CHAP. IX] SUM OF FlVE OR MORE SQUARES. 



The following formula is stated: 



307 





5=0 



AQ = 1, 



v _-i = 16 



c-i 



j, to, 4s + 2), 



A v . l = 16"- 2s -^ ( 



The cases v = 1, v = 2 correspond to theorems proved by Jacobi. 1 For 

 = 3, (l) gives N(2m, 12) = 264fr(w). It is stated that 



N(m, 12) = 8f 6 (m) - 16ro 2 ri(m) + 162s 4 = 24r 5 (m) - 2 12 AT(4w, 12, 12), 



where 2s 4 is the sum of the squares of the first terms in the various repre- 

 sentations of m as a sum of 4 squares s 2 + si + si + si. 

 It is stated that 



P2,(m) = Z B a N(2m, to + 2, 4s + 2), B Q = 1, B, = fr > 0), 



s=0 



B a being independent of m, but dependent on ?; 



po(m) = A^(2m, 2, 2), P4 (m) = iV(2m, 10, 2) + 64AT(2w, 10, 6). 



From the latter, N(2m, 10) = 12-17p 4 (m), when m = 3 (mod 4). For 

 such an m, Eisenstein 2 had given N(m, 10) = 12p 4 (m). 



Liouville 11 noted the existence of numbers a = 1, 



, a, v -\ = 16 



"" 1 



&=!,&!, , &_!, independent of m and a, but depending on v, such that, 

 for every odd integer m and every integer a is 0, 



v-l 



s=0 



2 2av p 2 ,(m) = 



+ 4, 4s + 4 ), 

 + 2, 4s + 4). 



='0 



These results and those in his 10 preceding paper hold also if N be replaced 

 by M, where M(n, p, q) is the number of solutions of 



n = 



(i's odd and positive, co's even) for which i'i, , co p _ 7 have no common 

 factor, and if $", p M be replaced by 



where P ranges over the distinct prime factors of m. 



Liouville lla noted that, if m is odd, the number of representations of 

 2 a+2 m by Q = x 2 + 4(?/ 2 + z 2 + t 2 + w 2 + w 2 ) evidently equals the number 

 4{4 a+1 ( I) (m ~ 1)/2 }p 2 (m) of representations of 2 a m as a sum of six squares 

 (Jacobi 1 ). The number of representations of n = 1 (mod 4) by Q is 

 p 2 (n) + 22i 2 npo(ri), summed for the odd integers i for which n = i 2 + 4s 2 . 

 Corresponding results are found for forms like Q in which however only 4, 

 3, 2, or 1 of the coefficients are 4, and for x 2 + 4(?/ 2 + z z + t 2 + w 2 ) + 16y 2 . 



11 Jour, de Math., (2), 6, 1861, 369-376. 



e*., (2), 10, 1865, 65-70, 71-2, 77-80, 151-4, 161-8, 203-8. 



