x PREFACE. 



of the beautiful theorem that every positive integer is a sum of four integral 

 squares. In 1621 Bachet verified this 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 Goldbach; 

 in vain did he convert the problem into an equivalent, but equally 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 1828-29 Jacobi compared two infinite series for the same elliptic 

 function to show that, if p is odd and a(p) is the sum of the divisors of p, 

 the number of representations of 2 a p as a sum of four squares is 8<r(p) 

 or 240- (p), according as a = or a>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 

 a(p) sets 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 a 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 A; is a sum of four squares 

 the algebraic sum of whose roots equals any assigned odd number between 

 \.'i/c 2 1 and V4/c. In 1873 Realis proved also that every number 

 JV = 4n+2 is a sum of four squares the algebraic sum of whose roots is 

 any assigned one of the numbers 0, 2, 4, . . ., 2/z, where ju 2 is the largest 

 square CAT. 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 ax^-{-by 2 -{-cz 2 =G (mod p), in which a, b, c are 

 not divisible by the prime p, while some determine the number of sets of 

 solutions. The corresponding question for n unknowns is discussed in the 

 brief Chapter X. 



