306 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ix 



residues 0, 1, , m 1 respectively. Write a m+t = a t . For m odd, 



m-l "'-1 



i=0 *=0 



when k = 1, t m 1. For m even, 2aia i+ j = 2aia i+k if j k is even, and 



m 1 

 i=0 



Lebesgue 6 proved his preceding results. 



Lebesgue 7 noted that tables of indices lead to integers ay such that 



P = /M/Cp- 1 ), /(P) = oo + OIP + + Om-ip"- 1 , P m = 1, 

 where p is a prime mco + 1, m > 2. Set 



= A + A lP + + A^p"- 1 = F(p). 



Then p fc = F(p}F(p~ 1 }. Hence if in the decomposition of 2p into a sum 

 of m squares we change a, into A;, we get a decomposition of 2p fc . 



J. Liouville 8 stated that the number of representations of the double of 

 an odd number m as a sum of 12 squares is 2642d 5 , where d ranges over 

 the divisors of m. The number of proper representations is 264Z 5 (m), where 



Z n (m) = {a na + a n(a - l) \ - - {c ny + c* 1 ^"}, m = cftf - c\ 

 a, , c being distinct primes. If D 2 ranges over the square divisors of m, 



23 Z n (m/D*) = d\ 



D 



Liouville 9 stated that the number of representations of 2 a m (a > 0) as 

 a sum of 12 squares is 



24 



^(21 + 2 5a - fl -5)2d 5 , 

 ol 



summed for the divisors d of m. Proof by Humbert. 48 



Liouville 10 denoted by N(n, p, q) the number of decompositions of n 

 into p squares of which the roots of the first q are taken odd and positive, 

 while the last p q are even and the roots are taken positive or negative 

 or zero; by N(n, p) the number of representations of n as a sum of p 

 squares. It is stated that 



(1) N(2m, 12) = 264{N(2m, 12, 2) + 224N(2m, 12, 6) 



+ 256N(2m, 12, 10) } (m odd). 



Let m be odd, d any divisor of m, 5 = m/d, and set 



6 Jour, de Math., 19, 1854, 298; (2), 2, 1857, 152. 



7 Ibid., 19, 1854, 334-6; Comptes Rendus Paris, 39, 1854, 1069-71. 



8 Jour, de Math., (2), 5, 1860, 143-6. 



9 Ibid., (2), 9, 1864,296-8. 



10 Ibid., (2), 6, 1861, 233-8. Proof by Bell. 686 



