CHAP. VI] SUM OF TWO SQUARES. 235 



C. G. J. Jacob! 4 * stated in a letter to Legendre, Sept. 9, 1828, that the 

 theorems relative to numbers represented as a (2] follow from 



= * , 



1-2 1+g 2 l--g 3 1 + g 4 



A. Genocchi 75 noted the conclusion that, if x 2 + y 2 = n has ]V"i (0 or 2) 

 sets of solutions with x or y zero, and N* other sets, NI -f 2iV 2 is double the 

 excess of the number of divisors 4m + 1 of n over the number of divisors 

 4m + 3 of n. 



Jacobi 47 gave the formulae 



2kK 4/ 2 4 3 / 2 4 5 / 2 



r n/2 , 



3 5 



TT 1-? 1 2 1 q 



where m, 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 + 1 of 

 m 2 n over the number of divisors 4& + 3 of m*n; 



( 



T ) 



A comparison of the square of the latter series with the former shows that 

 the number of representations of 2m?n as a sum of two odd squares is the 

 excess of the number of divisors 4A; + 1 of m?n over the divisors 4& + 3. 

 Jacobi 48 proved that 



= 1 + 4f"A<*> * (2K\ 12 = Y n * 



TT z=l \ TT / n= oo 



where A (l) is the excess of the number of divisors 4m + 1 of x over the 

 number of divisors 4m + 3. Although not explicitly stated by Jacobi, 

 it follows that the number of representations of x as a 121 is 4A (l) cf. 

 Dirichlet 52 ]. Evident corollaries relating to the evenness or oddness of 

 the two squares were noted by J. W. L. Glaisher. 49 



Jacobi 50 gave an arithmetical proof of his 47 first theorem: If p is odd, 

 the number of sets of positive integral solutions of y z + z 2 = 2p is the 

 excess E of the number of factors 4m + 1 of p over the number of factors 

 4m + 3. Let 



A R >A' iS' 



p = a ' p a ' ' ' a , 



where a, , p are primes 4m + 1, and a', , a' are primes 4m + 3. 

 The factors of p are the terms of the product 



(1 + a ++<*') (1 + P + + p*)(l + a' + + Q -. 



46 Jour, fur Math., 80, 1875, 241; Werke, I, 424. 



47 Fundamenta Nova Func. Effip., 1829, 106 (31), 107, 103(5), 184(7); Werke, I, 162(31), 



163, 159(5), 235(7). Cf. Jacobi 2 * of Ch. III. 



48 Fund. Nova Func. Ellip., 107, 184 (6); Werke, I, 162-3, 235(6). 



49 Quar. Jour. Math., 38, 1907, 7. 



60 Jour, fur Math., 12, 1834, 167-9; Werke, VI, 245-7. 



