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. Euler 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 1783. In the meantime, Lagrange 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 1829 from 

 infinite series for elliptic functions and proved arithmetically by him in 

 1834 and by Dirichlet in 1840. In a posthumous paper, Gauss left a 

 formula for the number of sets of integers x, y for which x 2 + ?/ 2 = A, 

 i.e., 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, Ahlborn 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 8w+7 is a sum 

 of three squares, a fact easily verified by Descartes. Fermat gave in effect 

 the complete criterion that a number is a sum of three squares if, and only 

 if, it is not of the form 4 n (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 P+cu 2 . 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 x 2 +y 2 +z 3 = n 2 ; a 

 simple expression for the number of solutions was given by A. Hurwitz 

 in 1907. The problem of the number of integers ^x 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 u, v, w for which u 2 -\-v 2 -{-w' 2 ^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- 

 tion as a sum of two or three squares in the^problems^where the latter 

 occur. Hence Bachet and Fermat ascribed to Diophantus a knowledge 



