CHAP. IX] SUM OF FlVE OR MORE SQUARES. 311 



where a ranges over the divisors SI =fc 1 of m, and b over the divisors SI 3, 

 while N' (or N") is the number of solutions of m = x z + 2/ 2 + z 2 + 2i 2 

 with x 2 even (or odd). For m = 81 db 1, A 7 "' = 2AT". He noted the recur- 

 sion formulas 



Vm 



mN(m, 5) = 2 (6n 2 - m)2V(m - w 2 , 5) = 10SnW(m - n 2 , 4). 



He proved (p. 48) for any odd prime p the statements by Stieltjes 24 con- 

 cerning F(p 2 ), F(p 4 ). 



E. Cesaro 26 stated that the number of ways of decomposing n into a 

 sum of p squares is in mean Cn p/2 ~ 1 , where 



c _ 1 ( I V" 



" 2(p -2)(p -4)(p-6) \2/ 



For p = 3, C = 7T/4. For p = 4, C = ir 2 /16. 



A. Hurwitz 27 proved and generalized the conjectured results by Stieltjes 24 

 concerning F(p 2 ) and F(p 4 ). If m = 2*pV "'> where 2, p, q, are 

 distinct primes, the number of decompositions of m 2 into 5 squares is 



23AH-3 _ 1 



- - Z - 10 - -rr -- 



_ P 3a+3 - 



- 1 



For proof, set m = 2 k n. Then by Stieltjes' formula, F(m 2 ) is K times the 

 sum, for all positive odd integral solutions a, b of a + b = 2n, 



(a, 6) = X(n 2 } + 2Z(n 2 - 2 2 ) + 2Z(n 2 - 4 2 ) + 

 But if 7, 6, e, are the odd primes dividing both a and &, 



SX(a, b) = E X(oO^(6i) - E 



Cti, &l flp &n 



Z 



where the summation with respect to a t , &< extends over all positive odd 

 integers a t) b t whose sum is 2n/t. By the known formula 



X(l)X(2n - 1) + X(3)X(2n - 3) + X(5)X(2n - 5) + 



viz., the sum of the cubes of the divisors of the odd number n, we get 

 (a, 6) = 



and hence equals [p, ][? )3] in the desired formula for F(m z ). Part 

 of Stieltjes' formulas follow from those of Liouville 5 of Ch. XI. 



26 M6m. Soc. Roy. Sc. de Lidge, (2), 10, 1883, No. 6, pp. 199-200. 



27 Comptes Rendus Paris, 98, 1884, 504-7. 



