl or3*d=2^; also if p, but not p», is a
■* BuH. Amer. Math. SocM 20,1913, 8a
*« Math. Que»t. Edao. Time«t Dew eeriee, 23,1913,17-18.
L^i6di dee math., 20,1913, 70, 98-100.
»»/ЬЙ., 112. Error noted pp. 183-4, 228.
«•Archiv Math. Phy*., C), 21,1913,143-CL
~ Proo. Cambridge РЬП, Soc., 17,1913,119-Ш). ;
ш Johns Hopkins Univensity Circular, July, 1913, No. 7, 35-40; abstract ш BuH Amer.
Math- Soc., 20,1913, 68-69, i
~Zeitechr. Math. Natnrw. Uoterricht, 46, 1914, 81-93. ;
"* Sitnmepber. Berlin Math. G^dL, 13,1914,101-101 See VoL I, Ch. IV,» of thur Hktory^

Crai\ xxvi] Febmat's Last Theobem, 773
divisor of a cumber of one of these four forms; and if p* divides one of the
four forms, provided h and m are divisible by p.
The congruence 5х+7*+Нж=0 (mod 13) was solved by several writers.21*
T. Suzuki2* found the 12 sets of solutions of 5х+8»Ч-11в=0 (mod 13).
L. Aubry*7 noted that, if m is prime to nt хш+уш*=2" has the solution
x=Auat у=АиЪ, z=Av, where nv—mu^l, ам+Ьщ=А. For m^3t n=2t
he gave a solution involving two parameters.
A. Gerardm*7" gave integral solutions of x2—y*=z* for 2^n^8.
И. S. Vandiver** wrote q(r) for (r1^1—l)/p and proved that if
is satisfied by integers not divisible by the prime p, then
)*0 (modp)
is satisfied by each of the six values t^xjy, —*, z(yt and either )
(modp*), ffC)=0 (mod p), or else ffB)=gC)^$E)i-0 (mod p) and, if
p^2 {mod 3), gG) =0 (mod p).
E. Swift2*9 proved that neither of x6^fcy* is a square.
H. S. Vandiver270 proved that if z*+yp-\-z*=Q is satisfied in integers
prime to p, then $E)=0 (mod p) and 1+§+Ц Ы/Цр/^И^О
(modp).
G, Frobenius271 proved that, if Fermab's equation has integral solutions
each prime to the prime exponent pt then q(m) is divisible by p for m = H
and *n = 17, and, in case p^5 (mod 6), also for m = 7t 13, 19. Moreover,
vanisbes identically modulo p for w^22 and m—24, 26» Here the symbolic
power hy is to be replaced by the Bemouffian number ^.
J. G. vander Corput272 proved the impossibility of a^+yB=As5for A = l
and other values.
R. Guimarae9m gave a bibliography and discussed the history of
Format's last theorem, including Wronski's* pretentious.
N. Alliston*74 proved that Format's theorem for odd exponents implies
that b<fH-*+c<"+2=n is impossible
"• Math- Queat. Educ. Ткаед, new sftriee, 2в, 1Ш, 101-3.
■* ТбЬокц Math. Jour., 5, l»14r 48-53. Further report in Ch. XXIII.1*
»»71/intermedbtire des matb., 21, 191*, 19-20.
■"■ SphlDx-Oedipe, 9, 1914, 136-9. For7»-1{F=3*, *Ш» 6,1911, 9U
** Тгапя. Ашег. Math. Soi?.r 16,19U, 202-4.
»• Amer» Math. Monthly, 21, 1914, 238-9; 23, 1916, 261.
^Jfour. fur M*th., 144, 19H, 314-8.
"i Sifxungeber. Akad. Witf. Berlin, 1914, 653-81.
™ №eow Archief toot Wiakunde, 11, 19l5t 68-75-
"RevieU de U Sooedad Mat. Eepaool*, 5. 1915, No. 42, pp. 3345. There in a great
пщ&Ьег of confusing robrprinte. Both СгеЦе'в Journal and Comptee Rendus Pans are
cited № С.Г-. the seeond being once cited as Ст., Berlin!
™ Math. QottL Ednc. Tin**, new series, 29,1910, 21.

774 History of the Theory of Numbers. [Chap, xxvi
P. Montel™ proved that if m, n, p are integers for which 1/m+l/n+l/p
< If it is impossible to fmd three integral functions of a variable such that
хт+у»-\-гр=О] in particular, х^+у^+^ФО if m>3.
P. Kckott2» proved that zll+y"+zll=O is impossible in integers prime
to 11) using residues modulo 11 of symmetric functions of x, y, z.
W. Mantel*77 proved that if n>3 and p are primes, z*+y*+z*=0
(mod p) is impossible in integers prime to p unless p = F/яг—n-3)/(n—3).
E. T. Bell stated and F. Irwin278 proved that if x"-y~ is a prime 2*r+l
for r a prime >2&ndn>2, thenn=3, x=2, y=L
A. Gerardin279 proved that lQk-\-l=z* is impossible in integers if n>L
H. H. Mitchell280 treated the solution of cr*+l = dy* in a Galois field.
A. J. Kempner*11 gave a simple proof that ain—l—2hn has only the
integral solutions a = ± 1, Ь =* 0 [Liouville,3* Lmd214].
A. Korselt282 proved, without using integrals as had It. liouville,10* that
xvt-\-yn-\-zr=O is not solvable in relatively prime integral rational functions
of a variable t if each exponent exceeds 2 or if one exponent is 2 and the
others exceed 3, the case282*1 ^+^+^=0 not being decided» In all the
remaining cases, the initial equation is solvable Cf. Vehnine,177 MonteL*7*
* X Schur281 gave a simpler proof of Dickson's1*9 theorem.
L. Aubry2*4 proved that alOk+l+zn if 0<a<10, fc>l, and n is a
prime .>1.
E. Maillet285 considered ап+Ът=см for m=n/p, where n, p are relatively
prime positive integers and p>l. It has integral solutions each Ф0 if
and only if
has integral solutions each 40 such that ab h^ Ci are prime to p and rela-
relatively prime in pairs, while a2j &*, c4 are relatively prime in pairs and have
no prime factors other tban those of p. The last equation can be given a
similar form in a}, h\f c), aj, b\, c\ which are relatively prime in pairs, while
any prime factor X of a\, b\ or c\ is a divisor of p such that m<l/(\—1).
In particular, if m>l/(>t—1), where /i is the least prime factor of py Fermat's
equation with the exponent m is equivalent to one with the exponent n.
This is also the case if one of a2, b2, c2t a5, 6J, c\ is an exact pth power and
hence if p has at most two distinct prime factors» Corresponding results
hold for aBh-\-bmt=c^1 with any fractional exponents, and with а, Ъ, с rela-
relatively prime ш pairs.
"*Аппд1евяс. Гёоо)е norm, eup., C)r 33, 1916,298-9,
*■ Arehiv Math. Phys.p C), 24, 1916, 90-1.
™ Wiskundige Opgaven, 12,1916» 213-4.
"'Amer. Math. Monthly, 23, 1916, 394.
1M L'mlermediaire dee math,, 23, 1916, 214-5; Spbinx-Oedipe, 19X7»
"•Tram. Amer, Math. Soc., 17,1916, 164^177; AnwOe of Math., 18, 1917, 120-131.
"l Arehiv Math. Fhya., C), 25, 1916-7, 2423
1Ш 8993
< 8993.
^^ Thia equation ia satisfied by the fundamental invariants of the ioosaeder group, ibid., 27,
1918, 181-3.
'" Jahnaber. d. Deutechen Math^Verdnigiuig, 25, 1916,
ш L'btermediaire dee math., 24,1917,16-17.
«*BuD. Soc. Math. France, 46,1917, 26-36,

Chap. XXVI] FeRMAT'S LAST THEOREM.
For reports on g^Cu*^1 —l)/p, see Ch. IV of Vol. I of hie History
There are additional notes by * E, Haentzschel2*6 on 2^x^l (mod p2)'
p = 1093, and H. E. Hensen237 on the computation of qu. *
L* E, Dickson288 gave an account of the history of Fermat's last theoren^
and the origin and nature of the theory of algebraic numbers.
F. Pollaczek289 proved that, if х*+у*+г*=0 has integral solutions prim^
to p, then?» is divisible by p if u^31 for all primes p except a finite number *
also, x*+xy+y*^Q (mod p) is impossible. *
W. Richter** proved Korselt's*** result for £he special case m=n=r^
There exist rational integral functions x, y, z of t satisfying/=яп+уJ4-z*^ ^
if and only if the genus \{n— l)(n—2)— d—r of the curve is sero, where ^
is the number of double points and r the number of cusps. But d=r=§
since dffdz^O, etc, hold only for x=y=2=0. Hence n = 1 or 2.
H> S. Vandiver291 gave an expression for the residue modulo Xя of Kum4
mer's" first factor kt of the number of classes of ideals in the domain
defined by a Xth root of unity» In terms of Bernoulli numbers we car^
infer necessary and sufficient conditions that kt be divisible by any giver^
power of X» He293 stated that if хр+у*+г*=0 holds for integers no^
divisible by the prime p, then 2ZP^1~ 1 (mod p2) for рф1 (mod 11), ano;
that the Bernoulli number B, is divisible by рЧог 5^(ф-Ь1}/2, t=p—Ак
p,p8, p10.
A. Arwin2*8 gave a method to solve (x-\-l)p—xp^l (mod p?), P a prime,
Vandiver294 derived from one source the theorems of Furtwangler2*9 аш}
the criterion of Kummer7* for solutions prime to p of х*-\-ур=г*.
P» Bachmano29* gave an almost complete reproduction of the papery
by Abel,lfl Legendre,17 Dirichlet,20 Kummer/1 Wendt,162 Mirimanoff,180' ***
Dickson/^' 1M Wieferich,211 FrobeniuV*> «• and Furt^^angler «3
Vandiver2^ employed the first factors hi and к of the class numbers of
the fields of the p*th and p^th roots of unity respectively, and the value
of ki = ht/k due to «L Westlund,2*7 and proved that kk is divisible by p if
and only if at least one of the first (p—3)/2 Bernoulli numbers is divisible
by p. Bernstein's234 first assumption m his second case therefore implies
that p — l is a regular prime (so that his result forms no extension over
Kummer*1), while the assumptions in his first case do not as claimed in-
include those of Kummer.76 It is shown that 101, 103, 131, 149, 157 are
the only irregular primes between 100 and 167.
*» Jahieeber. d. Dctitacbcn Math.-Vereinigung, 25, 1916, 2S4,
2П I/cnseignement math., 19, 1917, 295-301.
'« Annale of Math., BУ 18, 1917, 161-87,
**Sitzunpiber. AkadL Wm, Wicn (Matb.) 126,1]д, 1917, 45-59,
"• Archiv Math. Phys., C), 26, 1917, 30^-7,
»l BulL Amer. Math. Socfp 25, 1919, 458-61.
'« Ibid., 24, 1918, 472.
TOActa Matb,p 42, 1919, 173^190,
™ Ajuiale of Math., 21, 1919, 73-80.
"* Dae Fermat Pitiblcm, Vercin Wies. Verleget, W. dc Gruytcr & Co., Berlin and Leipzig,
1919, lfiO pp.
*» Pioc. National Acad. Sc., May, 1920.
™ Trana. Anier. Math, Soc.t 4, 1903, 201-212.

776 History of твш Theobt of Numbers, ПСилг. xxvi
The Encyclopddie des sc, math., I, 3, p. 473, cited the criteria gB)=0,
$) =0 (mod p)y without stating that the unknowns are prime to p.
On и*-\-&=КрГ, where Л ie a prime, вее Baer*4 of Oh. XXI. Thus** of
Ch. XXI proved that tf+jf+s», nbotiu&&+if^ if г is not diviable by a.
Reference* (all included In the present account) on Fermat'e bet theorem occur in the fol-
lowingphcee: Kouv. Oorwep» Math., 6t Ш»,90; Zeitechnft Math.-baturw. Untetridit, 23,1S92,
417-8; B&№ Math. Becreatitms and £evvya71892,27-30; «L 4,1905r 37-10; rintermedialze dw
zn4tb^ 2, 1805, 26, 117-8, 350r 427; 12, 1905, 11-12; 13, 1906, 00; 14,1907,258; 15,1908,
217; 17,1010, 34, 278; 18,1911, 255.

AUTHOR INDEX.
The number* refer to pages. Thoee in parenthesi* relate to eros^eferenoee. Those in
brackets refer to editois or translafcws (usually listed only iD the index of the first chapter ш which
the/ occur). The other numbers refer to actual reports.
Ch. I, Polygonal, Pyhamidal and Fighratb Nukbers,
XL,'' 38
Anonymous, 36
АЫН39
Abbe, 4
Archimedes, 4 A)
Artaudeau, A., 38
Abhtt4
Aiyatte,
Aflt, F., Г1]
Aubry, £., 34, 38, 39
ВаЛе^С. О., в, в, 8 <& 9,
Baehmann, P., 32, 34
ВЫЕ36
De Шшю, Rene F
Dilling, A., 22
Piophaatua, 3, $
16,36)
Doctor, G., B7)
35
Ball, W. W.R., 1
Barbette, В., 36
BariowJ\, 17, h
Bammue, J J.,
Bamie), £., 13
Ba*tien,L.,35
Beguolm, N.t 13 16 B2)
Benzine, JL 5
BWCL37
, S, 9
A> 6-7, 9,
^, ., П $5)
Engelhard, N,, 12
Engler, Ё. APp 35
&oott, E. B.f 33. 37
Euier, L.« 10-17 G, 22, 26,
27, ЗЙ7. 39)
, J. A., 12 A4)
Bible, б
виц, а, з4
BUk033
Fab p
Faulhaber, J., 5
Fauquembergue, £.> 26, 28,
Fermat, P., 6-8 A0, II, 13,
14, 17, 18, 29, 30, 32, 33)
Fontaoa,M.,E5]
Fid №
2535Л. 3i^
в^Гц^'ХГм. a, 5 Fuse, n: (iun <m
Bouniakow0kv,V.,23(l9,39) —, P. H,, [10]
Boutin, A^ 30, 31, 36, 39
Brady, C, 17 GaUimaid, M,, 15
Biaeainne, E,. [6] Gauss, С F., 17 B3)
Brocard, H., 26, 27> 34, 38 Gftardin, Aff 9, 34-39 G,
Gerbett (Pope Sylveeter II),
i 34
Cantor, H
Outer, M., 4 [1 8,41
СишеЙ, A., &>
Catolan, K, 26, 28, 29
Gersonne, J. D., 18
Gerfaiudt, C. I» [9)
Gill С,, 21, 22 B0)
Glauber, J.W.L.,2C C9)
GoIdbaoL Chr.( 10,11
OflbW26
chiuCi, м.^, за
Chnetie R, W. D.f $2f 33, 37
Ghti e^j-chttb, 4
Coete, L. M. P., 19
VAUi2
ШуЫ, Т., 34
Heath, Т. U [3]
ШЯиКВШ] \д-, О
777
Cunningham. A,, 32-34 C9) Henry, C, 26
CurHB.Wv> Й 1^9) Heppel^G.j 25
НШегг E,[l]
Iteidier, AbM,lO *" "
De Joncourt, EUe, 12
I>e Monimort. V. В., ^
De Murie, X, 5
Pe Hocquignyt G., 28, 29,
31,37,39C0)
s
Hoche, R.r 12]
Hoefer, F.f (I/
HromAdko, F., 33
Huntiagton, J., 20
Huttott, C^ 9
Hypeid«*,iC)
Iambfichus, 3
JaeobL C. G, J,, 20 A9)
Jolivald, P^ 36 C9)
Kfistner, A. G., 12 [6]
ШШ, Ь. S., 17
/GW., 12
t J. L., A5,19)
&. A», 28
Lebesgue, V. A^ 22-24
Le Cobte, J. L. A., 39
Legendrer A. M., 17-19 A1,
16,2St 25, ЭД
Leibnb, G. W., 9 C9)
Lemoine, Е„ 32
Leybourn, T.f 17 B5) [181
liozmet, E.p 24, 27, 34 Cd)
" *" J',2^24
S
25-27, 29 (l7) [4]
F. Т!, 22
Mariares, J ..
Marpur^ F. W., 16
Maser, Й, [\7]
Massoutie, G.t |3J
Mathiett, п. В», 38 C9)
Mourolycus, Fi, 5t 6
Mayer. F. Q 9
Mmetola, a,
M<mteiro,A.S^
Moret-ВЬлс, 26,27

778
Author Index»
, G,, 29
Neeselmann, G. H. Fn 12, 3}
Neuberg, J.p 27
Nicbolsoo, [17]
Nicobrtdee, N,,39
Nioolosi, G*, 34
Nicomachus. 2 F)
Niewiadomeki, R.> 38
Оздпат, J., 9
Palmfltrom, A*, 37
Paacal, B>,9
Peacock, G., 22
Р«рш, Т., 25, 29 A2, 39)
Picon, G., 38
Plana, J.. 24 A9)
Plutarch, 3 FP 12)
Pollock, F., 22-24 A3)
Posdger, KT., 18C]
Pythagoras, 1
Ramanujan S. 36
Rcalis,S., 11,24-28C9)
Remmelm, J., Ь E)
Rienaux, M.p 38
Roberts, S., 27
Roche, J. P. L. A. 39
Rodet, L., «I
Roman Codex, 3 F)
Rothen, pip [5J
Schnh, J. O, L., 131
Smith, D. E,, [4]
Soufflst, 39
Speuaippus, 1
St>Croix, 7 F)
Stegmann, F,, 22
Stifel, M., 5 D, 9)
Struvs, J., 18
Sturm, J. В., 23
Talir, J., 39
Tannery, P-, 6, 37 [1,3]
Tcbay, S., 28
Trilhet, P. F., 37, 38
Tenca, L. 35
Tenmdiu», S., [3]
Terquem. O.. 22
Theon of Smyrna, 1 <2. 18)
Tits, L.,35 C2)
Tnmson, A>p 22
TJnicomufl, I., 5
Vablen, K. Th., A1)
Valia, G., 5
Van Cleeff. J., 39
Van Schooten, F., 8
Vieth GVU. A 39
Von Graffenriea, J. B.^ 5
Von Schmtka, L., 33, 35
Von Sterncck, R, D., (Ц)
Wallie, J., 7 C23)
Waring, E., 10
W&tson, G. N.t 2$
WertheiV G., 3 13. 6J
West. J, W., 32
Whitley, J.. 20
Whitwonh, W, A>p 32
Wiegand, A., 39
W^hciuiiB, I., 5
Yang Huip 4
Zeuthen, H. G,, 4
Ch» II. Linear Diophantine Equations and Congruences.
Abu Kami], 77
Aiyar. P. V. S.L[41]
AlkarkhiT A., 71
Amoux, G.3 52,55,64
Ar^ftbhatta, 41
Ata Alia Rusheedec, [42]
Aubry, A., ,52, 87
, L., 70, 88
Aysa, R.T 76
Bachet, C, G., 44, 79 D5,
48P 52, 79)
Bachmann, P.. 52, 85 (92,
95)
Bwiow, P., 65
Berkhan,C> A. W., 50, 63, 82
Bernardi, G., 53
, V» 69
Bemoulh, Jean III, 48
Berteben, N. P., 52
Bertrand, X, 53L72{5H
Berwick, W,E.^., 98
Betti, E,, 74 1721
Bevcridge, W., 61
B<«zout, E., 81
BhAecara A., 42, 43, 59, 73
ВЫ6в2
В*апсЫ,6„в2
Bie, L. H., 50, 72
Biematiki, K. L., E8) 157]
Bindom, A>p 53
Binet, / P. M.p 60, 55, 56
Bbtkl«4
»,l4
BUchfeiat, H. R,97
BoQcompagni, В., 159]
BoDfantini, G^, 76
Bonnycastie, J,, 47
Borelr K, 96 (9^J
Boesut, АЪЫ, 68, SI
Bouniakoiraky. V., 51, 55,
56,71,73,82E1,56)
Brahmegupta. 41, 58. 73
Brocard, H.p 51
Brown, A», 98
Burn^idc, W.,92
Busche, E., 55, 91
Cahen, E., 53, 86, 96, 99 (92)
Calvitti, G., 53
Cantor, M., 58, GO
Carhni, L,, 53
Castcrmun, L.p 53
Catalan, E., 49, 66.68 E6,
67,68)
CaUldi, P. A., 79
Ctuchy, A., J55, 56, 7J ^57;
Cay]oyp A,, 84
Сеааго, Е., 66. 67, 68, 70 F7,
68)
Chang Ch'iu-chien, 77
Chafeelet. A., 86
OieviHard, A. J., 53
Ch'in Chiu-fthao, 60
Chinee, 57
Chryetal, G.; 70, 85
CJaeea, В. 1„ 53
Colebrookc, H, Т., [41]
CordSer, H,, 57
Cotes. R., D7)
CreUe, A^ L., 50p 51, 56
Crocchi, L., 69,70 F5)
Cullcn, J.. 64
Curt», M^ 59 [Щ
Daniels, M. F<T57, 64
Da via, A., [62]
, К W., 62
De Comberousse, C« 53, 72
&e Gvbrgyfexentmikloe, D.
90
De Jonquiirea, E,, 85
De la Bottiere, 46, 61
De Lagny, T. F., 46, 61t 7L
80 D7)
DeiaRocbe, E.t78
DembHChick, H., 53
De Murr, С, Т., F0, 78]
Denioy, A., 96
DePaoli^G., 55 G1)
De Polignac, c^
De Rocquigny, G*, 64
DeSluBe.R.F.,79
Dieu. TS| 53
Dioptumtus, 77, 80 G9)
Dinchlet, G. L^ 53, 94 (96)
D'O М8в
г
gB, A,, 64
Dorn, W., 53
DoBtor» G,( 64
D*Ovidio, E., 83
Duccir E>p 53
Eg«n, P. N. C., 53
Ewenst^in, G., (84)
Elliott, E. B>, 76
EIowbou, G,, 53
Emerson, W.. 47
Ens, С., 60
Euler, L.t 46-48, 61, 71, 80
D8-51,53, 58, 61, 63, 75)
Fekete, M,, 89
FermatrP,, E5, 56>
Ferrent> 53
Fowler, R. H,, 98
Frobeniua, G., 84. 90 (83,
85)
Fuji warn, M., 99
Fyzi, D2]
Gamier, J. G-, 53, 82

Author Index.
779
Gati* C. F., 49, 54, 55, 68P
63, 73, $0^61, 61)
Gegenbaner,L., 87,01 F7)
G*nocchi,A., 49, 77,78
Gergonne, J. D._ 81, [49]
GUlet.J '66, ад
Giraud,G.,98
Giudice F-,73,74
Glenn, О. E4 88
Grace, J. H.,85, 99
Gram, J. P*, 50
Grebe, E. W,, 50
Gr,M^.,53
Grunert,* A 51, [56r 71)
Guatten, R., 53
GunthertS-,66\87
Hankel, H.,58
Hardy, G, H.. 97, Й8
НаувлЫ,Т>,44
Некет, i., 82, (83, 84, 86)
Heflbronner, >. С, [79]
53
,Е.Л53)
k, J. С, L., 53
Нед1,К.,91
Негтея, J., 53
Hermite, ht 65P 67, 94-5,
F9, 70, 96^99)
Hilbert Ь., 96
Hmdenburfc K. F., 63
Hindus, 41-4, 58-9
НосЪе. Н.Л58]
Huberti, tf., 60
Humbert, G., 35, 98, tt*
HurwitB, A., 95P (96,99)
Huiton, Q, 62 F1) [80]
Huygene, Chr., D7)
Ibn aI-НшШп, 59
Jacobi, C. G. J.f 74, 93 G5)
Jaufroid, By 73
Kakeya, s!, 97
KAetner, A;G.,62P
Kei!l,J,,62
Kempner. A» J.p 98
Келеу, У., 45
K^gelTG. 8-, 64
Khirr, Jt», 53
KGnig, J., S5 (92)
Km W, 53
Kroaeeker, L-, 54, 85, 91, 94
(84, 97J
Lacrpix, S. F-, 53
Ladd, C-, 54
Lauremberg. J. W,p 79
Lebeegue, V. A., 53P 64P 70,
72, 74, 83, 86 (87)
LegeDdre, A. M-, 88
Lehmer. D. N., 76. 88
LeotiAido Piaano. 59, 77 F0»
Щ
Lereh,Mv54 F7)
Levi, B 96
Ley bourn, Т., [621
L*bri> G., 49, 55, 65
likwati, 42
Uttlewood J. Б., ЭТ
Lorents, 63
В,.6
er 62
Mac Mabon, P. A., 52 G1)
Mahavtr3c3rya, 41, 77
Mantiactipt» 60
Marchajjd, D., 64
Mathews, G. В., 7l, 91
Matthiefloen, L., 53, 5SP 64
E7)
Mer»y,Ch.,84
Meyer, W. F., 75
^БбО
М^Б-.бО
Mlkami,Y4 157, 60,771
MikWri/H., 95,96 (95-99)
Morale, M^ 53
Moriconjr C., 46
Moas, T^ 73
ki, J. J,, 5a
Nicboleon, P., 49
Nioomachue, 68
Norton, H-T. J.p 97
Ochitowitech, A, ?.p 76
P»ciuo[o Liica, 79
PAlatini, JF4 53
PliPSSei^l
Peiott, J., 67
,p
Pilatt«p 4<)
PinKue, 60
Pisano (eee Leonardo)
Reskot, A-, Ж, 72
Poineot,L.p56Ei)
Porcelli, О.. 63
Pratoiiua, J., 79 (81)
p 70
60, 78
Hcuecble, C. G., 51
Riee», Rp 92
Robmaon. R.T 62
Rolle. M., 45 (
Rudolff, C., 79 (
Bunkle^J, D., [50)
Room, В., 73
52, 53)
)
Safdi, C, 54
S&rtofi» A., 53
Saundereon, N., 46, 61, 80
D9* 93)
Scarpu, U., 93
Sch-Sfer, J. C, 64
SchSwen, V,t51,73
ЗсЬеШег, Н, 53
Schoentjee, Hx 66
ScbonemanTi, Т., 86
ScbubertH., 53 (86)
Scbukr, W. F. S,, 52
Sctorenter. D., 60
Shod» Б, AftUm, 77
81* J51
81ад*, J^51
Smith, H. J. S., 51, 72, 83.
89, 90 G5. 83-5, 92)
, D, КЛ77]
J Go
Sommer, J.,
Soacbino. 8.. 53
Spdta, C, 46
Steiiut», E., 92
Stern, M. A., 65, 86
StiettWT 1&475
StiettWT. 1,&4,75,В5
Stifel, M., 60
Stracbey.E., [Щ
StJuiete,X., 53
Stnive. J.r 82
Studniika, F. J», 89
Sun-Tau, 57, F0)
Suter, ft, 1771
Sy]veet«r, J» J., 51, 66 G9,
93
Swmc, 64
TaxtagJia, N., 79
Taylor, J., [42J
Tchebychef, P. L., 54, 94
Teunme, J. A., 53
Teraue»i;0.,77, 78 [57, 79]
Теяй, G. M., 63, 72
Thaaiup, F.. 53
Thacker, A., 62, 80
Uspenaki], X, 96
Vacea, G4 57
Va» C^a, S, L., 9в
VSbte;£\
Yon Cg, ,
Von Speyer, X, 78
Von Stemwfc, K. D., S7r
W*lHfl, J,. 61 D7)
Weber, H., 83, 92, 97
Weih»uch,K,,68,75
WelLrteili, J., 97
Werebruaow. A, S,, 68
Wrtbi О ДО S0^
«rtbeim, О,, ДО, S0^
WeyLH-,97,96
Wheder, A» D^ 50, 63, 65t 72
Wiedemanit, E., J59]
M., E5-57>
F,, [77|
H, J>r 64
. . > 1571

780
Attthor Index.
Yih-hing, 58 F4)
Yotmg, A., 85
Agionomov, Nm 160 A39)
Anonymous. 127
Arbogaet,L.F. A., A20, 129,
133, 158)
Bacbmann.P., 147,169
Bancroft, D., 142
Barbette, E., 152, 159 A39)
BattaglmL CL 121 A1G)
Bella^tiB/G., 123 A24)
Ве11епяЛ45
Betti, КД12П
Bignon, L. 126
BoniaUi, P., 118
Boechi, R, 132
Boecovkh, P. R,, 103 A06)
Bouwakowsky, V,, 129
Brianchon, C- J., 113
Brioschi R, 120, 121, 130
A19, 137)
Bninacci, V., 112
Brunei, G., 149
Brueotti, L,, 155
Candido,G., 161 A12)
Capeffi, Jl. 134
Catalan, K, 114, 126, 142-3
A40, 156)
Cauchy, A., 115 A21, 134,
156)
Cayley, A., 119-23, 126-7,
130-1,133, 138, 14ХЫ, 147
A17,119,121,161)
Ceaaro,E., 140A64)
Christie. FL W. IX, 158
Ctuystal, G., 141, 145
Ceorba, G., 153,161
H, M. F., 145 A27)
Darling, К В. С, 163
David, E. А. А., Ш
De Montferrier, 127
De Morgan, A., 115 A17,
126, 135, 145, 151, 160)
ВегЫя, J-, 124
DMDcagne, M,, 154 A45)
Durfee, W. P., 136A37)
Eamoneon, L., 148
Ely, G,S,, 136,141 A42)
Euler, L., 101-6 A03,112-3,
115-8, 120, 123-4. 126-8,
130,133,136-8,140,14fr-7,
149, 155-6, 159, 162)
Fai di Bruno, F., 130 A20)
Farey, J. A49)
Fer«ob, E., 124-5
Ferrers, N. M,, 118 A24,
137, 139, 154,158)
Fofsyth.A. R.t 150
Frand, J,, 152
Franklin, F., 131, 133, 137,
139 A37,138,146)
, J. G>, 57
H.G., [79]
CH, III. PARTITION3,
Fregierr П2 A18, 127, 130,
161)
GambardelK F., 128
Gsun.C.F*. 155 A49)
Gi^i.D., 155A33)
Glaieber, J. W. L., 129, 130,
133. 140, 158-9 A37, 140)
Gloeel, К.Л51 A46)
GoldacWdt, L., 146 A16)
Goormaghti^ti, R., 162
Greenetreot, W. J., 16О
Hammond, J., 162A54)
Haidy,G.fc 162
Hermes, J 149
Herroite.C., 126
Heiechel, J, F. W.r 117 A19,
145)
Hindenburg, K. F,, 103, 106
A03)
Jackson, Г. Н» 155
Jaoobi, С G. J.. 113r 116,
143 A30,135r 139,140,146)
Jelinekr L.t 128
Jenkins, M., 139, 141, Ш
A39)
Kempner, A. J,, 160
Kirkman, T. PM 118-9, 128,
147 (lfe)
Klugel, G. S,, [1031
Kramp, ChrM 106
Kroneckei-, L., 104
Kuechniriuk, M,, 150
Labey, J. В.. 11041
Хлсгой, S. R, 112
Lagmnge, J. L.. A54)
Lagueme, Б.{ ldO
" Lanavic^nsiB." 124
Landau, E», 1$5, 158
Legend», A» M., 113 (H6,
147, 152)
Le Heux, J. W. N,, 161
Leibni^G^W., 101
LemoiDe, E., 127 A12)
Leonardo Pieano, 105 A10)
MacMahon, P. A,, 141,143-
5,147,150-L153-1,156-*,
160-3 A49)
Mahnke, D., 101
Malfatti, G.F., 110-2
Mansion, P,, 130 A12)
Mareano.G. В., 127,133
Maaon, Т. Е.. 160 A39)
MathewH,G.B.rl50
Meieselr E., 127, 143 A27)
Mefo&aer, OP, 156
МкпеЬеиlJ, a. C, [104]
Mifinoai, G..156
Minetola, S., 156-7,161
Zeigmondy, K., 87 (92)
MitcheU, О- Н^. 1Э6
Mortara, E., 123
Muir, Т., 155
Naud6, P., A02)
Netto, E,, 134,154-5, [150]
Oettinger, L., 124
^P» 107-10 (П0-2,130)
Pascal, E., 144
Pimna, С» М-, 132
Plainer, G., 145
Pomey, J. В., 14
Ramaiiujan, S., l62-s>
Roberte,S., 124 A19)
Rodriguee, O», 114
Rogere, L, J^ 148, 163
"Rotciv," 150
Saduo, E., 144
8ardi,C, 125, 128
Schlegel, V., 126
Schubert, 145
Schur, J<f 159,163
Scona, G,, 161
Sharp, W. J. C.t 141, 1434,
148
Silldorf, G., 128
Smith, H. J. 8., 113
Stem, M. A., 114, 140 A40)
Stifel, M., 105
Sylveeter, J. J.f 117, 119,
122-4, 128, 131, 134-5,
137, Ш, 140-1, 144 A06,
116,120-1,123-5,131.133,
136, 142, 151-2, 154-5,
159, 160)
Tait,P. G., 144 A41)
Tanner, H. W, Lloyd, 143
Tanturri, A,, 164
Tebay, &, 145
Thorm, A., 149
Trudi, N.. 125, Ш A03,
106,119)
Vachette, A., 126 A26)
Vahkm, K» Th., 146 A50.
152,154,162)
Van Schooten, F., 105
Verniory, 145
Vince,S.,H2
Vo^celh, P., 118A12)
Von Schnjtka, 1^, 162 A46)
Von Stemeck. R. D., 150»
152, 154, 160 A64)
Von Wanerecbleben, 128
Walter. Т., [1301
Watburton, H., 117 A18)
Weihrauch, K, 127 A43,145)
Werebruaow, A. S., 155-6

Author Index,
781
Wolff. Я..153
Wdkn, H. J., 148
Zetier, Chr., 140
Zeigmondy, K,, 148
Zuchrbtian, J., 146 A51)
Aida,JL,lftB
Airy, G. By 190
Alhocain, Ben, 166
А1кдгШ,166
ABmmii, G. J. Ц65]
Ano&yzoow, 166, 171, 175-6
АрэдЫыспЬа, 165
Си. IV. Rational Right Triangles.
J.L.U85)
АиЫу,А., 171,187
Bachet, С ,GN 175, 177, 188
A76,' 181)
Bachnunn, P., 183A83)
ВаЫег, В., 172, 174, 176,
Berichan, O. A, W., 168
,A,,166,177
к т
Collme, ^173A^4)
, А., Ш, 178,
DavCT J,, 180
De Klhr, J^ 173, 175, 177-
8,185
Be Combmjxisee, C^ ЦБ1
De Jonqii№& £ № A72)
01т7КЙ7(Ш)
(Ш
160 A84)
517217
._._ ,172,174-
7. 186, 188 <lfcf 173^4,
'*- 181-2)
Р. О., Ш, 190
x^peibn " 180
Eeoott, К. В, 184
Euclid, 165
Euler, L,, 167. 17?, 175, 179
A7L 185)
, А. В., 181--2,187
F&uquemb^rgue. E,. 179r 186
FermsiX, P., 172-3, 175-8,
Ш, 184-7 A72, 179, 181,
183-4)
,Э.,[
Fitting 170
3*TN
Fontene, G-, A83)
Fremcle, 171,173,178,183-5,
Ш (та. 185)
1«11,а,]1б
О&шЫоИ, Р., 160
Gedlkmg, J., 170t 190
GezUkimaU0lNv17O
Genoochi, A., 166,176
Gerardin, A., 188
Qill0tt176
Girard, A.f 131
Gbih
Gneber, 169
Grm»oble, E. В., 171
Gnib«, M. A. ,171, Ш
GruiJ, 190
Gnuwrt, J. A., 171
Griiaon, J, p., 167,173, 182.
HaenUecbeb E., 170
Hmhted, G. B.r 189
Hart. D. S., 160, 174. 182»
190 A72)
Heald, a, ISO
Heath, T/Im [165, 1771
Hdbj. £.,№&, mi
уГ,, [165,171,181]
Нвгод of Alexandria, 1165)
НШуег, С. ВЦ 174
НсДш, А,, 171,Л75,181, 185
HopkU, G, H t 182, 184,
190
Huitan, a, 182 AS3)
17Й
lntfifcy, С. М., 171
, 188
. E., 170
Kersey. J. 173,180
ia«nf k, A70)
Шиле, W» 170
Ejkiaefr, V. J., 131
Koecbezo, С А,, 167
i, W., 190
Lasnbeft, P.. 170
Lefamer, D. NP, 172
Lentbenc, P., 171
Leorado Piaano. 1^-7 A75)
Levy, A,, 171,183
Leybourn, Tb., 179
178]
Iieber.BL, 190
Iiouville, ^168,171
Lucae, E., 176,181
[166,
, i79
R, 177, 184
Manuecnpt, 175-6, 1S0
Mvlio, A,. 174,1S2-3,190
Martone, M., 186
MatamAgq, 167 4
Meyer, K, 170 A67)
.Tyl9ft
L A. J.F.. 168
ftiiYJi67)
,J)
MilbornJ. C., 170
Monok, H. 8., 184
Moutu^KJ.K., 1S2
МогЫ-ВЬшс. 1$1 A82)
Muir, T-, 1Й
Nidta, Т.. 183
Nipeue, M. J., 165
ОйЫп»,а.А,.г8а
Oughtred, O., f 71
Cfcaniini, J., 178, 182, 187
A83)
G. M.. 190
Psgoim, G. M
P«det,F., 171
Pftpta, Т.Л84-5
Perkins, 6. R.r 181 A80)
Fiiuna, С. М., 169
Pbto, 165 A67, 171, 177)
Poineot, Ъ.. 1вГ, 171 A71)
]
ygs 165 A66-7, 177,
187)
Quintili, P.t 170
Вдо, Т. 8-, 171
Rath, Н.Л39
RouUel, P., 160
EkhrtP17l
> 187 A78)
S»n& E., 189 A90)
Senjmna, K- J, 186
UfcLl8a
Sohotten,

782
Author Index.
Schubert. H., 171
Schul*, J. О L, 186
Sehuke, J. C, 189
Scott, Judge, 182
Stifel, M,, 167 A70)
Svechnikoff, 188
Tannery, P.. 186 [165]
Taylor, J.. [166]
Tebay, S., 184, 188-^9 A90)
Thaarup, F., 171
Thacker.A., 171
Tbeon of Smyrna, A67)
Tiebe, A., 190
Tucker, R>, 187
Turrifcre, E., 188-9
Tweedie, C*, 174
Unicorn us, I.j, 167
Van der Bupg C. J., 170
Varali-Tbevenet V,, 171
Vermehren A,, 171
Victa, F,, 167, 172, 175
A70, 173, 176)
Volpicelli. P., да, 168 A69,
172)
Von Luhman», F*, 190
Walte, J., 184
Wheeler, A. D., 171
Whitley, J,p 179
Whitworth, W. A.p 190
Wiegand, A.. 189
Wildbore, C, 178
WilkiuBon, Т. Т., 182, 184
Woepcke, F., [166]
Wobtenholme, J., 182
Wright, W., 180, 187
Yanagihara, K.. 167
Young, J. iL, A73)
Zcuthen,H.a,
Ch. V. Triangles, Quadrilaterals and Tetrahedra with Rational
Sides.
Alliflton, Ny 215
Anderson. J., 194
Ankum, B20)
Anonymous. 213
"AtticuB," 304
Aubiy, L>? 200
Bacbet, C. G., 191-2, 205,
209A93)
ВасЪшапп, Р., [1961
Baker, Т., 194
Bambaa, M., 224
Bailmen, Б. N., 200, 220
Beard, W. F., 221
BeU, E, Т., 201
Berton, 214 B21)
BeeeeU, F., 224
Bhaecara, A,, 216
Biddle, D., 199, 212
Bille, В., 214
BlichfeldtJI. F.. 198
Botteher, H.r 200
BrahmeKupta, 191, 216-7
B20)
Brancketr Т., [201]
Caraot» L. N. M., 217
Ce&aro, £.t 215
Chartres, R, 212
Chaelee, Mf, 216 B17)
Colebf©oke,H.T.,[l9l]
Скюк, А., 194
Cimliffe, J., 193, 201, 204,
210-1,219,221 B05, 212)
Curtius, S., 191 A95)
Daibouxr G.T 219
DaveyrJ., 193,212
De Lagny, F., 206
Deecarte9t R., 201
De Virieu, J.t 214
Diophantue, A91-2, 205^6)
Dolgufiin, P., 212
Euler, L., 193, 2024, 221
B01, 204-5, 207-8, 219,
221, 223-i)
Evans, А. В., 197, 215, 219
Feldhoflf, 212
Fermat. S., [191]
Fu«, N.,204,210B12)
Gad6eap 216
Gau»tC.F.t 195 A91)
Gepnim&t&s, N., 200
Geraidin, A.,200,220
Gerono. G. C.r B16)
Gill, C,, 194^5, 205, 211
Girardr A.t 214, 217
Gould, B- A., 206 B06)
Grebe, 217
Grebe, E. W., 195, 205
Gruber, M. A., 206
Guoteche, R.T 208, 224 B09,
224)
Haentzschd, E. 205, 209
216, 220. 224
Halsted,G. В., {1Щ
Harmuth, Т., 199
Hart, D. a. 196-7, 219
A98)
Hecbt, В., 200
Heinnrhfl. J.p 213
H6tE2O6
^p, ., 198 B00)
Heron of Alexandria, 191
A94, 19S-9, 207)
Holroyd, J., 2i2
Holt, C, 194
Hoover, W.p 201
Hoppe, R., 197, 221 B0L
224)
Johnson, R. A,, 215
Jonee, S., 211
Kastner, A. G., [191]
Kersey, J., 209
King, W, F., 206
KlobaBsa, C. 216
Kommerell, K., A96)
Kummer, E* K, 191. 217
B09r220)
Kunze, C. L, A., 195, 205
Kunubima, 192
Lampe, E., 220
Lehroer, D. N., 199
Lenhart, Wm., 206
Iigowekj, W,, 196, 219
Martin, Af, 197-200, 205,
215 A96, 198)
Matsunago, 192
Maurin J,, 206
McKenxie, J- L>? 197
Mikami, Y-, A92]
Monck, H, S., 197, 214
MuUer, R,, 198
Nakane Genkd, 192
Neim, F.p 220
Neuberg,J,, B15)
Newton, L, 220
PeU. J., 201
Рерш, Т., 198
Peters, С A.F., [195]
Rahn, J. H,, 201
Rath, H,r 196
Rignaux, M., 201, 213
Hitter, F-Л W2i
Roberts, C- A., 198
Robins, S., 198, 219
Rutherford, W*, 212
Sachs, 3,199
Safford, Т. Н., 199
Scherrer, F. K, 201 A96)
Schlonulch, O.T 195, 198,
219
Schubert, H,, 199, 205-8,
213, 219, 221, 223-4 B01)
Schub, O., 213, 224 B20)
Schumacher, H. С„ 195
Schwering, K., 2l3-5t 222
§imerka,W., 196B01)
Smith, D-E.J192J
Tebay,S., 196A99)
Terquem, O*, 216
Teach, J. W,, 205
Turricre, E., 201, 205. 213,
215, 221 B21)
Van Schooten, F., 192, 201

Author Index.
783
Vieta, F., 192
W&teon, G. N.,221
WeUl, M., 216
Wetetvtra», K<, B20, 224)
Wbitworth,W.A.,l98-9
WolflteiboW J., 196
Woo&oaae, W. S. В., 214
WorpHfky, 198, 212
Wright, W., 211
Yatee, В., 195
Zaitben, H. G.r;
ZQge, 215
Ся. VI.
op Two Squahbs,
Ahlbom, H~, 236
Albocain, Ben, 225
Aradt, F.( 237
Aubry, A., 245l 255
Bachet, С G*, 226-7 B29)
Bachman*, R, 243 B34)
B641
Baririen, E- N.. 253-1
Barlow, К 234
BarteUb,A.L..256
Bastien, 1^256
Bezuelin, N,, B32)
Beflavitie,G.,239
В А 249
да, „
Bertram!, J, 244
Bhaacara, Av B29)
~ ош, G., 253
Bock, J. т. ,-»'-'
Bonfantini, G., 256
Boret and Drach, 12401
Bougiuef, N. V., 246
Boumakowaky, V., 239
Brahmegupta, B55)
Biicaid. JL, 255
Biocarf, П, 254
Cardan, H. C, 226
Carmichad* R. D., 266
Caialan, K.> 247, 249, 251
B57)
СаисЬу,А.,234,236
Cayley. A,, 240-1 B36)
Сшаго,Е.,246
Chalauz, M-, 257
Chaeiffl, M-, 226
Chavannef], F. L. T,t 243
Chelini, D-. 239
ChryBtal. O-, 240
nimninyhmn A. 241 252
B57)
De Longcbimpe, G., 251
Deltour, A., 254
Deecartee, tL, 228
DickeonTt. E., 252, 254
Diophantua, 225-6 B27,229,
231,236-7)
Dirichlet, G. LF 236, 242
B33, 235, 240, 248-4)
Eisenstein, G^ 236 B34,
239,241)
Ешпю, V., 242 B51)
Eufer L., 229-232 B25,
229, 232-4, 239, 240,243,
244, 260, 256-7)
EL. 246
Fermat, Pn 227^8 B29, 230,
240,^56)
Fleck, A,, 256
IbiUa252
aum, C, F5 233-4 B36-
241, 243, 247, 249, 250,
252-3,257)
Gmnb^uer, L, 247, 250
B49)
GenoochL A., 226, 235, 239,
240 Г2&. 240)
235,
Geron^G/O. 2^2
Ginurd, A., 227-8
Glaieher, J. W.
243,250
Goldbacb, Chr, 230
GoldBchdder, F,, 250 B34)
GoldflcKmidt. L , 241
Gruber, M. A. 251
GuDther, S., 246
Haentuchel, E.} 256-6
Hlb G. It 244
Hermite, С 237, 239, 247,
249 Л34, i60)
He«erW
HeydonJ. K, 255
Hoppe, R., Z43, 261
Jscobi, C. O. J., 225, 231,
236-7 B36, 239, 241 243,
246,248,2^0-1)
Jmootetahl, E., 253
Jaquemet, €., 229
8,, 241
R, 232-3
Kulik, / В., 237
Ьадешде ^- Ь., 232 B31-
% 244, 264)
Lit С А
Laiemut, ^ 5
Lambert, R, 2S5
I^DdftO, E, 25M, 257 B53,
256-7)
ЪЫР 8.232
e» V. А4 237, 239
, A. M^ 233-4 B37V
Leonardo Fbuio, 226 B26)
Lrelie, J., 232
L6vy, A.; B53]
Lbflfe, J,, 241
ucee. E., 244-5, 250 B46r
257)
Malebr*ncbe. N-, 229
Marched, 2^
Matn>t, A., 251
MUunK22Q
Mwipia>G^227
McDonald. J. H., 251
Meyl,A.,B57)
Moi; J., [248]
Morit», В- Е-, 252
Muir, Т., 244 B54)
Nwdniofr, P. S., B57)
Paciuob, L., 226
PabmtrCm, A., 262
Pentemak, P^ 252
Paulnuer.255
*
Kdffer,
Plan*, J., 241
FhMihet, E., 239
ЕЫш.8.,24&248
ReoBchJe, G. G., 241
Riux, M., 257
ffiS57F
Roberts,
Schubert, H., 252
Setting, P-, 243
Secretr J. A,, 237
Sierpmflki W.^253 B55-6)
8iffiin«DLB:, [236]
Smith, H. J. S.,240
вттвг. J» 253
iHie^TJ24e
St6nn№C.,25l
Suhler H., 239 B36)
4,257
TiurndT, R, 225
Тегашть О» 226
TfflBuPF2^
241-2

784
Author Index.
Vaoca,G.,227
УлЫеп JL Th,, 250 B54)
Van der Corput, J. G.,
256
VermehreD,A.,242
Vieta, F,, 226
p
B40)
, P4 234, 238, 240
., 257
254
Wantielr L., 237
Weber and W&lbiem, [231]
Weber, H., 251
Сн. VII. Sum op Three
Aubry, L, 272-4
Bachet, C, G.r 259
Bachmann, Р.. 270
Bahier, £L, 27*
Вапнк*, Й. N.. 273
Begueiin.N*, B61)
BelUcchi, 265
BiwondAi, G., 27X
Burhenne, Н., 264
Catalan, E., 266-270 B66,
270)
Cauchy, A,» 262
Cesaro, K, B74)
Chrirtie, R, W. D., 27i
CliK261
Daiflelli, U~ 266 B67)
D'Avillez, J.F.,270
Da^ifl, R.F., 2?3
De Murr, С. Т.Л259]
De Rocquigny, 270
Deebores, A, 269
Drt R 259
Descartee, R.r 259. 260
De Tanne&berg, W., 274
Diht 259 261
&beg, , 2
DiophAntufl, 259, 261
DincMet, G, L., 263 B64)
Ее11и,Т¥.С.,274
EieenBtcin, G., 263
Ibcott^E. В., 271
Euler, L. 260-1 B65-6, 270)
——f J» A., 261
Fauquemberguc, E., 268
^ Й, 269,260 B61,265)
nuDD, С F._ 262 {261,
263r 265, 268)
Genocchi, A., 261, 265
Gerardin,A1L272,274
Gerono, G, C., 268
Gill, C, 262
Glftfeher, J. W. L., 268-9
СНеш*, Й2
Goldbach, Chr.± 260
Grigorief, £.. 271
Gruber,M.A.r270
Haas, 271
Halphen, G. H., 266
Hennite, C.r 265,269 B73)
Hoflel, X, [263]
Hromadko, F4 271
Humbert, G.r 271, 273
Hurwitx, A., 271
Jacobi, С G. J., 262 B70)
Kronecker, L.r 265, 268 B74)
Lagrange. X L4 261 B60)
landa^K. 271-3 B72)
Lebespie, У,А., 264-6
Legcndra, A. M.p 261 B62-3)
lionnet, E.t B67)
Iiouville, J., 264-5
Lucae, E., 272
Maliatti, G.r B66)
Mathieu, H. B.,271-2
MatHunago, 260
MikamCy.r [260]
Miot, E., 274
Monck, H, S,, 267
Aid*, Ay C02)
Aubiy, L, 302
Rachet, С G., 275
Ch. VIH. Sum of Four
ChriHtie, R. W, D,, 299
Corey, 8. A, 302
Beguelm, К, B76)
Berkhan, С. A, W, 290
BLaocht,296
Bokazxo, B4301
ВоадйакогЛу, V., 285, 292
B86)
Catalan, E., 293-e, 29S-9
C00)
Conchy, A, 284-5 B78,
288,300)
K
Davis, R. F., 299
Ddtour, A, B86)
De Murr, С. Т., [276]
Deeboves, A., 289
Deecartee, R., 275^6 C02)
ГЯскаоп, L. E., 300, 303
р75B76)
DinSlet, 6, L., 290 B96,
302)
Dubouie, E., 302
^ ., 287
Euler, L., 276-9, 281-2 B79,
281-2, 284,286,291,297-
300,303)
Wheeler, A. D 23«
Woepcke, F., 225-6, 241
Xylaader, G., 22G
MordeU, L, J.t 274 B65)
Muhle, Ьм 273
МШ1егг J,, 259
Neuberg, J., 266-7
Pendae, N. B-, 272
Pocklington, H. C», 273
Proth, F, 272
, 266^8
t 2
giomota 259
Rignam, M., 273
Samle-Croix,259,260
8ап1апа,К. J,, 2*2
SchierTo.^267
Schubert, H., 271
SieriMnaki, ^.,272B73)
Smith., H.! J.S., 262
Turriere, E-, 274
УаЫеп, К. ТЪ., 270
Yon Sterneck, R. D., 270
Welech,! _
Werdbrueow, A, S., 271, 274
mitworth, К 271
Young, J. R., 262-3 B70)
Zolotaref, G., 265
Fcrgok, E., 295
Fermat, P^ 275-6 B82)
Frattini, G., 296
C, F., 282-3, 300
,297)
-_Jauer,L.,300
Genocchi, A., 2&» B83)
GttlA27e
Gttawl,A,27e
GUieher, 1 W. L., 285, 293,
295-7, 301 B83, 302-3)
GoldbacK Chr-, 277И* B84)
Hmnite, 0,288,295 B83)
Humbert, £., 299 [299]
С Q. J^. 285-в B82-

Author Index.
785
Kauder, C. F.t 276
Ugrmnge, J. K, 279-281
B81-2,286, 299, 301)
Landau, K, 302
Lebesguej V. A., 293 B82)
Legendre, A, M., 282, 286
B83-4, 291, 298-*)
Libri, G,, 286
iionnet, £.. 293
Ldouville, JL 291-2 B85,296)
Iipechitc, R>, 297
MaKatti, G. F-, 283
Martin, A., 302
Matrot, A., 299 C00)
Mtood, G., 302
ding, F., 2S6
Neuberg, J>, 293
Pellet, A, E., C03)
P«?pin, Т\, 296, 298 B90)
Petr, к.. 300
Picard,E,,289
Flana, J.,292
Pollock, F-, 288-9t 291-2
B82)
Puchta, A., 297
Riatis, S.p 293-4, 297 B93)
Regitej 276
МЫкГ^.гЛ
Mordetf, L. J>, 303
MQller, J., 276
Adcock, R. J.p 321
Aid» Ammei, 318 C19.321)
Saintedoix. 275
Schonema»nt Т., 287
Smith, H. X SM 292, 295
Sotferuneki, В., 300
Souitlart, C.p 291
Stem, M* A., 298 B91.
299, 301-2)
3tudtu&m F, J., 300
Sylvester, J, J4 290, 295
Tannery and Molk, 285
Tchebychef, К L., 28S
Torelli, G., 294
Unferdinger, F., 293
Vahlen, K. Tb.r 299 C298)
Vaxk E. Tengbergenj C», 302
VonSpeyer, J., 276
Von Slerneck^ R, D4 301
Weil]. M., 290
Werebruaow, A. 8.r 303
Werthdm G.p 300 [279, 299]
XyUnder, G., 275
Zehfuaa, J. G., 291 B93,
296, 208-9
Сн. IX, Sum op n Squares.
Euler, L4 C18)
S., C21)
Mikami, Y.p [3181
Minkowski, H, 312 C05,
Ш 3U)
ВасЬтыш, P., 314
Bb K, 323
,,p
Barlow, P,, C09)
,(
i, G., 318
Boulyguinej В., 317
В«2АШ
В«2п,А.,
Brocard, H.,
ri, L., 319
Candido, GM 319
Catalan, EM 309, 318, 320,
321 C13)
Cauchy, A., 318
O*aw),E.r 311,312 C20)
Cbrietie, R, W. IX 321
Golligoon, Ed., 320, 322
Cunliffe, Ji, 319
Delannoy, H4 314
1)еРао1и,Н„320
Dick«mr L Й., 323
1>*О<»впе, MM 310
Doetor, G- 320 C22)
Dubouis, E., 316
Eiaenstein, G r 305 C06^8,
312, 314-6)
Eecott,E.B.,l323)
Fermat, P., <309)
Ge«eDbauerr L.. 313
Gl»iflherr J. W. L.p 309, 313-
6^ 320 C171
Goornugntighj R., 318
Gray, J. A,, 320 C21)
Hsrdy, G. H., 313
Hsrt, D. S4 318. 320
Hermite, C4 310 C18)
HitomirC., [318]
Humbert, G., 315 C06r 318)
Hurwitsr A., 311
Jaoobi, C» G. J., 305, 316
C06, 307, 308r 310)
JaCobrthal, E4 314
La Marcat G,, 322
Lapdau^ E., C23)
Lebeegue, V. A., 305, 306
Lemoine, E.t 309, 310, 3H
liouville, JfJ ЗШ-9 C07,
308,311,314,316,318)
Lucae, E., 319 C20-3)
iQj A., 318, 320-2 C22,
323)
Mathewi, G. В., 313
Meiaaaer. 0., 314
М^уиг, Т., 32a
Miot, E.. 323
Monck, H.S.,319
Mordell, L. J.,317,318
MoriUj R. E^ 314 C14)
Moureaux, M>, 319
Naeimoff, P. 9., 312
Pepin4T 310
Pelr.K.314,316
Plulferiji,RH.,321
PoUock, fc, C1Й)
Ramftnujan S*,317r318
Ru№n{, F*P.P 319
SantomaurOj plj 309
Sardi,C43O9
SierpWi,W.p315ClS)
Smith, H» J. S., 308, 312
C05,314,315)
Stieltjea,T.J,,310,3l2C11)
Tano, F, C21)
ТогеШ^.-ЗОв
Ttirriire, E4 C23)
Uepenskij, J. V., 316
Ch. X. Numbeb of Solutions of Quadratic Congruences ш п
Unknowns.
Aubry, L4 C25) Fnkttini, G., C25) Jordan, C4 325, 326 C27)
Bachmann, P., 327 Gegenbauer, L., 327 Kbts, J., 327
Dickeoa L. E., 327 C25) Hermite, C, C25) Lebesgue,V. A., 325-6 C25-7)
51

786
Author Index.
Le Уатааеешг, В., 327
lAn.G,, «25,326)
Iipechiti, K,, C25)
McAtee, J. E,, C27)
MinkowAi, H., 327
Pepin,T.,326C25)
Schtinemann, Т., C26)
Smith, H. J. В., 326
V&nKTeugbergen, C-, C25)
Zragmondj, K., 327
Ch. XI. Liouville's Series or Eighteen Articles.
Bachmann, P., 339
Baakakov, S. J., 338
Bougaitf, N. V., 338
Bouniakowaky, v., 331
Deltour, A., 339
Diricblet, G. L., 337 C33)
Fergota, EM 338
Humbert, G., 337
Jaoobi,C. G. J., C29)
Lebeegue, V. A., 338
liouville, J.. 329-338 C37,
339)
, G. В., 337
Meiesner, Бч 339
Nanmoffj P. S., 339
Pepb, Т.. 338. 339
Piuma, С. M., 338 C39)
Smith, H. J. S>, 338
Toreffi, a, 338
Ch. XII» Pell Equation;
+c made л Squabb.
Adr*in, R-, 367
Alkaleadi, 350
Alkarkhi, A,, 347
AmthorlA»344C46)
Archibald, R. a, 345
Archimedes, 342 C44-5)
Axndt, F,, 374-^5 C66)
АясоК G.r 397
Aubry, AM 395
Auric, A.f 39в
Bacbmann, P., [3711
Baines, J., 369
Barlow, PM 368 C82)
Baudhiyana, 341
Bell, А. Н„ 345,390r 395
Berkhaa, C. A, W.j 377 C67)
Bh&*carar A., 347-8 C46.
349. 351)
Bickmorej С. К, 389 C68,
394, 398, 400)
Bilk, S<, 381
Bobyninr V. V., 342
ВоЬшёек, 8Ц 398
Bony, J4 396
Bortolottij E-j 390
Boutin, A., 391, 393, 396
Br&hmeguptaj 346 C55r 399)
Brancker, Т., [3541
Brocard, H.r 383, 391 C86)
Brouneker. W.. 361, 353
C46, 352, 364,363)
oj J., 351
Cantor, M.,344f 346
Cardan, H. C, 351
СМ4Ш
Савя1П,М.4Ш
Catalan, E.T 385
CaUldi, P. A4 351
Cayley, A4 377 C89, 399,
400)
Cbabert,374
Cbaalee, M4 355
Chatelet, A., 396
Chrielte, R. W., D., 392,
394-5.307
ChJTBtal, G.r 387
Chuquet, N^ 350 C50-1)
Colcbrooke, H. Т., [346-7,
350]
Collina, M4 381
Coesali. P., 367
Croweli^ E. В., 346
CunntDghatD, A., 302j 394-8,
400 C58,366.368,377,389)
Curtze, M., 342
DrAndrea, C, 372
Dedekind, R., 371, 378 C73,
399)
Degen, C, F., 368 C73, 377,
388-9, 394-5, 398^9, 400)
De Jonqui6rear E., 386,391-2
De Khanikof, N.. 379
Delambre, J. B. J.r [3431
De la Roche, E.t 350
De la Vallee Pouean, Gh,.
390
De PH6pitaI, 354
Do Longchampe, G. 385
Demme, C4 342
Be Ortega, J., 350
De Polignac. C*,392
De Sluae, Rene F., 361
De Virieu, J., 384
Didon, R, 380
Dbphantue, 345-6 C46,363)
Dinchlet, G- L4 369, 370-3,
376-7 C65, 367, 376, 378,
381, 393, 396,398)
DubouiB, E», 398
Du Haye. 374
Dupuis, J., [3421
Durfee, W. P,, 385
, P. N. С, 369 C79)
atin G.r 374
Euenatein, G.r 374
Е1-Наюагг 350 C51)
Elphmetone, M., [3461
Et^ G 354
Elphmetone, M., [3461
Eneetr^m, G., 354
Epetdn2 P., 395
Eecott, fe» В., 393-4, 397
Шег,Ъ., 354-8, 3№J-1 C41,
343 352 354 3612367
,, , (,
343, 352. 354, 361-2,367-
369, 373, 374, 378. 381,
385, 392, 394-5, 400)
Етапя,А. B», 380, 384
Faber Stapuleneia, J., [350]
Fermat, P., 351, 353 C41)
Ferrari, R, 397
Fontcn«f &, 396-7 C97-8)
Frattini, G»p 387-8, 39Or 394,
397
Friechauf, J., 379
Fueter, 1Ц 398
GaUBS. C. F.T 343. 367. 37l
C54r 357t 360r 378-9,
391-2, 398)
Genoccbi, A., 376 C70)
Gerazdin, A,, 395, 397, 400
C66)
Gexono, G. C.T 378
GUI, C, 370
Girard, A., 355
Gfipel, A., 376
Got, Th., 399 C79)
Getting, II. R.,36^
Goulard, A., 392
Greeb, 341-2, 346
Gunther, S., 345, 350, 385
Hankel, H., 346 C54)
Hart. D. S^ 381-3, 385 D00)
Heath, ТЛ L., 342, 345
Beiberg, J, L., 345 [342]
Heine, H» Б., Э94
Hellwig, X a U, 367
Нешу, a, 342, 353
Hermann, G~, 343 C43-4)
Hermits, C, 376
Heron Ы Alexandria, 342.
347
НШег,£.Л342]
Hindus. 341 2t 346, 386
Hochbflim, A., [347]
Hoffmann, J. J. I., 343
——, К» Б», 383
Holm, A^ 300, 395
Holmes A. H.r 395
Huhech F.t 342. 345 [341]
Honmthr K, 342, 345

Author Indkx.
787
Hmwiti, Лм 386,390 C81)
Hnygens, Cbr.j 353
Indian» (see Hindus)
JacobL C. G- J., 369, 370.
375 C7&-G, 389)
Jahn, J. C.J343J
Jaquemet, C, 354
Jom N, 350
Kauder, С F-, C44)
KWeticmtcbi, 4Й&
Knirr, J,f 3*»
Koenfe, J. F., 375 C69)
Konen, H.f 342 C52, 354
353)
Kixmp, Chr.t 368
КгоП, G-, [341]
KiOuecker, 1л. 378 <399)
Krumbiegel, В.. 344
Kunerth, A,, 382-4
Ugnnge, J. L, 358-9,360-5,
379 <354, 356-7, 362, 364,
367, 378, 383j 385)
Legends, A, MM 364-5 C57,
359j 369, 370, 374, 386,
391, 394-5, 398, 400)
Leiete, ChrM 342 C43-5)
lemotne, EM 389
Leslie, J., 367
Leasing, G. K, 342 C44)
Uvy, A.,397 [396)
lionnet, E., 384
Ъогем, L»r 380
Loria, a, 345
Lucae, Е„ 386 C98)
Luce, J. В., 375
МаЫег. К, 342
"MilBAT>3»6
, NM 353
Мл*тв, А„ 354 [347]
Martin, A\, 368, 380, 382-3
C68,38^
Mathews. G. Bv 389
Mattfaieeeen, I,, 380
Merriman,.M., 345
Metttd, G., 399
Meyer, A», 379, 390 C99)
-,С.Х344
Miller, W. УС., 38\
H,, 376
e, В., 381 C85-6)
й347
Ейц347
, К. В., 343
Мопй% C.j 380
Moret-Blanc, 380
Motitt, R. E., 381
Мшг, Т., 381 C358
Naah, A.
Neeeel
34d
381
C358)
,r 381
G. H. F., 344,
i E,r [360]
dri, В., 39в
Ono, Т., 399
O L
4 379
Paeiu0lo, L^ 351
PiM^i, A,. 390,
364367
J367
P&olyWiwowa, [345]
Peirce, B4 370
Pell, X, 354 C4L 355, 366)
Perott, J., 350, 3&S-6
Ретю, О„ 399 C99,400)
Fetetsen, J.y C80)
PteiF367
PteMi,F^367
Pieam, K., 384 C84)
Pktor, T» L., 369
Plans. JM 356 C57)
Plato, C85)
Pirf, H., 384 (Ш)
г, F. Т., 369
341
^ 341
RahiL J. H4 354
НЫ, 8^384
к, Н., 399 C99, 400)
1дв&,351
Gy 392
, C., 378
, Hy 386 C66)
Roberta, C. AM 388
, SM 383, 385
Rocchetti, M., 384
RdL350
RodetrL.,350
Rometo, J., 393
Scbldmikb,O.r3S0
Scbmidt, W., 381
ScbmiUj Т., 400 C99, 400)
Sehoenborn, W.. 342
Schroder, J,, 395
Schur, X, 400 C99)
Seeling, P.r 379, 380 C69)
Simon, M>j 399 C46)
Simpson, K.j 355
Smith, H. J. S,, 352,357. 378,
382 C76, 378)
Sommer, X> 396
Bpecknxann, G4 390
Stern, M. A,, 370, 377, 379
C85, 391)
Stevin, Simon [355]
Stormer, d 391* 396
Slrachey, K.t 349
Strong, Т.. 370
.K. L*, 343
Suter, H4 |350J
Tannery, P., 342, 345-6,
35О-1/354
F
TariBte, V.a.39ft
Tartaglia, K-, C51)
Tchebychef. P. L., 376 C87-
S)
Tebay, S., 381
T6denat, P., 368
TeObet.P. F.r393
Tenner, G. W., 372-3 ($58,
382)
Terquem, 0, 344
Teeeanek. J.L 366
Theime, F. Б-, 344
Tbeon of Smyznaj 342
Thue, A,, 391
Unger, E. S., 342
VanAubeLH.»386
Vincent. A- J. H4 344
Vivante, J4 386
Von Schaewen^ P., 393
W&Uia, JM 351-5 C51, 35£-6,
358,395)
Waring, EL, 363
Waeteek J., D00)
Weber, HI, 369, 393
Weill, M., 3S5
Weiseenbora, HL, 342
Werebrusow. A, S., 393j 400
Wertheim, G-, «2, 351, SM
C52)J347]
Wesel, J. L.t 369
Whitford, В. Е., 342r 398-9
C50, 352, 368, 396, 377.
396,400)
Сн. XIII. Fubthkb Single Equations of the Second Degree.
Aida, A,, 429
AiaaaTaj427
Alkarkhi, A., 419 .___. , ,.., _.
Anonymoue, 433 BarteUs, A. L., 404 Bt&hmegupta. 401
Arndt, F., Ш7 Bedoe, P., 401
Aubry, A.. 406 Bertrand, J., 405 Oaben, E., 418, 428
, L., 427, 434 Bervi, N. V., D01) Caleolari, L.t 424-6
B&chet, С G^ 404 D34) Bhiacar», A^ 40lr 428
ВлеЬш&ш, P., 418. 426, 431, Bisconcini. G., 404,418
433 D33) Bonfantim, G>, 427
Tt .. л-t i-. к X АЛЛ D^_l .л.. ЛЛ-t

788
Author Index»
Candido, A. ZM 404
Canterxani, S., 402
Cantor, GM 424
Catalan, E4 D28)
Cauchy, A., 429, 430
Caytey,A.,430Dl6)
Cesaro, E., 4llr 432
Chrietie, R- W. D.r 412
Ciamberlim, C, 418
Claude, R. R L,, 403 D04)
Clebech, A.p D34)
Cloves, T.p 403
Comacchia, G., <40S)
Cimmngham, X., 406, 412,
426
Da Motta Pegado, L. R, 404
De Comberouaee, С, 418
Degel, О., 434
De Jonquieree, K, 418 DGS)
DeU VaU6eP^i«in,C. J.,406
Be Longchampa, G., 426,433
Deeboven, A,, 405, 428, 432
Descartes. IL 402
Deflmareei,E.,4l7D02>
Diophantue, 402, 404, 419
. G. L., 405, 409,
423 D12.426)
Dujardin, 4Л6 D16)
Eiaenstein, G., D31)
Euler, L,, 401, 404, 407-8,
412, 414-5, 421 D04, 408,
41*4 421, 428, 430-1)
Fennat, P., 404
Ferrari, F., 427, 434
Ferval, 412
Focke,418
Fornari, U,t 419
Gaues, С F., 407, 416, 422,
428 D17-9, 426, 432)
Geoocchi, A,, 419, 423 D20)
Gerartm, A,, 406, 427, 4334
Gerono, G. C, 403
Gill, C., 405 D04,419)
GoldecheMer, F,t 426
Copel, A., D11)
Gttnther, 8„ 401 <402)
Guzel, В ./«И
Hansen, H. E.t 404
Heppel, G., 426
Hermite, C, D30)
Hilbert, D., 408
Holm, A., 426
Holmee, A. H.r 402
Hopkins, G. H., 403
Hoppe, R D28)
Humbert, G., 433
Hurwits, A., 412
Jaoobi, C. G. J., D08, 411)
KauBler, С F., 407
Kluge,W,, D11)
Kunze,C, L.A.,410,420
Lagmge, J. L,, 402, ,
420, 422 D00, 416, 418,
421,423)
Legendre, A. M., 405, 409,
416, 421-2 D06-^8, 423,
425-6,430)
Leonardo Нншо, 402. 419
D20}
Leslie, J-, D12)
L6vy, A.7 412
libri, Q., 429
Lore»*, L., 403
Malfatti, G. F,? D28)
ManeioD. P., 402
Mantd, W-, 434
Marcolongo, R», 418
Mathew«,G. B.,406,418
Mataunago, 420
Meyer, A., 431-3 D33)
M&ami, Y.r [420t 4291
Miller, Т. Н., 404
Minding, E. F, A,, 409. 423
Minkoweki, H., 433
Nash, M., [403]
NaeimoffpRS., 411 D08)
Nejedli, X J., 410
Neuberg, J., 405-6
Falmsttfm, A., D08)
Paoli, P., D23)
ParaDjpye, R, P., 426
Pepin, Т., 405, 418, 426
Pistor, Т. К, D19)
Plana, J.,424 D27)
Pomeot, L, 403
Poletli, G., 430
Rautenberg, 418
НЫн1 S., 402, 4G7, 410, 411,
425, 431-2 D33)
"Rifoctitlee, H.," 404
Kignaux, M.T 419
Roberta. S.T 411, 426
Kocbetti, 432
Ruggeri, С, 412
Scheffler, H» 410, 412
8ien>ineki, WM 404
Smith, Н„ J. 8., 413, 431
D08, 412, 417)
Sommer, J,, 412
S6e, E., 406
Stunner, С„ D04)
Studniika.F-J-, 411
Sfkora, A., 404
Sylveeter, J, J*, 410
.,^)
"Tarate, V. G*,'> 434
Teilhet, P. F., 427
Thue, A., 427
Tumere, E,, 428
Vatoff, L4 406
Volpiceln, P.,403
VShP^l
Wasteels. J., D12)
Weber, H^, D08)
Weill. M.s 428
Werebn«ow> A. S., 426
Wertheim, G., 418
Westlund, J.p 418
Weael, J,K,417DO1)
Wijthoff, W.A,, 419
Woepcke, F.T 420 [419]
Ch. XIV. Squabks in Arithmetical ob Geometrical Progression.
D, Ii.,436D41)
"Amicus," 435
Anderson, H. J.T D41)
, J-, 441
Andre, D., 438
Aubiy, L., 440
Bahier, E4 440
Barbwt P<f 440
Beha-Eddin 441
Bills, S7 43^
Biaconcmi, G., 439
Botto, C, 439
Boutin, A^ 439
Bnrawm,B.,440
Campbell, C, 436
Cantor, M., 435
JiCiviet" 438
Clay, H,, 437
СоШов, M., 440 D40)
Cunliffe, J., 436
Cunningham, A., 436,
440D35)
De Morr, С. Т., [435]
Depreb, J., 439
Diophajitua, 435
Bpeilo, 437
Eufer, L., 435 D40)
Evane, А. Б„ 437
Fennat, P., 435, 440
Ferrari, F., 440
Fremcle, 435 D36^440)
Faman, J., 440
Geoocchi, A., 440 D41)
439, Gerardin. A., 439, 440
Goannagbttgh, K-, 439
Griffiths, D. Т., 437
Guibert, M. P. A., 437, 440
Hart, D. S., D38)
Ivory, J., 436
Jones, A. E,t 436
— ,S.,4^6
Jordanue, N., 435

Author Index.
789
Kirkwood, D., 438
Mure, A.r D41!
Martin, A., 438, 440
Matteeon, J , 438
M6trodf G., 439
МШег, W. J., 437
Muller, J., 435
Neeselmann, О- Н. F.t 441
[441]
Neuber*, J,, 43»
Newton, I., 441
Perkins, G. R., 438
ШШра, F.p 436
Plakhowo, 439
Pocklington, H. С, D40-П
Fh*tet, X, 435
Regiomontanue, 435
R«*, JL 439
Rytey, S., 436
Saint. W 441
Smyth, 4ЭТ
Stodd&rd and Henkle, 43d
Surtees, J,, 436
Tafdmacber A., 440
Turrtere, E.p 43D, 440 D3&)
Vondiver, H. S,p 440
Viet*, F.P435 D39,440)
Ward, S.p 441
Welsch,440
Whitwortii, W,A,P439
Willianw. J. D.p D38)
Wright, J.p 436
f VT-, 441
Young, J. R.,436
Ch. XV, Two or More Linear Functions made Squares.
Adr&in. R, 452
Albrkii, A., 444
Anonymous, 449
Bachet, C. G^ 445
Baker, TfJ 455
Benedicts. N4 454
Bh&ecara, A.T 444
Booibelli, R.^ 444
Bourp^ne, Due der {447]
Brahmegupta, 444
Brocuid, HM 466
BucJmer Fr., 455
Bumpkin, Cr 447 D47-8)
Chelucdi,P.jU448
Christie, R. W. D., 456
Coc<JOi, 456 D47)
Cobalt P., 451
CunMe, J. 451-2
Cunjunghanij A»r 457-8
De Billyr J., 44$ D57)
De Ugny, T. T.t 447
Diophantu9r 443 D44-6, 443,
4&Ч 458) ^^^
Fermatj Р„ 445 D47, 457)
Flood, P. W.M 456
NeseelmaDn, G- H. F.r 443
NiblD, R.r451
chij A.. D58)
Gewudin, Asp 457-8 D56)
Gill, C, 456
Goeeeliu, G., 444
hd, E,r 457^-8 D48)
Шкк^Р 447
Hart, D. 8.t 456
Hutton, C., 448-9
Johiwon, S., 452
Jonee, S.p 455
Katnickr H. R., 458
Kausler, C. FPJ 453
Koreelt, A., 458
Lagrangt, J. L4 448 D52)
Landen, JM 447 D50)
LMbiiii,G. W.,447
Leonardo Pisano, D51)
Leslie, J4 451
LeyboiiTD, Т.. 448
Lowry, D48)
, D., [447)
, 447
,, 456
, J.r 446
Petrus, M., 446 D46)
Pockliogtou, H, C, 457
Poa*lgerp F. Т., 454
Pfreetet, J., 446 D54)
RenAJdinL C4 446
Rignaux, M,. 458 D47)
Rolle, M., 44V D50, 456P 458)
Rly, S,, 455
Scott, Judge, 456
Turriere, E,, 468
Van Schooten, F-, [444]
Viet», F.p 444 D4tt)
Л-оп Schaeweo,
D46) [445J
Waring, E.,450
WiUbC44
Ь, А., 456
Euler, L.p 448-*, 450 D46-7,
454^456-8)
Сн. XVI» Two Quapratic Functions op One or
Squares.
Aboal Waft, [4591 Bill*, S., 475
Adr*jn, E», 478, 482D81) ~
Aiyar,K,470
Albocain, Ben, 450
Alkarkhi, A., 460,^478 D63)
, r , ()
Anooymoue, 459 D62, 466)
Arab*, 459, 462, 472 D66)
Aubry, L., 471, 477-^8, 48l,
Auric, A., 478
Bachet, C. G. 482
Barlow, R, 465
Bastien.L., 471
Befaa-Eddin, 463, 480 D69)
ВЬамага, А., 479, 4S3 D79,
480)
KacoL, 471
ВошЬеШ, Й., 472, 478
Boncompagni, В., 460-L
463^, 466 D62, 470) [460]
Brooke, С. Нч 475
BuchPer, Fr,, 483
Oandfdo,^},, 471
Card»n.il. C, D63)
CarmicW)JR.D.,471
Christie, R, W. P., 470
Clere, E.. 479
CoUine, M4 465, 475 D68,
475-6)
Goeeati, P., 464 {4ft4, 482)
Ol^ J., 465, 479
Wright, W,, 454
Young, J. R>, 454
Two Unxnowns made
Cunningham, A<, 470-1
DeBUly, ^482
Degen, C. F>? 474
De Jonqui^ree, E., D77)
Desboves, A.f 470
DestoumeUes.G. LeSecq.,467
Diophantus, 459, 472, 478,
483 D59, 465, 472, 476,
482-3)
Drummond, J, И., 477, 480
Emereon W.p 478, 483
Eecott, E. В.. 480
Euler, L., 4Й4. 472^4, 478,
481 D64-5,468,471-2,474,
476, 478)

790
Author Inimcx.
qguftj E., 483
Feliaano, $.k 463
Fermat, F „464, 482 D65,
460)
Foraide],P,,463D63)
Film, N,, 4&
Genocchi, A., 460, Ш-&,
467, 470, 476, 479, 480,
D61-2. 468-9, 470-1. 481)
Gft&rdin, A4 47b 477, 481,
484
Gerono, G. Cr 477
Ghaligai, Ъ\ 463
Goormaghtigh, R., 477, 484
Goeeelin, O.. 463 14631
Gunther, SL, 469
Hidcke, R, 483
Hartley, j' 464
Heppel, G., 470
Hid 459
Jenkins, M., 470
King, W. F., 477
Ugrange, J. L., 482 D65)
iAisant. С A., 478
Leonardo Pietno, 460-2 D60,
462-6, 46S. 470-2, 476)
"L Pbanue," 644
Libri,G^ 463-4 D60)
Lucas, E., 468-9, 476, 480
D67, 477)
Mane. A., 480 [463]
MathewB, G, В., 470
Matbieu,H. B.,471,481
Manrin, J., 471
McCIenon, R. В., 460
M&rod, G., 471
N&aeknann, G. R. F*, 46St
480
Ono, Т., 47Х
Oi&nazn, J., D82)
Paciuolo, L., 462 D63-4,468)
Ffinormitanue, J., 460
Рат&шруе, R. P., 480
PeUTj., 483
Рерш, Тч 477, 480 D76)
PetruB, M., D74)
Quijano, G., 478
Bahn, J. H4 483
Школив, Mr, 483 D77)
RAerta, 8., 470
Tartaglia, N., 463 D63)
Teb*y, B.t m
TeOhot, F. R, D83)
Terqtlem, O., 460
Tumfire, В., 471
« Umbra/' 465
Wateon, В.. 476
Welmin^ W., 482
Welsch, 4711 483
Werebrusow, A. S., 481
,484
Weet. J» W.. 478
WUkinwm, A. C. L., ^
Woepcke, F.. 459, 462, 464,
466 D68) [459]
Xyl&nder, G., 460
Young, J. R., D65)
Zeir,G.B. М-, D80)
Flanude, M., 486 D86-7)
R^alis, S., 4S8
RiM., 489
Сн. XVII, Systems of Two Equations of Degree Two.
Gerono, G. С, 488
Hammond, J., 485
Henry, С, 487
Heron of Alexandria 485
D86)
Hultech, F4 [485-6]
Lemaire, G., 487
lionnet, E.f 48»
Lucaa, £., 487 D8S)
Marrej A., 485
Martin. A., 485
din, 485
Bbiecara, A.f 489
Bini,U.,487
Boutin, A., 489
Brierley, Щ 485
Cantor, M., 486 D86)
Catalan, &, 488
Cataldi,P.A.,485
Cunningh^^ A., 489
De Jonqui^ree, В., 488 D88)
EBCOtt, E. В., 487
Fennat, P.t 485,487
Genocobi» A., 485, 488 D88)
Gerhanlt, СЛЧ [486]
Meyl, A, J. F*, D88)
Minrhead, R. F., 489
Neeeclmaon, G. H, R, 485
Pappus of Alexandria, 486
БсЬдпе, Н., 1485]
Steen, Ad., 486
Tannery, P., 486
Turri&e, E,, 488
"Umbra," 489
Valla, G., 48Q
Waeochle, H.T [486]
Zeutben, H. G., 486 D86)
h* XVIII. Твдее or Mobe Quadratic Functions or One or Two
Unknowns made Squares.
Brabmegupta, D92)
Cimliffe, J^ 491, 493
Eukr, I*., 493-6 D93)
Erans, А. В., 491
GomperU, B-, 493
Hampeon, J., 492
Hart, D» В., 491-2 D92)
493
Mallock, K, 492
Matteeon, J., 491-2
Pterkinfl, G» R., D92)
Uj Т., D93)
Von Scbaeweo, P., 492
WhiUey, J^ 493
Williams, J. D., 492 D92)
Wid493
Winwwd,493
Wright, W., 191, 493
Zerr, G. В. М., 492

Author Index.
791
Сн. XIX. Systems о* Тнеее ов More Equations of Degree Two
Three or More Unknowns.
AdraiiuR., 525
Aiyar, N. G. 8,, 513
" ,EJ 525
Anonymous (A. B. L., J. B.)j
512,521,525
Avery, C, 531
Bachet, C. G., 614, 519
В&Ыег, E., 615, 520-1
Bain», 52ft
Barfoien, E. N.. 521
Barlow, К 49a
Baston, L, 531
Bhaecara, A., «22
Bilk, 8., 499, 504, 508
Blunder, I*, 49S
Boije af Gennfe, С. O., 618
E18)
Bombelli, R., 517. 524
Bonoycaitle, J., D98)
Brancker,T., 527
Вгос*1т1,Н.,502
Buchner, Fr., 515 E16)
"Calculator," 499, 605 E06)
Cbaband, С.. 600, 502
Cofebrooke, H. Т., 1522]
СоапИ,Р„515~б
Стой. С, С- 618
Cunliffe, J., 499. 505-6, 606r
610-11, 631, 525-8
5,R, 501.513
De Billy, Jaj 609, 519
De Lcmgcbunpe, G., 532
De Slaae, Rene F4 514 E15)
DiopbantUB, 513, 515-7. 522.
524-5, 527 E00, 500r 514.
516, blS-Q, 521-2, 624)
Draugbon, H. Wo 530
Drummond, J. H^ 526
"Epetfon," 625
Eecott, Б. В., 4№
Euler- L., 497-^ 502-3, 505,
507, «W, 514, 517-9, 520,
522-4, 526, 530 (<Ш,
60O-L 503-6, 511, 518,
621, 525f 531-2)
Етпшв, A. В., 613, 515, 526
Farquhar C, 525
Fauqaembetgue, E., 610
¥аш«, Ну 509
,632
7ешМ, „ 516&, 524 E0ft)
Ferrari, F., 601
Fmen4532
Fu*, N., 532 D97)
Gerardm, A., 510, 613,
524,^7,530
Gill, С., 4Ц 504, 510 E01)
Giub«7M.
Guntber, E,
Hakke, P., 497 E00)
HrtTb S 503, 513, 521,
^L, 618
G4 501
H. 610
HoUp С., 5ЭД
Hatton, C^ [6261
Joneu, S., 521-2, 527f 629
K, С L. A., 600
, 3 F09)
Lenbirt, Wm, 409, 506, 512
Leonardo РЬшю, 609, 527
E09)
Lecdie, J-, 615
Leudidarf. 0^.501
Lucm/б., [518J
Lynn, J~, 626
Martin, A4 500-2, 504^5,
612-3, 53U F00)
MatWeoo, J-50I, 528, 530
иакя«Гс~Ъ02
Mukhopfi5hyly, A., 602
G. H. F., 516,
518,526
Olmn, H.r 501
(yRibn, M. 8. 503
510
.1Ц 531 E31)
Rallier dee Ourmee, 522
i V^ 514, 625
SOl^SlB
, S.,
Saund, N., «7, 514,
516, 519 (S15. Б2Й
Schwermg, K, 501 D97)
Scott, Judge, 500, 505 E01,
604)
Sr, V. M, 502
Sion. Я5
512, 626
2
" 621
ТеЬаУ,« ч ,
Тгвивоп^ А., Г
Turri&re, К, 513
Уегйсй^К^б!
VieU, F., 614
, W., 627
, 499, 606, 524, 526
J5ill2624
F^ 527
., 610, 521, 525-6,
Ytmn&J. 1^525,52»
Zerr, G. B. M.p 608
Ch. XX. Quadratic Form made an лтн Poweb.
ye, 637j 544
Aubry, L., 528, 538
Bacbet, C. G., 533
Baiim^n, E. N» 543
Bafetien, I*f 5ЭЙ
Bini, tT., 637, 642
Boutin, A., 536
B £L, E33)
Brocard, H., 534^8 E34,537)
Qmdido, G., 542-4
Carmicboel, R. D.r 544
Catabn, E., 635
CIA E40)
D542
A., 637,639,542
CtuaoI, 539
Davis, R. F^ 637
Degd, O. 542
De Jooqm^i*, E-r 634, 536
F85,539)
Delaimoy. H4 536
De Longchampe, G., 537,644
De^xivee, A., 638 E42)
D^tcartcs, R., E36)
DiophantUB, 633

792
Author Index.
Dirichlet, G. L,, 540
D'Ocagoe, M.r 541
Eecott, E. B,, 536-8, 542
E34,538)
Euler L-, 533-4, 53», 540
E34-7r 541-2)
Fauqueipbergue, E., 535-6i
538F43)
Fernmt, P., 533 E34, 536!
538)
Ferrari, R, 543
CerardiibA., 53S-9
Gerobo, G. C., 534
Goulard.A., 536
Griffin, F. L,, 544
Hayaahi, Т., 539
Ivanoff, I., 53d
J&nseen van Вшу, W. H, L.,
544
Jordanii8fN., E33)
Kronecker, L.r E39)
bigrange, J, L.t 539 E42-3)
bmdiui, E,, 536, 538
Landry, P.. 640
Lebeegue, V. A., 534, 540
F43)
Legends, A. M., 534. 540
E36-7)
Lucas, E., 534 E42r3)
Luce, J. В., E40)
Mehmed-Nadir, 544
Mordell, L. J., 638-^9
Neuberg, J., 543 E44)
Ottinger, L*, 534, 541
Pepin, Т., 534-7, 54J-2 E42)
Fftrtb, F., 534, 638
КеаЬн, 8., 535, 543 E39)
Roee, J.,544
Stunner, C, 536 E42)
Swift, E., 543
Tait, P. G., E3ft)
Талвегу, P.. 536
«TariBte,V> G.,r'544
Tbue, A., E38, 543)
Vandiver, H. S,, 542
Waring, E., [5331
W^eHlTM., 542
Webch, 637
Wbw, A. S,, 537, 542
Tf G. В. М., 544
Ch, XXI. Equations op Degree Three.
иАЬшаа,81
Alhocain, Ben, 545
АШ&гкЫ, A.p 612
AJkhodjiuidi, 545
Ambler. J. K.t 601
AmfiEer, R,, 562
Andereoo, J.j 601, 604, 61Й
AnanymouSj 609
Ashcroft, 601
Aubry, L.7671-2,585,588,596
Baba, S., 560-1
Bachet, C- G., 551-2, 607
E51, 607-9)
Bachm&nn. P,, 5501 59S
Baer, W. S4 578 <58lp 599)
Bainee, J.,612
Barbette/E., 565
Barlow, F.. 548
Becker. J. W.. 547 E73)
Beha-Eddin, 545
Bendz, Т. К., 576
Bennett, J, 609
Bhascars, A. 595, 599
Bills, S., 604. 610
Biiiet, J- P. M., 554 E55,
55»9560)
,)
Bini,U,,613 E91)
Biaconcuii, G., 597
Bombelli, fe.,,599
Bonoompiagni, В.. 583
Bouniakowaky, V., 556, 566.
583 E63, 581)
BourgogHe, Due de, [6021
Boutin, Am 568
Вк ТЛСЮ
Candido. G., 669 E75)
Cantor, M,p 584,686
Cannicnael, H- D., 550, 560,
566, 577, 595
CashiDore, 58lj 598
Catalan, E., 556, 563, 566,
576, 586 F66, 581, 586,
599)
Cauchy, A.p 588 E62, 566,
672, 589, 59(M, 593, 597)
Chancy, LM 562, 592
Christie, R» W- D,, 664, 577
Chuquet, N., E99)
Cole, W., 606 F01)
Colebrooker H. Т., [595, 5»!
Collignon, Ed., 564
Cunliffe, J,t 595, 600, 604
E98)
Cunningbam, A,, 577 587,
596,697
Curjei, H. W., 601
Davey, J4 601
Davifl, R. R, 571, 572, 606
,R.,en
De ВШу, J., 566, 569, 602
De Helguero, F, Ш
Be Jonooort, "&., E87)
Be Jonquieres, E. 596
Belannoy, И., 576
Delaunay, В., 578
De Rocquigny, G., 571
Drabovca, A., 562, 666-7,
571. 575, 591 E77. 598.
613)
Brocard, H.r 561, 571, 588,
606E87)
Brouncker, W., E52)
Brunei, G., 566
Bungua, P^, 550
Burneide, WM 550
Calrolari, L., 548
Bickeon,L.E., 563,613
Diophantus, 650, 599, 602.
607 E45, 551, 605, 607-9,
610-2)
Dirichlet, G. Ь„ 574, 693
E78)
Efeenriein, G.r 694
Elefanti, R, 663
Emereob, W.. 554
Eecott, E. В., 561, 571, 685,
597-8,606
Buler U, 545-6, 552-4, 567,
569, 570, 572, 570-9, 582,
699, 603 E47-9, 550, 554,
559, 560-1, 569r 672^4,
579, SSCM, 597, 599)
, J., A. 546
Evana, А. В., 661, 584, 606,
611 '
Fauquembeigue, E.» 661-2,
565, 572, 580-1, 60£*
Fennat, P., 545, 551, 566,
668-9, 572, 602. 607 E52,
661, 567, 572, 575, 583,
603,608)
FrtrarLR,^
Flood, P. W., 613
Foster. J4 604
Frenicle. 545, 552 E60)
Fucha, C, H.j 579
Fueter, R.t 550
Fujiwara, M., 559 E58)
Fues, N, (I), 546
GambioK, D-, [5481
Gauss, C. F.. 548
Genocchi, A., 545, 583-5,
613 E87)
G6rardjD, A., 558-9, 562,
564-5, 566, 671-3, 581,
300,597-«, 602, 606, 612-3
F80,596)
Gir*rd,A., 551, «У1
Glenie, J., 546 E47, 673)
Godfray k., 601
Gokai, Ampon, 560
Gomperti, В., 600
Cfeormaghtigh, R., №t 612
GraMmaDn/H.,666
Greenfield'W. i., 580
Grigorief, £.T 585

Author Index.
793
Gruber, M. A., fiS7
Griteon, J. P4 554
Gttnther, 8., 540
HaeflUechel, E., 568, 572
Hagner, 1. К., 547
Hampeon, J.. G09
Hart, D. 8., 563, 584, 605,
611 F11)
ШцЬет, С. F., &47
Havaahi, Т.. 660, 577, 698
E99)[563J
Heath, Т. К, 608
Henry, С„ 576, 602
Hefmite,C.,555r596 E57-g)
Ш1ГД554}
g, F^ [5471
Himao.Y.. 563
"Hob,J '^602
НоМев, H,, 559 E50)
Holro, A,, 558, 591,611
Holzftr, L., 569
Hoppe, R4 679, 596 E61)
Hromddko, P., 585
Huberti, N., 608
Hurwita, A.. 555, 593 E74,
592, 59ff, 09»)
HuttOD, C.f [599, 608-9]
Jaoobi, С G. J., E81)
Jand&sek, J4 659
Jtmeeen тал lUfirYi W. H. L,,
581,593
Joffroy,X, 574,596
Jonee, 8., 554, 600
у S.r 560
Kaetoer, A. G., 547
Кяш1ег, С F., 548
K&w&kitbj 560
Keieey, J,r 608 [545j
Xing, W. F>, 597
Korneck, G4 556 E50)
Kraut, W. L.. 567r669
Krey, H.p 549
Kroaecker, L.. 54S
КШше, Н., 558 [559)
Кщщцег, Е. E., E49, 550)
Lagrange, J. L., 570-2, 595
E71, 575, 503, 597, 599)
Laisant, C. A.> 574
Lam^G.j 548, 573
L&ttdsu, Kf 568
Lander J.. 609
Lebeflgue, V. A.t 583 E85-6)
Legendre,. A. M., 545, 648,
570, 573, 589 E50, 574-5,
588.591)
Leibnli, G. WM E90)
Unhmrt, Wi», 561. 673, 582,
696, 604, 610 E61, 598,
611, 613)
Leslie, J., 600
Levi, B.t 592 E93)
Lexell, A, J., 56», 595
Leyboom, Th.. 609
Ubri, G^ E93)
Lowry, 600
Lucae. В„ 566, 672. 574-6,
580,587,590 E75-6,588^9,
592, 597, 599)
"MachAirt, L» N.r" 585
, E., 599 E94)
, К А,,*' 688
602
Schopie, 548
Schufamachcr, 549
Scbwcring, K,r 557, 576, 581
E56)
Msrre, A., [545]
M«rtin, A., 557f 563-4, 584,
606 E65, 606)
Mathews, G. В., 594
МдШеи, Ем 594
Mtteoti, J., 561, 584,605
eieen, LN 584-5. 588
ч 594
i, У., [5601
noffj D., 558
MordeD, L. J., 571, 592 E81,
693)
Moreau, C., 557
Moi*l,A.,596
MoT^t-Blanc, 574, 580, 590
E7S)
n, G. H. F,T [545]
Nichobon, J. W., 560
Nobljt, M.t 610
Nonifc» R-, 5SI9, 565, 5»T
E91)
Otobom, G., 560
О, J., 602, 612
PaoK,.P', 567
Pegorier, F. 581
РЫетж, Й., 596
Pell, £.608
Fepin, Т., 549, 574, 576, 596
Ы) ' '
Perkliis, G. R.r 5B7
Perrin, К„ 549
Peteieen, J,, 594
:Pt*T, K.j 55»
Foinc«H, H., E66)
Poeei*err F. Т., 554
Prtetet, Jv 552, 572 E74-5,
577,589)
Rahn, J. H,, 608
Reali*, S., 556. 663, 575, 584,
590,596-7 E98)
Richaud, G, 655, 580, 586
Kignatix, M.? 578, 606
НоЫдооп, Т.. 595
Rychlik, К,, E50)
Ryley, J., 609
, S>, 563
Saul, J., 600
Sawin.A.M., 575
Schier, O., E61)
SchWl, V., 657
()
Sensimig, D. M., 567
Shlraiahi, C, 560
Smith, D. E., [560]
ЗпеШив, W,t {60S]
Soip, W., 601
Soimner, J^ 549
Sond»t, P., 556, 591
eouOtert, C.. 594
Spieer, W., S99
Steg^J. KA.r5G0
Stevenson J. 585
StwskhiUB, H.> E50)
Strong, Т., F11)
Sylvester, J. J. 549t 574-6,
589, 591, E75-6, 591)
Tafelmacher, W. L. A., E81)
Tait, P. G.r 549
Tanner, H. W, Lloyd, 594
Tannery, P., 545, 568, 602,
606
Tebay, S., 563, 605, 611
Tdbet, P. F.t 558, 580, 601
E57)
Terquem, 0., E99)
Tbue, A., 580, 592
Tig, L\, 566,, 595
bCE^)
Van Ceulen, L4 60S
Van der Corput, J- G.t 578
Vandiver. H. $., 594r 597
Van Scborten, ¥,, 545, «Ш
F08-9) [550]
Varchonr L., 578
Vieta, F.p 550, 552 E51-2,
554, 557r 607)
Viacbeta, J. N^ 585
Von Schaewen, P>p 568 E67-
«)
VonSr. Nagyr J.P568
Walker, L. C, 571
Wailie, J., 545r 552 F08)
Ward, S*, 601, 612
Waring, K, 552
Wateon, W., 612
Weil], M., 561, 565, 578, 580,
593r 597 E81)
Wefecb, 597
Werebrueowr A, S., 557, 562r
564, 566t 568, 577, 581,
592, 602 E58)
mitley, J<? 563, ЬЪЬ, 600-1
E64)
Whitwortb, W. A,s 587
Woepcke,F., [545J
Wriglit, w.,600,612
Young, J>t 600
Young, J. R*, 554, 582 E60,
606r 613)

794
Atjthor Ьгаях,
Сн. XXIL Equations or Degree Fotm.
Adcock, R, J., 658
Adrain, R., 636 F37)
Aids, X., F06)
Anderson, H« J., 636
Aubiy, L., 638, Ю8> 665, 671
Auric, A., 694
Bachet, С. О., 615 F20)
ВЬ P 615
9
Е7
Barbette, Вч 652
Вагшеп/к N., 655, 659, 660
Barlow, У, 618
Baetien, L., 656, 665. 671
Beha-Eddin. 667F36)
Bends, Т.К., 619 F28)
Bertaind, J 619, 62ft
Bill», S., 6ii 666
BinLU'65^F56)
ВьЗс, dt 656
Bottari, A., 620
Brehm,665 F63)
Brocari, H., 665
Oakolari, L., 641
GarmichUl, K. D,. 620, 634,
656, 660, 665 F28)
—, F. L-. 665
Cauchy, A,, F461
Christie, Я. W. D., 663
Cipolla. M.p C21
Шит, А '671
Couturat, L., F17J
Cunliffe, 5.t 666 669
CunninghaxD. А», 627. 634,
638, 646, 653-4, 656, 659,
664Ц 671 F35)
ВеВШу,Х,(б18)(ад
De Lagny, T. F.t 639
Deebora/ A.. 631-3, 637,
642, 64b, 648, 663 F31-2,
639,646}
Dickeon, L. E., 616
DiophantUB, 6571 667 F15-6,
620,657-8)
Doetor. G., 657
Dutoidoir, H., 615
n, G.r 662
Eecott,E.B.,633,658
Euler, L., 618, 621-2^^4-5,
627-8/634A 639, 648*
644-5r 64&-9. 660-lr 666-в
{618-55,620-1.623.6Ц №9
632r 635-8, Й4Ь2, 646-8,
652[ 660, 663. 660, 671)
, J. А., Ш
Evane, А. В., 666
Ffiuquembergi», E.,622, 634,
Ш, 64^7. 650, 655, 662,
664, 671 (Й57)
Fermat, P., 615-7, «20-1,
639, 667 F17, 621-4, 62^
631-2, 634, 643. 647. 657)
Ferrari, F., №, 654 F54-5)
Frobeniua, G., 640
Fum, N., 036
Gambioli, В., 619
GenoochL A., 619, 621, 637,
667 F24, 636, 630)
Gerardm. A., 627, 033-4,
6387^7-8, 654th5. 65ft,
663-5, 671 («52, 656, 660,
671)
Gffl. &, 670
Gl638
,3
Goormaghtigh, R., 667, 671
Griffin, F. L., 064
Giigonef, E.? 647, 663
Granert J. A., 619
GGnteche, R., F40)
d, E., 634, 643-4
F39)
Haldeman, С В., 650, 653,
658 F5Ь 655)
Hart. D. S.t 649, 6Й2, 666
HbiT646
HayaebiT.,646
Heath, T. L., 1616]
Hilbert, D., F20)
Homer, J.i 618
Humbert, G., 644
Hurwitx, A», 663
Jaoobi, С G. J., 641
Kaualer, С F., 618t 635-6,
662F36)
Kempner, A. J., F57)
Kommereli, K., 663 FЙ0,
665)
Кгапшг, Л. Бм 628
KroneckerjL*» 619
Kummer, & К, F40)
J. L, 623-4, 628,
\, 625, 631-633,
637)
Laieant, С А., [657, 663}
Ubeonie V. A., 619, 622,
625^6, 637, 662 F21-2,
626-7, 630, 639)
Legttkdns, A. M., 615, 618t
628,656,639
ЫЬпмТб! W.p 617, ЯЛ, 665
(«21)
d
L6vyr A., [620]
tibnGj№
G.,
E
, 670
Frenicfe, 616-7, 621 F28)
Locae, E., 621, 636-7, 630-1,
636-7,643. Ш, 670 F31-2,
634, 643, 060)
Maffett, R., 670
Mahitke, D, 665
Mantel W,, 634
Mane, A., [667]
Martin, A., 649,650-1,653-i,
658,667 («50.6S2)
Maeont666
Matbieu, H. В., 638, 664
Mehmed-Nadir, 650
Miot, EL, «54, 656, 659
Mordefl, L. J.r 643
MraH-Blane, 630, 637, 662
Newelmmtt, G. H. F,, [6671
Norrie, R^ 646, «62, 655.
659 F47, 657)
Ntrtihorn, F.r 620
Огапаш, J^ 618, 621
Pepin, Т., 619, 626r 630,
632-^3, 637-8, 641-2 F33,
639, 640, 643)
Petras, M., F71)
RetotxjoU, С,в38 F38)
FockUngton, H. a, 634, 638
F36)
Preset, J., 667
IVoth, ¥., 657 («50, «59)
Р1кц Jn 641
felis, &., 627, 632, 637, «57,
663 F27, в34)
B%iaux, M., 634r 647, 671
RoUrU, t>^ 670
Roroeitx P., 643
Rychlik, JLt (£20
г, L., 641
Schopie, 618 <630)
Sclmrmg, K. F40-1)
Sommer, J«, 620
StaLJ. E A., «47
\, 636
ff; С. а, «29
Swift, E.,&65
Tafdm&^ber. A,, 656
, W. L. A., 619
Tannery, P., «16, 638
"TbriHteiV. G./656
Tolbet, P. F., 063
Тещиет, O^ 61ft, 629 F19)
Thue. A.. 633
Torrioelb, E, 621
Tuniere, E.t 621
Van Schooten, F,, 668
Vieta, F^ 615
УЬЩ, P., 619
J«41
Walker, L. C, 627
Walmood,666
Waring, £., 618, 628, 657
WeJ0ch,654
Werebruwm, A. 8., 638,
646-S, 65l 656 F38)
Wetbeini, G., 617
WhiUey, J., 670
Wilder, C-, 670
Woepeke, R, F7l)
WdCc^eeo
2ехт,СБ.М.,619.
ZetsUien, H. G,, 615

Authoe Index.
795
Сн. XXIIL Equations о* Degbee n.
Agronomof, N.. 683-4
ApcW,I>-,68O,6$9F90)
Anonymoue (Mile. A* D.,
PJLG.),679,68O,688
АяеоН, G., 688
Aubry, 1^ $80,685,686, бв*,
702
Bachxnann. P., 678, 681
Barbette, К, ей
Bftrinen,E/N., fi&B, 702
Вегвег, A.,686
BemoulK, Ь., 687
Berton 680
Bertrand^ J,T F7^681)
Beverley, Т.. 700
Bianca, C, 683
Biddle, D., 691
Вмтап, С 686
ВЬ^Уб8а
у,,
Bottari A,, F83)
Bfwoiakoweky, У,, 685, 687,
603
Bourget, H., 680
BoutSL A., 674
Bricani. R., 680
Brocardfja., 681,6S3,685,699
Brunei, Ъ., 700
Candido, G., 698
Cannich&el, R. D., 691
Catalan, K, 686 F86)
Cauchy, A., 700
Cfc/E685(
Ceefcro/Ev685(ff7
"ОшгЬошег/' 697
Chaundy, T. W., G88
Chrirtae, tL W. D., 697
Colling К 682
СЫ G flSft
Оатаееимц G-, 673
CtLb^MT
^.M.TOa
m, A., 696-9, 70S
Davie, R. F., 702
De Fariuw, A^S84
Do Jong, L., 700
Delorme, H,, 70l
Be Hocquigny» G., «81, 683
Deeboye*, A., 684, 691, 701
Dmchiet, G, L. 677-8 {691}
Di San Robert, P., ^^
DjdeSg
EdaJjn, X, 703
Е^ий G
EDurttrRie84
ЕюД ВJ681,683.6
68^684, 687,
(r&,,7)
Era», А. В., 680, 701
Faore, H., 679
Fermat. P.% 687
Filippov,A.,683-i
FlecbaeDbaar. A.r 687. 691
Fortey, H., 6Й6 F85)
FrobeniiiB, G,, 699
Fl»v,P»8,ree7
, N. A), F96)
G&iia, C. F.. F93)
Gegenbauer, L, 686r 701
(Ш, 686, 700)
G«iin, АЬЫ, 685 F83)
Ger»rdiii,A., 682,701-3 F99,
700,702)
Gerono, G. C, 680-1. 693
GiU, C.r 700
GolJbach, Chr», 679
Goormaghtkh, R,, 700, 703
Guibert, M. P. A., 680
Haentzechel, B-, G00)
Haldeman, C. B.r 684
Hart, D. S., F82)
Hayaehi, Т., 680
Hertet, С„ 687
Heraute. C.. F73)
Heron of Alexandria, F93)
Hubert, D4> 676, 695,700
Hill, G. W., 680
—. J., 684
, W., 691
НоГюаш), W., 691
Holmes, A, H., 699
H}№
urwitB, A+J 675, 697 F77,
693-1, 699, 700)
P., 700
Jacobi,C. G. J-,G00)
Janculoecu, R., 691
Кошпл, Т., 609
Koraelt, A., 700
Kuhne, П., 696 F97)
Labey, J» B,, [687)
Lackerbauer, P^ 7Ш
Lagputineky, M.r 689, 690
Lftgwngv J. I*, 673, 691
F73, )
Landau, E^ $№)
ЬЪ, V. A., 680
Эге! A. M^ 673, 688
шж, G. Wj. 699
Lemaiie, G,, 691
Leonardo Pieano, F89)
ti B F77)
Je92
b,A.J.,e92
bn, G.. 678, 684, 602-3
F7b efo)
Lkmvffle, X, 678^9, 680, 700
F81, 701)
i G> A^ 703
Lucas, E~, 681, 685-6, 694
F85-6<698)
Maillet, E-, 674-5r 677, 695
F76)
Mantel, W,, 699
A
MfcrfKjff, A., 604 F
Martin. A., 682-3
Mathews, G. В., 696
Mathieu. E.r 679
Meissd, K, 678
Metrod, G.» 679
Meyer, C. F., 703
Minetola, S., F79)
Mlnkowski, Н„ F90)
Miot, E,. 656, 699
Moreau, C, 681. 689 F79)
Moretr-Blanc, 681, 686 F86^
в)
Moutton, E. J.. 687
5ЛЛ Л 6*4
Narumi, ff., 680
Neebitt, A» M^687
Nicholson. J. W^ 684
Wiewen^oweki, P., 688
Noether, M-, 675-6
Padoa, A., 688
Pahnetrom, A.v 679, 689
Paulet, F., 682
Poeecbkor703
^ H., 673, 676, 678
Kadoet G., 700 F77)
ШЬя, S.f 694
x/M.r 699, 7(Ю
tvM694
RochettvMx694
Runge, C», 674
Sadkr, J.. 689
gatunovHfey, S, O.. 696
gchier, O., 882
Schilling, Г-, 689
Schimmack, R., 687
^chulte. E.t 691
всЪ^вгшь К,, F97)
Slavk J.,703
S6e, K, 680-1
StelJ, Jv E. A., 688 F84)
a696
a,696
S* Т., 687
Swift, E., 702
Tannery, P., 687
Tanturri; A., 687
"Tariste, V. G./1 6Qftr TOO
ТЪШю*, Р. F., 681-2, 696
Thorin, A., 689
Ibue, A., 675r 678, 608 F75)
VftnHengel, J., 687
Verhagen, И., 703
Von Sa. Naey, J,, 677 F76)
Ward, 8^G00}

796
Author Index,
Wekch, 686. 609
Whitworth, Т., «90
WillUw, J. D-, 702
Zerr, G. B, M-, 683, 702
Zmna. A,r 703
ZX, 689, 690
Zuhlke, P.. 600
ZMg, H., 703
Сн. XXIV- Sets of Integers with Equal Sums of Like Powers.
В. В., 706-9, 710-1,
Agronoinof, N., 713
АПтлп, W' 715G14)
Aubry, A., 712-3 <716)
, L, 709, 710
BarieieD, E. K., 709r 713
Bastien, L., 712 G07)
ВЫ. U' 707-S
Birek, CL 71(Ъ1
В'шпйп, C.t 700
B<wd*, J. С <fe, 714 G15)
Ceriro, E., 705
Chrirtie, R, W. D., 700
Cruaeal, 711-12
Cunningham, A., 706 G07)
Ш Fftrkaa, A^ 709
Delanibre, J. B. J.r 714 G15-
6)
715-6
Euler, L,, 705 Gl6)
FUlippov, A., 713
Frolov, M., 706 G05, 709}
Martin. A,, 706 G0S)
Matbieu, H, В., 707-8г 710
Miotr E.. 709, 710 G16)
Multer, J., 714
Nichobon.J. W., 706 G05.
716)
in, A.- 707-9, 710-L
713 G16)
Goldbach, Chr.r 705 G06,
710)
Goorma«htigh( R., 713
Иьгое, G14)
Knigbt, Т., 715 C716)
Ыстоа, S. T./7X4
Me, J.KT.,714 G15)
Proth, F.t 705 G05)
Pnmhet, E.r 705
e,V, G^710
Талу. G,, 708r^> 710, 712
G08,710,716)
Тлу\от, H. M-p 706
, J. H., 706 G16)
ВасЬпшш, Р», 721, 723
Baer, W. ^7^4
Bou^d, O23 G28)
Brttacbneider, С A<f 717
Brocard, H,, 728
Ch. XXV- Waking's Problem and Related Results.
.,724
725)
Jacobi, C-G. J.r7l7G17)
Cauchy,A., С
Cunningham,
, 729
Daee, Z., 717
Davie, R, G26)
De Rocquigny, G*, 726
J. A,, 717 G17, 719,
-, L., G20)
Evaae, А, В., G26)
Fauquemberguc, E. 729
Fleck, A., 720, 723 frl*-
Frobeniiw, G., 724 G25)
C. F., 728
, A.| 723, 726
Hardy, G. H4 725
Hauddorff, F.. 722 G22, 734)
Hilbert, D., 722 G22-5) Olti»ioarer G., 719, 729
Си. XXVI. Fermat's La$t Theorem, (
Abel, N. H., 733 G45, 755,
757, 769, 760, 762, 764,
787,771>6)
Kempner, A. J., 717- 721,
723 G19)
Kurecb&k, J., 723
Lftiaant, С. А., [718]
LUu, E., 720, 722-4 [72Ц
V. A., 728-9 £717]
j, ., 721
Leniaire, G., 720
Lexnoine, E.. 719
Lenhart, Wm,, 726
Libri, G,, 727-9
Liouville, J., 717 G18, 723)
littlcwood, J.E.,725
Lori*, G., 724
Lucas, E,, 718, 729 G20)
MailJet, E.t 717-9r 720-1
G17, 720-1, 723)
MathewB, G. B.r 719
Minkoweki, H,, G24)
ii,G.,726
n, S-, 726
&., 718
Rebout, £., 728
Heroak, H., 724 G24-5)
Ripert, L.. 719
Rouve, L., 723
Ryley, S,, 726
Sdunidt, E.r 724
Schur, JM 721
Stridebere, E,, 722, 725 G24)
Strong, Т., 726
Tardv, P-, 728
TebayT 8., G26)
Teilheir P. F,r 726
G., 719 G17)
Von
Wiefcncb,
Wolff, Т<
717 G22-4)
72(Ь2 G22)
Sttbjbct Index
Allirton N^772-3
Arwin A-, 775
Aubry, L., 773-4
ff, A. R,, 717
f', AND THE
Bachmwin. P<% 767, 775
Влет, W.S,,G76)
BaD, W. W. U, [7761

Author Index,
797
Бй% А,%_7й6
Barbette. EL 767
Bybw, Pfi 733 G33, 745-6}
Bell, E, Т.. 774
Bendi, Т. к, 759
Berliner, H., 771
Bernetein, В., 771
F,, 7& G76)
Km. ,Й,7в1
BirkboftG. D.,772
Bobnieek, St., 771
Boaoompagni, B, [732-3]
Bortetti, F., 754
Bottari. A., 762 G62)
Botuuako^fiky, V>> 736, 73S
G37, 764)
:t,R,[75lJ
Cailler, C.r 766
CaUolari, L,, 743-1, 746
G60)
Candkfo, G., 760
C*Wl731
6, 771
rltni L. 77t
G47, 772S
CataJaD, E., 738, 747, 751-3
G34, 747, 752t 754 760,
766)
Calta»«>, P., 762 G62)
Cauchy, A., 737^8, 740 G38,
740, 742, 745-6, 748,752^
754-5,758, 760)
CeeAro, E., 751
Cornaochia, G., 762, 765
G36, 762-3, 765)
L nuixvfchfttn A», 763
iiid, tt V.p 7il, 755
Dedekind, R., G42)
De Jonquils, E., 751, 757
G^3,753,759)
ЬоюйЬа
759)
тр?, G.p /751J
P W
Pe Remilly, P. W., 757
Dasbovte, X., 749
РТШ751
7
L. E., 762-3, 775
7И7'774-5)
)
phus G3f)
Dincblct, G. L., 735-6, 738,
741 G35, 738r 760, 771,
775)
Itixon, A, L., 7&3
РгасЬ; S. M,, 738
DuWis, E., 769
rhipuy. M.c75l
Dutordoir, H*, 766
Ermakov, V. P., 760
Euclid ,G40,75ft)
Euler, I,., 732 G32, 736, 753,
759, 764, 766)
$, K, 751
Fftbry, E^. 784
Fermat, K, 731
, 6., Г73Ц
Ferreri F., 764
Fmera.N. M., 747
Fle<AeenbaftrI A.. 766
FJeck, A.. 762, 764, 766, 760
Fri*m7&
G., 767-8, 773
G65, 7ft>
Furtw&ngler, Ph., 768, 771
G75)
Fuai, N. Ш, 732
Gambicfc, B.t 7&Э, 760 G3ft,
di74ti
d,7
Саадв, С, F., 733, 753 G32,
740, 750)
G*geobauert L., 753-^ G52,
765)
G«vo«y, A<( 742T 746-7
G37.741)
Gtfrardm, A., 767, 773-4
Germain, R, 732, 734-5
G57, 763Л69)
GKHmo,G.fc,,744,747 [706)
Glaieb^t J, W. L,, 748
Got, Th., 17571
Gram, I. P, 7^9
Gnmert, J. A., 744 G7Д)
С«1т(№йев, И., 773 G45)
Gunther, S,, 732
HaetiUechel, E., 775
НауавЬ, TM 769
Песке, Е„ 768
Heine, H.E., 743
Henry, C, [7311
c,E^7T
Heppcl, G., 752
Hilbert, D., 757 (Г42)
Hoiwel 747
Hurwit*, A., 764 G63)
Irwin,
I
F4774
^ L I., 762
J^ckeon, J.fi^747
Jacobi,fc. G. J^739 G63)
Jaquemet, C^, 731 G55)
Jif Z 769
apfeer, H<f 772
Клив1ег, С F.,733
Ketnpner, A. J,, 774 G66)
Kokott,P.,774
K<irkme, A., 750
Konieek, G., 766
Kojsdt. Aw 774 G60, 775)
Knonedccr, L,t 741
Кшпщег, Е. E.y 736, 738,
7«K2, 744-5 G39, 740-2,
745,755,757-9,760-2,764-
57fe74775)
Landau, E.p 772
Landrv, F., 743
Landabe/K. 0^754
L*porte, E.t747
Leb^gue, V- A., 737^9 G35f
737^)
37^)
LefSbure, A>, 750 G50)
Legendre, A- M., 784-6 G37,
743r 755-8,76Й, 763,766-7,
775)
,
Leen, S., 755
:Uvy, A., ITO2]
Lewi ban Gervont 731
2
in™., 764
ind. В., Уб*. 766, 769 G44,
756,755,774)
Lniden*nnr F., 759, 76Й
G60, 762i
ionnt E., 738
J!, 738-9 G39, 760,
)
, 7)
Lucas, VШ, 754т i <757)
, F., 7Й
LubalF./74S
MacMabonrP. A., 752,762
Maeuncfrenj "Ph., 772
Maillet, K, 7S8-9, 761, 774
G56)
Malebnuiehe, Kt 731
Malvy,766
ManaonLP., 751, 753
Mantel, W., 774 G36)
MkffV757a&5>
Maikoff,V.,757a&5>
Marre, A., 1731]
Martone, M., 753
MatheWB, tf. В., 752, 757
G53)
Mewsner, W., 772
Mention, J., 739
Mercier, F., 771
F!, 745
/a, 772
eyi,X. J. F., 747
Miot,E.,772'
Miiixnanc^ D
ic^, .f 55,7,
764t 766, 768, 770 G64-5,
767-8* 771, 775)
Mitebd], H.k, 774
MonteL P., 774 G74)
Мога^Йапе, 7*7, 751
Muix, Т„ 748
M
J.,76S
., 750
Nby, J., 764
Newton I.t G43)
NiewiAdoroaki, R., 772
Norrie, R., 772
ЬалеГТ. 737. 739, 740
G37-9, ^40, 7^0 7, 760)
PfcuM, F., 737. 745
Feflet, A.B, 749, 751, 753,
770 G36, 751, 763)

798
Author Index.
Fenin,T,,748, 750, 758 G»,
fto 7Й, 769)
Picud, E., 756
Pi, а М^ 751
, 771
n, EL C-, 772
а., 756
R, 775
Quetelet, А., [737]
Banned, б. N. 771
Hicbter. w., m
Bieke JL, 746-^ G55)
Ki fcb 7ft5
Ki«f fcbr. 7ft5
Robots, Й. 751f 753
BothbobД, 754-6
Rowe, J. K.. 772
Rychfik, К I 767
Sauer. R., 761
Schenng, K, |733]
Shi^S.f75O
E 76
SwopiBj, 735
Sehnr, L 774 G63)
Smith, Ef. J. 8.r 745 G33)
Samnwr, J.,762
8bj
ooe,7boj747)
Зрескшмт, G^ 757
Stickel, Rt 7ЙО
Stoekb&us, H., 767
SkLT773
So«ikLT.,773
Swift, E,, 773
Sylow, L» 733
§lrt^JJ
Tafelmacber, W. L. A.r 755r
i
RG.f74e
ТяОАл, В/*1,, 745-6 G33)
Тану. Н.. 758
Тесяе Н. 755
Teffiet, P. R, 758
Тегчиет»О., 739Р 740
Thorn», K., 745
Thne, A., 758, 7*4, 770,
776
Todhuuter, I., 746
Turtecbaninov, A.t 764
V&cbett*, A., 730, 745
V*n dar Corput, J. O.. 773
Vandhrer, H. 8,1731, 745,
773, 775 G64, Wr>
ri D., 754
P V 7
Varidco, D., 754
Velmine, P. V., 760 G74)
Von Vitry, P.f 731
Wantiel,L,f740G40)
WdBch.W
Welmin, W. P. (вес Velmine)
Wendt, E-, 756 G62, 767,
769,^75)
Wer^rurow, A. a, 760, 762
W«rthmcLJ.t76ttG75)
Whitwortb, P 760
WieferichД7 741, 764 G66-
& 771, ^75J
w3faklR763
w3fakenl,R,763
Woodall, Й. i., 750
Wronaki, H., 743 G73)
Zerr, G. В. М, 766

SUBJECT INDEX.
For classification of Diopbantine equations, вее Table of Contents.
Abd'a theorem. 501, 581
Abeliau integral, «1
Abrade, 42, 348
Additive, 41-3,346-8
Affected «quare, 348
Algebraic number^ 237, 253,
271, Ш, Ш, ЗвЗ, 396,398,
309, 412, 421, 477, 534,
536-8,54K2, 646,549,550,
567,669, 570'573-4, 5^8-9,
6934>, dl5.674,677-nS, 6fll,
ВД6,738-742,745,755,757-
fl,761-5, 767-8, 7ЙМ, 775
(see cbw, norm, unit)
— —> conjugate develop-
development* of, 674
Analytic functions, 162, 725
Apocalyptic, 5
Apocopatton, 137
Approximating fractions, 390,
412
Approximation, 83-9, 361,
378
— to fractions, 47, 50, 93r 9d,
88
• aquare roots. 341-2,
346-7, 35СЫ, 354-6, 378,
390, 4Ш
Arithmetical progression, 1,
2 6, 36. 47, 32, 107, 116,
124, 128, 143. 286, 323,
419, 440-1, 443-4, 446,
450, 452, 454, 456-3, 522,
582Ц_595. 602, d<H-5,
612, tfro, е&ы, im, 710,
716, 762,181 (eee rational,
equazttt, Hiun)
, eum of terms of, equal
to a». 112, 118, 127, 130,
161, {66
— вепс» of order n, 18
Astronomy, 41, 61-2
Aeymptotic, 162, 172, 246,
Ш, 253. 255-7, m-3,
318, 705 fee limit, mean)
Augment* 41
Antomedian, 205, 439
Automoxph, 248, 297, 430,
Ш, 677
Bachet'» theorem, 275
Balloia 154
Bemouiliaa function, 153
— numbera, 127, 161, 741-2,
744-5. 761, 768, 773, 775
Bertnmd'epoetulate, 679-681
Ве&яЦ'а function», 256
Bible, 5
Bicuraal curve, 676
Bilinear formJ568
Binomial coefficient, 4, 5, 7,
9, IS, 20, 39. 106-7, il4,
136, 127, 144-7, 149, 155,
164, 599 (dee figurate, tri-
angle of Ea«cal)
every number a nun
of, 144
Bipartition, 153
Bicniadtate, 437, 717 (see
difference,, wim of equ&fea)
Biquadratee. relations be-
between, 564, 647-657, 663
— and equares, 489,
646-3r 656-660, 665
—, sum of 2, not a square,
615-620
— three, 38, 684, 753
—. two equal sums of two»
644-7
Biquadratic residue, 370, 630
Birational tranHformation,
568, 692, 676
Bonds, 149
— between atoms, 131
Cattle problem, 342-5
Cede (eeeCoeci)
Chessboard, 162r 245
Chinese problem of remain-
remainders, 57-64
Circle (вее rational)
Circulating function, 115»
117, 11§, 161
Circulator, 119-121, 159
daee number, 550,741, 745,
755, 763, 767-8 775
Coeci rule. 79-81 (aeevirgina)
Colonne, о
Combination*, 114, 124. 128,
132-3, 144, 154-5, Щ
Combioatory analyme, 36,
141, 145. 151, 161-2 (see
partition)
Complex number, 95, 97,
119, 121-2, 125, 162, 166,
170, 20L 231, 234, 237,
239242247^ 26« 274
239-242,247-^, 26«f 274,
283, VSt, 289, 296-7, 315,
371,373-4,376,389,397-8,
41L 431, Б48, 572r 594,
620, 678, 084, 692, 716,
725, 740, 767
Composite, 12, 15, 2$, 229,
37o
Composition, 145, 147. 149,
Ш, 1ST (p«e product)
— into equaree, 205, 305
— of two forms 3? — erf*,
346-S.355
Concoroant forma. 473-7
Configuration, Ш, 592
Congruence, J9, 33, 87, 408,
Ш, G7B, Ш, 6S7, 692-3,
700, 768, 761, 766, 771-2,
775 (see lineor, quadratic)
Ч^"^ 734,
799
736, 750r 753,756-7,762-5,
769, 770, 774
— 1 + jT ** rf 739,743,756
— 4t« + Bu* + Ct>* ш 0,
749, 751, 764. 773-4
Congruent number, I74t 435,
459-472, G15
of order n, 472, 701-3
— to а ашп. every number,
86,87
Continuant, 183. 244, 254,
381
Continued fraction, 19, 21.
32, 47-49r 51-54, 68, 72,
75, 93-94, 99, 149, 182,
198. 219, 233/237-8, 240.
242-4,250,267-8, 295,344,
350-3, 355-^8, 361, 364-^5,
367-372,374-401, 408r412,
417-8, 473, 540, 587, 641,
673,686, 762
-——, bracket notation in,
49, 149, 240, 35Г
— —, negative, 379, 381
Contravariant, 431
Covariant, 120, 133,161, 568
Cremona transformation 85,
320
Cube, 6, 8f 23, 25, 28, 33, 35,
38, 177-Ш, 187, 246,
251-2, 255-7 (see differ-
difference, every, numbers, sum)
— not a sum of 2 cubes,
545-550
— <A sum of n numbers plus
a&y one a cube, 607-12
Cub&, equal sums of, 550-
562
<—, relations between, 562-6,
613
Cubic equation. 220, 545-
613, 661 (see discriminant)
— form, binary, 53Й, 575,662
— — — mads a cvibft, 56&-Я
— a eqtzifcre, 569-
572,640
, ternary, 588-595, 695
— function made square, 457
— residues, 575
Cuttaca. 42. 58
Cyclic method, 63, 34fr-8,
349,386
Cydotomy, 243, 251, 295.
370-27376, 398, 736, 750,
753-4,763, 766, 769, 770
Decomposition into maxi-
maximum squares, 309, 310
— — — triangular numbers.
32
— powers, 686
Definite polynomial. 720,722,
723-4

800
Subject Index.
Denumerant, 123, 134-5
Derivatives of Arbogaet, 120,
129, 133, 168
Descent, indefinite. 228, 256,
276, 353, 466, 633, 54в,
548, 550, 574, 576, 615-9,
632, 634,636-4 ™> 748
Delermmant, 75, 78, 82-U4,
£9-93, 95, 121, 125, 244,
291,3U, Ав, 566,594, 679
089, 695, 7*f-2, 756
Diagonal number, 342
Diametral number, 167, 170,
342
Difference of 2 squares, 402-
5, 420r 426, 535, 549
— biquadratee not a
square, 615-20
Differences of order», 13,14,
15,117, 124,231-2,366
— — three squares, 446, 450,
452, 454, 505
two biquadratee, pro-
product of two, equal to third,
470
505
-a square
cubee, 535, 540, 578-
581,607,662
Differential operator, 134.
154,162
Digits, 20, 27, 33 38, 52,138,
166, Щ 3ld, 379, 459,
685, 699
Discordant forme, 473
Discriminant of cubic, 548,
695, 662, 736
— equal unity, 695-6
Distributions, 145, 162-^3
Divisor with odd conjugate,
130
— product, 701
Divuleions, 101
Dodecahedral, 16, 23
Double equamiee, ВО, Ш, 445
Elementary divisor, 83, 90
Elliptic function. 25,113,146,
209r 220, 224, 235, 246-7,
249, 250, 262, 265, 285,
293, 295-6, 30i, 305, 309,
ЗЮ, 315-8, 333 378 389,
411, 457, 482, 501, 569,
572, 640-1, 644
— integral, 482, 641
— parameter, 592-3
Epicycloid, 254
Equation linear in 1 ud-
known, 401, 690-1, 700
— quadratic in х and in y>
474, 625, 639-642
— with like first n coefficiente,
714-6
polynomial coefficients,
695, 743 (see Pell)
— sr -Ь У" = zl with x, yf z
polynomials, 760, 774-5
—, ««-i* ^ lr 731 738,
744, 747, 752-5, 76^
— ox» + V = <**, 698.737-
8,749.754-5,757-9, 761-2,
764, 766, 769, 771-6
— x* = y, 687
— l/xi + - - - + if»* -
I/a, 688-691
Equivalent, 91, 96, 677
Euclid's g.c.d. process* 45
52, 53r 61, 93, 95, 352
Every number а вшп of cer-
certain numbers, 14, 22-5. 30,
144, 160. 429, 717, 719 (see
binomial, congruent, figur-
ate, geometrical, polygonal,
pyramidal, sum of squares,
triangular, Waring)
— ' — — four positive
rational cubes, 727-9
■ nine cubes,
23,717-725
— three rational
cubes, 726-7
— ■— represented by «* +
20* + Зт1 + W1. 263, 270
з« +**
з* +v
3t*« 728
— odd number represented
by 2x* + э* + Л 260, 264
Exceptional prime, 74l
Factorial, 9, 124, 126, 143,
232, 679-682, 722
Factoring, 395-7, 763
Factors of (x + y)+ — x* —
V, 737-8, 745-8, 751-5,
760-1
Farey number?, 149
Format's theorem. 55, 171,
238, 25L 281
FermaVe last theorem x* -j-
tr + л 731-775 (see alge-
algebraic, BernouiUian, class,
congruence, equation, /ac-
/actors, polynomial, regular)
_ , n - 3, 545-550,767
, n - 4, 615-620,
757, 761
, n := 5, 732-3, 735,
738, 755, 760-1, 771, 773
. n=6,732,75», 772-
3,776
-_ n = 7, 663, 737,
739, 746-8, 761
,n-10, 772
— , я - II, 774
— , n - 14, 736
-, n = 37, 745,755-6,
769
—, ft - 59, n - 67,
745
"-,neven,736-7, 754-
5, 765, 772-3
— — —, n fractional, 738,
756, 774
,х,у.* polynomials,
749, 750, 757, 760, 770-1,
774, 775
, criterion of Kum-
mer, 744,761,765,767, 775
—' — — — -— Mirimaiioff,
761, 768-9, 771
— Wieferich, 764.
766-8, 771, 775-6
Feueibach's circles, 201
Figurate number, 5, 7-9 13,
14,17, 18, 22,23, 29, 33-5,
39
every number a sum
of, 13
' generalized, 9
Finite aumber of solutions,
674-5, 764
Fourth power (вве biquad*
rate)
Gamma function. 95, 97,721.
725 \
Generating function 76, 137,
139, 148, 152-3, 157, \Ж
A01-164, implicitly)
Genus of curve, 169, 592-3,
675-7, 775
Geometrical methods 55-6,
64, 66, 68P 73, 87, 95, 98,
166, 174-5, 219, 255. 274,
297,318,341,388,396, 425,
434,471,613,542-3, 546-Y
555-7, 576 5«ЬЗ, 596,642,
644, 675-6, 687 (see genu*,
lattice, tangent)
— progression, 105 441.
443, 451, 453, 456, 602-3
— — every number a eum
of terms of ar 105, 110, 112,
.164
Girard'B theorem, 228
Gitter (eee lattice)
Gnomon, 1
Gonal (see polygonal)
Greatest common divisor,
50, 51. 73, 74, 313, 772 (see
Euclid)
Group, 696, 774
Harmonical progression, 435-
7
Heptagonal, 2, 6, 10, 28
Hennitian form, 289
Heron angle, 199
— parallelogram, 207
—, problem of, 485-7
— triangle, 191
Hexagon, central, 545
Hexagonal, 2, 6, 8, 18, 19,
20, 25, 31, 38, 39
Hexahedr&l, 16
Hyperbolic function, 389
Hyperelliptic curve, 677
Icoeahedrai, 16, 23
Ideals (see algebraic)
Indices, 305-6,357, 394
Infinite products, 13, 28, 31,
101-5, 113-6, 137, 140-lr
146, 148-4, 163, 234, 245,
313, 317
— seriee, 13, 16, 19, 20 24,
26, 31, 33, 71, 98, 101-6,

Subject Index.
801
108, 111, 113, Иб-б, Шр
120, 127 137! 1*0-1, №,
146t 148-9,165, 160,162-3,
235-6L 240, 243, 245, 254.
262, 265, 277, 283, 285.
292, 297, 305 313. 333,
337-8, 380, 399, 403,411,
562, 714-6
Integration. 51, 256, 266,
566,685,722-3,749,751
Invariant, 92,433,571,592-3,
643-4
Inverse tangent 395, 716
Irratiocal, 93-99
Irreducible, 674, 695
Isobaric function, 132
Jacobi symbol (see Legendre)
Julian period, 61,62
Knota, 144
Latin square, 154
Lattice, 5L 52, 66, 6Sp 87,
95, 98, lfeo 162, 168, 236,
24IP 245, 257, 724
Legendr&Jacobt symbol
{mjn) 248, 253, 263-4,
305, 308-9, 312-4, 325-7,
370, 372, 386, 431
Lemmscate, 389, 398
Like powers, relations be-
between, 585, 648, 682-6,
705-716
Limit, 97r I55r 158, 254,272,
377,399,400,404,675,68$,
723
Linear congruence in one
unknown, 47, 52, 54-64,
693
— — — 1 improper or
imaginary solutions of, 55
щ several unknowns,
86-88
— congruences in n un-
unknowns, 83, 88-93, 423-4,
426
— difference equation, 107-
111 115, 158,368
— differential equation, 144,
393
Linear equation in 2 un- \
knowne, 41-71,91
— , number of
solutions, 50-1, 64-71 130
in 3 unknowns, 52, 55,
7W3
— number of
solutions, 121, 126, 128,
130, 132,141,145,152,160
A01-164, implicitly)
— — in n unknowns, 41, 43,
73-77, 133, 143, 161
► — — —, number of
solutions, 109, 114, 119,
122, 125-7, i30-l, 134,
140, 142, 155-6, 164
Linear equations, system of,
in 2 unknowns, 66-8
in n unknowns,
73, 77-86, 150, 342 (see
multipartite)
— — number
of eolutione, 106, 123, 144,
149
—- — — — — f fun-
fundamental solution. 75-6.
834
— — — , irreduc-
irreducible solution, 85
— form, 83-99
— forma, system of, S4--86,
90, 93-98
—- —, sum of powers of, 95,
96
— functions made squares,
440, 443-458
, product of, 677-9
— inequality, 52, 135, 153,
156,161
Logarithme,7l4-6,741,744-5
Lucas* u*, ия, 542^3, 698
Magic square, 6, 118, 134,
154, 156, 163
Malfalti'e circles, 197
Matrix, 82-85,90,91,95,134,
139, 153
— augmented, 83, 84, 90
—, rank of, 84, 90, 91, 95
Mean value, 28, 32, 140, 234,
246, 3UT 411, 751 (see
asymptotic)
Minimum, 94-99, 288, 364,
374, 408, 540, 673
Mobius1 function ji, 701
Modul of Dedekind, 92
Modular equation, 389
— system, 85, 91
Multipartite, 123, 145, 147,
150-2, 160-2, 163
Node, 139
Norm, 283,296,373,398.570,
593-i, 677-45, 740
Number and its square sums
of 2 consecutive squares,
477, 488-9, 670
Number of divisors, 67, 129,
139,142,162,235-6,238-fl,
242-3, 248, 250, 265, Ш,
313,403-4
Fibonacci, 439
~- — form iab (a» - b*), 459
— — — —, quotient a
square, 474, 494-5, 661
■— — sete of к integers < i
and prime to z, 701
solutions of щ . . . Uk
- ft, 167, 403, 679
m
Numbera a sum of 2 rational
cubes, 572-8, 729
cubic functions,
578
unlike powers,
725-6
Oblique triangle, 214-6 (see
rational)
, automedianT 205, 439
1 linear relation between
angles of, 213-4
— —, rational angle-bieeo
tors, 209-213
medians, 202-9,212,
428
Octahedral, 16, 23
Optic formula \/x + 1/y =
I/a, 688-691
Orthogonal substitutions
530-2
Pack of cards, 157
Parallelogram, rational sides
and diagonals, 205-6
—, Heron, 207
—a, ratio of areas given,
486
Parallelopiped, 267, 487, 502,
613, 670
Partition, 101-164^ 250, 270
(see bipartition, circulating,
circulator, combinations,
combinatory, composition,
distributions, generating,
multipartite, permutation,
separations, symmetric,
variations, wave)
—, compound, 123, 141
—, conjugate, 121, 124, 136,
139, 154, 103
—, double, 123
—, graph of, 118, 134, 137,
139, 141, 142, 147, 151-4,
158, 160
— into squares, 296
—, modulo M, 88
—, new definition of, 161
—, perfect, 143, 145
—, regularized, 139, 141-2,
152
—.self-conjugate, 136-7, Ш
Pell equation, 341-Ю0
— —; upper limits for solu-
solutions, Ж&9, 400
ш polynomials, 389,
390, 393^, 397, 743
Pentagon inscribed in circle,
202,221
11,13,16*, 18-20,22,2^ 25!
26, 28, 29, 31, 33-35, 37,
113, 147
Permutation of letters, 114,
131, 147. 155, 157, 162
Pile of bullets, 25
Planude, problem of, 485-7
Polygon, 56, 147, 386, 543,
68S
—, rational inscribed^ 221
Polygonal divisors 32
— number, 1-7,9-22,24,28-
39, 117, 147
, central, 5, 10, 12, 16
equal square, 3, 10, 11,
13, 38

802
Subject Index.
* -. flumber of wayfl a, 3.
7,11
— — of second order, 5, 10
— .sum of power* of, 30,34
~- numbere, g.e.d. of two, 36
of m riaee, every num-
number a eum of я», 6, 12—14
16-18. 29-32, 309
— —. нолю of, 2, 5> 19,20. 22,
25,26, 28,29, 34,35,37
~- readuee, 33
Polyhedral. 16, 17, 24
Polynomial in а -\-Ъ and
об equal to л* 4- bm, 737,
739, 745 (see equation)
Primes, 11.3ft. 148, 234, 252,
260-1,264,266,396,681
—, sum of two, 282 28»
Frae, 312, Ml, 734, 742,751,
7f3
Probability, 28, 158, 2l4
Product of 2 forms a like
form, 431. 470, 570. 594-5,
678, 601, 697, 727-3
consecutive ia
9, 56, 393, 679-682
itt ( 4 l>/
( 4 >/,
Products by twos plus n
made squares, 513-520
Pronic, 5, 6, 232-3, 350, 407
Pulveriser, 41-44, 59, 348
Pyramid, rational, 221-i
Pyramidal number, 1-5 7 9,
1Q, 14,16-18 30,25, fc>, 34
—~ —, central, б
— —t every Dumber a eum
of, 14, 23, 30
, not a cube» 25, 34
Fyramidi^pyramidal, 7
Quadratic congpraice, 19,
279, 280, 282-8» 295-7^
299-303, 325-7, 384, 408,
693,749
— equatiato in x} y, 412, 485
(вес equation}
— —- in n ^ 3 unknowns.
34
Quadratic form, binary, 11,
17, 26r 94, 184, 233r 236,
239, 242, 247, 254, 280-3,
265, 269, 273-4, 331, 356,
359r 362, 364-5. 367, 370,
372,374-fi, 381,386,391-2,
39fc 4012 4ЙА 408
, , А ,
4КЫ, 417-8, 430, 507,
636-7, 541-2, 649, 653-4,
559, 57S-9, 619, 637, 667,
697, 699, 707, 712, 733,
741, 760 (see composition)
, ternary, 17,260,264-5:
272-3,2Й2,287 2Й1,3&4-&,
379, 422, 424, 429, 430-1
— — in n^4 variables,
263, 268, 308. 311, 313,
330, 332, 335-7, 387, 431,
433. 643, 724 (see every)
— function *= cube, 28,
— — - power, 533-544,602,
764
— — - square, 10, 13, 19,
341-401, 404-7, 43S. 445,
459-532
— —^ constant. 407-412
— residue, 231, 241, 253, 279,
282r 284, 287, 292, 296,
299,30t, 325-нб, S65,370-1,
378, 421-4, 432, 476, 630,
751r766
Quadrilatersl, 201. 211, 2Kb
221, 223, 255, 449 680
Quartic equation, 465, 485,
615-671
— form a equara, 567. 632,
627-644, 732, 746-8
,571
Quaternion, 297
Quoixty, lift
National group of pomte,
676-7
Kational oblique triangle,
191-214, 220, 668
— 7 circumscribed curde,
191, 195-6,200-1 211
-— —' —, eecribed circlet,
195, 20O-lr 213
—» Feuerbach'e circled»
201
— — —, inscribed Circle,
194-7, 200-1, 208, 211,
215, Ш
—f ieoscelea^ 20], 211
— 1 Malfatti^e circlee,
197
— -~ —, perimeter. 193
195-7,199,200-1,212
-— — —, rational angle*
tnaectttB, 210-3
medians, 207-9,
212-3,511
— — —r aides in arithmetical
pfogrefflion, 192, 196-9,
2<ХЫ» 214^5
ratio of area
Recurring series, 36, 75, 110,
366, 391, 674, 695-6
Reduced multiplier, 41
Begular prime. 741, 746, 757,
759^ 762, 767, 775
Remaindered 57
Represe&t&faoa by aquarea,
296,305
Eight triangle, 165-190, 273.
459,463
— -—, area given, 465, 615
— not a gquare, 462,
615-620
— — formed fejia 2 num-
numbers, 166
, primitive, 166, 459
— — problems involving
area of, 166, 175-181,
498, 533, 617, 634
~- *- — «idee, but not
uc*, 1ЗД-&, 620-7
~- —, product of eidea divis-
divisible by «ixty. 171
, rational angle-bisec-
angle-bisectors, 188-9
— triangles areas in given
ratio, 174, 474. 494-5
—■ — number of, with given
aide, 172
> <J. «qual area. 172-4.
525
— hypotenuse*. 225
— —, цЫев of^ 189t 190
with given difference
Or шш of iegsr 181-^
Roots of unity (see cyclot-
omy)
Satin, 37, 245
Seminvariant, i24, 142, 571,
*43
Separations, 151, 156, 16]
Series of L. Pisaoo, 170
Slide ruJe, 694
Sophien, 769
Spherical trianzle, rational,
224
&raare, \-Ar 7, 8, 12, 16, 20,
26, 27, 31, 32, 176 {see
decomposition, diSerefioee,
every, number, sum)
-~ roots {see approximation)
Squares in arithmetical pro-
progression, 186,435-440, 443,
446, 4S9, 492, 498, 527,
604, 617, 635
— — geometrical progres-
progression, 441
—, sum (Л two Jese third ft
square, 507-9
— euma by twos squares,
«97-5BZ, 529
—T linear functione of, m&de
squares, 502-511
Sum and difference of two
equ&res not squares, 618
— of any 2 of 4 integers a
cube, 561
ti integem,
139, 159, 160.162
—-~ cube» 01 numbera in
arithmetical progression a
cube, 583-6, 685
-a square,
585-8, 6S5
— — 2 tube* л ъаряхъ, S72
678-581, 587, 600, 752,
758
— — 3 cubes a square. 566,
605-6
-— — — — & sum of 2
squares, 588
— -~ n cube* ie» <* а юа&1«,
607-612
divwoiB, 19, 24, 31,
125, 140, 244, 246, 250,
264, 266. 269, 285, 2$9,
296,305-317,329-339
l

Subject Index.
803
2 squaw», 11, 15, 23,
27, 32, S3, 36-38, 68,
225-257, 308, 317, 321.
322,346, 348-4,359,364-5,
372,373-4,383,385-7,3M,
399,400,406,407,533,703,
716
a square. 23,190,
446, 466t 483, 497, 527,
601-2, 613 (aee right tri-
triangle)
^tf 234, 236,
249r 253-7
— — 3 squares, 11, 16r 17,
21, 23, 233, 244, 259-274,
276, 27S-&, 282, 322, 335,
350, 385, 605, 7X8, 720
— a square. 222
249, 260-2, 265-9, 271-4,
443, 45$, 502-5, 68S-&
■— а eum erf two,
244t 267-8, 272-274, 3S4,
434
^N, 271-3
— — 4 squares, 6, I5r 18, 24,
83, 233, 250, 261, 267, 270,
273, 275-зоа, 307-е, зш,
312,3l4r 316-7,320-2,329,
330, 339, 428, 529, 649,
666, 718-9
urith given alge-
algebraic вит of roots, 25. 278
-9, 284, 289, 291H*. 319
— — n squares, 17, 22.
25-27, 32, 37, 253, 273-5,
305-323,329,331-2. 334-5,
339, 453, 544, 585, 660,
664, 705-712, 720
—— a equaie, 318-323,
425, 434, 509, 612
-, power of, 319,
321-2
— — squares a biquadrate,
620-7 665-7, 670
Summation over divisors,
329-339
Sums of 2 equarej?, two equal,
206, 225-7, 220, 232-3,
237.245. 248,252-4,256-7,
263, 273, 283. 345. 347,
426-8, 436, 447, 450, 432,
455, 457, 492, 494-5, 497,
499,500,504,525
— — — —, quotient of,
252-3
— — 3 Bquarefi, tw> equal,
260,264,267-4,2f7O-4,2S4,
299
— — ; prodU<lt Of IWO,
a sum of 3 equates, 263,
267-271, 273
■ ~~-—:—not aeum
of 3 squares, 261
— -~ 4 equates, t*o equal»
277-S, 284, 231-2,
303
-~- —--, product of two,
a mun of 4 squares, 277,
281, 283, 2£l, 293, 298,
302-^,530-1
n squares product of
two, 285, 318, 323
■ —' -t quotient o/ two,
314
Symbol, pYm, 1, 3;P'M, 2, 4;
& ДЙЛАУ 6;Л_^ 16;
(Г), 7; № 54; O(x), 255:
(a/6) of Legendre and
Jacobi, 3ft6. S70; (a, b.
. - . ) and j«t b, . . ♦ J
id continued traction^ 49,
357: (afbLt 370; » 705;
ЯЫ) 758
Symbolic, 30, 154, 722, 703,
773
Symnjetnc function, 101,122,
144-6, 154, 160
Sysygy, 76, 161, 571, 643
Table, 2, 12, 15, 2Q, 34, 38,
51, 104-5, 117, 125, 1^7-9,
133, 136. 141, 144, 155,
163, in, 183, 189, 190,
Ш, 195-6, Ш-200, 219,
225, 2324. 236-7, 241,
251-2, 257, 271, 274-5,
290, 297, 353-358, 363,
366, 368-370, 373, 375,
377, 379, 380, 386; 388r
389,391,394, 3%, 398-400,
407, 435, 44S> 459, 462,
464, 471, 473, 485, 494,
538-9, 541, 564, 573-L
634-5 650, 653, 656r 683r
68S, 717,1П, Ш, 722
T'ai-yen rule, 67; 59, 60
Tangent method, 174, 55S,
576, 690-2L595, 661
Tangential, 591-2
TetradecAgona], 5
Tetragone, 7
Tetrahedral, 4 7^ 23, 25, Зб
TetrahedifOD, 502
—-, rational, 220, 222-4
Tbeta functions, 113, 254,
273-4, 30Q, 303r 314, SL6,
378,393
Toroid, 566
Transvereion, 137
Triangle (see oblique, ra-
rational, right)
— of Pascal, 4, 9
— — TartagUa, 36
Triangular number, 1-39,
103, 130, 152, 248, 260,
276. 289, 2fl2, ДО-Ч, 401,
407,435,574,587,664r 707
equal ta a difference oC
biquadrateff, 36
— — cubes. 36,
37
— — — -~ a pentagonal
number, 18, 19, 22, 33, 35
a square, 3, 7, 10.
I3r 16, 26,27,31, 32,38
— ■ —. of a trian-
triangular number, 27, 36, 37
not a biquadrate, 8,10,
17,22,25,36
— — — a cube, 10,17^ 25, 33
— numbers, consecutive, 2.
27, 30, 33, 38
— — difference <rf, % 24, 26,
33-35
, product of. 12, 29,
31, 33-36, 38
■. sutn of squared of. &,
12, 23, 26, 27, 33
— — sum of two, 2, 9 11.
12.15( 24,26,27.30-32.31
36-38
— — — — three, 6, 11-13,
15-18, 20, 24, 26, 28, 29,
31, 33-35, a7, 38
four, 19,20,23-26,
32
— nt 4, 19, 22-24,
28, 30, 32, 34-37, 130
— — — -^ certain and
squares, 2r 5» 12, 15, 20, 23,
26,36-38
■ —■ — give»
every number, 11, 15, 23,
24,26
Triangular-pyramid, 2> 4> 9
Triaaguli-pyramida^ 7
Triansulo-triangular, 7
Trigone, 7
Trihedral as^\t Totiotial, 221,
224,502
Triple equation, 445
Unicureal eurve, 642, W6-7
— surface, 677
Unit, 396, 399, 412, 636, 578,
594, 741-2, 755
Variations, 140,154
Virgma. 79, 106, 123 (see
Waring's probfem, 717^725
Wave, 119, 121, 124-5, 135,
155,15ft
WiboDre theorem, 56-7, 232,
243, 255
Weierstr&sa' normal form of
qtzartlq, 224, 644
Wonder Zahi 666, 5, 6
Zeta. functioti, 725