CHAP. VIII] SUM OF FOUR SQUARES. 303 



their g.c.d. is ra, but a, -, d have no common factor. Combining numer- 

 ous cases, he obtained Jacobi's 24 theorem on the total number of solutions, 

 and the theorem that, if N = 2pf p* and a ^ 2, the number of 

 solutions in which a, , d have no common factor is 



8/Kpi + !) (p.- + I)??' 1 p}-', 



where h = 1 if a = 0, h = 3 if a = 1, h = 2 if a = 2. He showed how 

 to find the 4n sets of solutions of x 2 + y z + 1 = (mod p), where p is a 

 prime 4n 1, also the solutions for any composite modulus. 



L. J. Mordell 111 proved by use of theta functions that the number of 

 solutions of x 2 + y* + z 2 + P = m is 8 {26 - 2(- l) c c}, where b and c 

 range over those divisors of m whose complementary divisors are odd and 

 even respectively [equivalent to Jacobi's 24 result]. 



Mordell 112 proved the conjecture by Glaisher 100 (p. 48) on the derivation 

 of all representations of 4wim 2 as a SO from those of 4mi and 4m 2 . 



A. S. Werebrusow 113 gave the general solution of SO = OB . 



L. E. Dickson 114 gave a history of the proofs of Euler's 7 formula (1), 

 its interpretations and generalization to 8 squares. 



For Pellet's proof that Ax 2 + By 2 + C = (mod p) is solvable see 

 paper 104 of Ch. XXVI. 



For minor results, see papers 12 (end), 31, 49, 106 of Ch. VII; 13, 26, 30, 

 39, 52, 76, 84, 94, 95 of Ch. IX; 159 of Ch. XIX; 434 of Ch. XXI. 



i" Mess. Math., 45, 1915, 78. 



112 Ibid., 47, 1918, 142-4. 



113 L'interm6diaire des math., 25, 1918, 50-51; extr. from Math. Soc. Moscow. 

 " 4 Annals of Math., (2), 20, 1919, 155-171, 297. 



