CHAP. IX] SUM OF FIVE OR MORE SQUARES. 309 



where 77 = 1 or ^, according as a = or a > 0. Next, 



For d = 3 (mod 4), 



where 77 = 30 if a = 0, A = 3 (mod 8); 77 = 74/3 if a = 0, A = 7 (mod 8); 

 77 = 140/3 if a > 0. For 5 ^ 3 (mod 4), 



where 77 = 1/3 or 5/12 according as a = 0, a > 0. 



J. Liouville 14 stated that, if m is of the form 8k + 7, 



where i ranges over the positive odd integers < Vm, and d ranges over the 

 divisors of the odd number n. 



E. Catalan 15 obtained by means of elliptic functions the result that the 

 number of solutions of i\ + + i\ = Sn in odd integers ii, - , is equals 

 the sum of the cubes of the divisors of n. 



J. W. L. Glaisher 16 stated that, if R m is the number of representations 

 of N as a sum of m squares (attention being paid to the signs of the roots of 

 the squares), and if P is the sum of the reciprocals of the odd divisors of N, 

 then 



Ri - i2 + ifls - IfR* = (- 1) N ~ 1 2P. 



C. Sardi 17 stated that the numbers of the form 40m + 63 are decompos- 

 able into seven squares which end with the digit 9. Cf . Santomauro. 19 



G. Torelli 18 noted that the preceding result follows from Fermat's 

 theorem that every number is a sum of m m-gonal numbers, in the equivalent 

 formulation by Barlow 90 of Ch. I, which implies also that 200m + 14283 

 is a sum of 27 squares ending in 29, of which 23 equal 529 or 729. 



E. Santomauro 19 proved that every integer 40w + 9k is a sum of k 

 squares which end with the digit 9 [if k > 1, as it fails for m = 2, k = 1]. 

 Cf. Sardi. 17 



E. Lemoine 20 called N = a\ + + a>l a decomposition of N into 

 maximum squares and n the index of N if a\ is the largest square ^ N, 



14 Jour, de Math., (2), 14, 1869, 302-4. 



16 Recherches sur quelques produits indefinis, M6m. Ac. Roy. Belgique, 40, 1873, 61-191. 

 Resume in Nouv. Ann. Math., (2), 13, 1874, 518-23. Cf. Berdelle. 33 



16 Mess. Math., 5, 1876,91. 



17 Giornale di Mat., 7, 1869, 115. 



18 Ibid., 16, 1878, 167. 



19 Un teorema d'analisi, 1879, 8 pp. 



20 Comptes Rendus Paris, 95, 1882, 719-22. 



