CHAP. VII] SUM OF THREE SQUARES. 265 



L. Kronecker 33 proved by use of series for elliptic functions that the 

 number of representations of n as a BO is 24F(n) l2G(ri), where G(ri) 

 is the number of classes of binary quadratic forms of determinant n, and 

 F(ri) is the number of classes of such forms of determinant n in which 

 at least one of the two outer coefficients is odd. This result gives the 

 theorem of Gauss 20 since G(ri) = F(n) if n = 1 or 2 (mod 4) ; G(ri) = 2F(ri) 

 if n = 7 (mod 8), 3G(ri) = 4F(ri) + < if n = 3 (mod 8), where t = 2 if n 

 is the triple of an odd square, t = in the remaining case. 



J. Liouville 33 " noted that, if m = 3 (mod 8), the number of solutions of 

 m = i 2 -f- i\ -f- il, where i, i\, iz are odd and positive, is 



/m-! 2 



= p'(m) + 2p'(m - 4-1 2 ) + 2p'(m - 4-2 2 ) + , 



where p'(ri) is the excess of the number of divisors < ^n of n which are of 

 the form 4ju + 1 over the number of such divisors of the form 4/x + 3, 

 while p(n) is the corresponding excess for all the divisors of n. 



V. A. Lebesgue 34 stated that every solution of t 2 = x 2 + y z + z z is 

 given by 



t = G(e z A + / 2 C), x = G(e*A - / 2 C), ?/ 2 + z 2 



where G = g* + h 2 , A = a* + 6 2 , C = c 2 + d 2 . In the identity 



2 _ 2 = y2 + z ^ 



set gr = 1, h = 0, and replace ae, 6e, c/, df by a, /5, 7, 5; we get 



(3) (a 2 + /3 2 + 7 2 + S 2 ) 2 = (a 2 + /3 2 - 7 2 ~ 2 ) 2 



+ 4(a5 - 



a special case 35 of Euler's formula (1) of Ch. VIII. Since every integer n 

 is a S), ?i 2 is a sum of three squares ^each 4= 0? m general]. 



A. Genocchi 36 proved Fermat's statement that the double of any prune 

 8k - 1 is a SJ. 



J. Liouville 37 stated that, if m = 1 (mod 4) and F is any function, 



summed for all the decompositions i 2 + co 2 + 16s 2 = m = ij + co? + 8s i 

 in which i and ii are odd and positive, while co and wi are even. G. Zolo- 

 taref 38 gave a proof by use of elliptic functions. 



33 Jour, fur Math., 57, 1860, 253. French transl., Jour, de Math., (2), 5, 1860, 297. Cf. 



Mordell. 112 For n s 3 (mod 8), C. Hermite, Jour, de Math., (2), 7, 1862, 38; Comptes 

 Rendus Paris, 53, 1861, 214; Oeuvres, II, 109. 

 330 Jour, de Math., (2), 7, 1862, 43-44. Cf. Liouville 7 of Ch. XI. 



34 Comptes Rendus Paris, 66, 1868, 396-8. 



36 Also given in Bellacchi's Algebra, 1, 1869, 105. 

 36 Annali di Mat., (2), 2, 1868-9, 256. 



17 Jour, de Math., (2), 15, 1870, 133-6. 



18 Bull. Acad. Sc. St. Pe~tersbourg, 16, 1870-1, 85-7. 



