310 HISTORY OF THE THEORY OF NUMBERS. [CHAP. IX 



a\ the largest square ^ N a J, etc. Let y n be the least number of index n. 

 For n even, y n ends with 67; for n odd, with 23. Also, 



/ -. I O \ 2 



==3). 



M. d'Ocagne 21 stated the empirical generalization that, if m ^ 3, the 

 last [(m l)/2] digits of y m are the same and in the same order as those 

 of y m+ z- Lemoine added the remark that the only possible final squares 

 are R z , R 2 + 1, R 2 + 1 + 1, B 2 + 1 + 1 + 1, R 2 + 2 2 , # 2 + 2 2 + 2 2 , 

 # 2 + 2 2 + 1, R 2 + 2 2 + 1 + 1, R 2 + 2 2 + 1 + 1 + 1, where R > 2. 



T. J. Stieltjes 22 noted that, in view of Jacobi 25 of Ch. VIII, the number 

 of decompositions of N = 5 (mod 8) as a sum of 5 positive odd squares is 



AT - - (M - - I") 2 1 



- 8-1 2 ) + 2/(# - 8-2 2 ) + - - -, 



where <r(ri) is the sum of the divisors of n, and 4f(m) = 2( 

 summed for the divisors d of m. 



C. Hermite 23 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 



Jx(#) + x(N - 2 2 ) + X (N - 4 2 ) + X (AT - 6 2 ) + -, 

 x (n) s Z(3d + d')/4, 



summed for all factorizations n = dd f , d' > 3d. 



Stieltjes 24 noted that the total number F(ri) of solutions of 



n = x\ + + x\ 

 is 24A(ri) + lQB(n) for n even, and 8A(ri) + 4&B(ri) for n odd, where 



A(n) = X(n) + 2X(n - 4) + 2X(n - 16) + 2X(n - 36) + , 

 B(n) = X(n - 1) + X(n - 9) + X(n - 25) + 



X(ri) being the sum of the odd divisors of n. He expressed A(ri) in terms 

 of B(ri), and (4ft) in terms of B(ri), and therefore F(4ri) in terms of F(n). 

 He verified for each odd prime p < 100 that F(p y } = 10(p 3 p + 1), and 

 for p = 3, 5, 7 that 



T. Pepin 25 expressed the number N(m, 5) of representations of m as a 

 sum of 5 squares in terms of N(m, 4) in the evident way of considering 

 m x 2 as a 31 . By use of elliptic functions he evaluated NI N 2 , where 

 NI (or ]V 2 ) 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(- 1) 



21 L'interm6diaire des math., 1, 1894, 232. 



22 Comptes Rendus Paris, 97, 1883, 981. 



23 Ibid., 982. 



24 Ibid., 1545. 



88 Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 9-48. 



