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 n squares whose sum is a square. 

 Following a hint by Jacobi, Eisenstein stated in 1847 that the number 

 of representations of an odd number as a sum of 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 

 Legendre-Jacobi 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 1919, and were proved 

 by means of theta functions by Humbert and Petr in 1907. In 1867 

 H. J. S. Smith stated general results on representation by 5 and 7 squares. 

 This paper was unknown to the members of the commission whose recom- 

 mendation led the Paris Academy of Sciences to propose for its grand prix 

 des sciences mathematiques for 1882 the subject of representation by 5 

 squares. Prizes of the full amount were awarded both to Smith and 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 Stieltjes, Hermite, Pepin, and Hurwitz (pp. 310-1). Mention should 

 be made of the papers by Gegenbauer (p. 313), Boulyguine (p. 317), 

 Mordell, Hardy, and Ramanujan (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 expansions of elliptic functions) 

 which express many equalities between sums of the values of quite general 

 arithmetical 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 ax z -{-bx-\-c = y 2 . Diophantus was led to such an equation in at least 

 forty of his problems. He was content with rational solutions, which he 

 showed how to find if a or c is a square, or if b = 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 ax 2 +c = y 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 earlier, the Greeks had given approximations to 

 square roots which may be interpreted as yielding solutions of ax z +l = y 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 ax~+l = y 2 , where a = 4729494, and has 

 been solved in modern times. 



