326 HISTORY OF THE THEORY OF NUMBERS. [CHAP, x 



of signs in 



V. A. Lebesgue 5 gave two proofs of Jordan's formulas, not using induc- 

 tion. The first proof uses his 1 results for reduced congruences. The second 

 proof is based on his 2 amplification of Libri's method. 



H. J. S. Smith 6 proved that if p is an odd prune and m any integer, 

 xz y z = m (mod p) has p{p + ( mfp)} solutions. Each of the con- 

 gruences xz y* = 1, 3, 5, 7 (mod 8) has 48 solutions in which x and y 

 are not both even. If p is any prime and i > 0, i' =^ 0, 



xz y z = mp i (mod p i+i> ) 



has p 2t '+ 2i ' (1 1/p 2 ) solutions in which x, z are not both divisible by p. 



C. Jordan 7 proved that x^y^ + + x n y n = (mod 2) has 2 2 "- 1 + 2 n ~ l 

 sets of solutions, while Xi + 7/1 + x\yi + + x n y n = has 2 2 "" 1 2"" 1 

 sets of solutions. 



Jordan 8 determined the number of sets of solutions of / = c (mod M], 

 where/ is any homogeneous quadratic function of x i} , x m . The number 

 is the product of the numbers of solutions for moduli which are the powers 

 of primes whose product is M . Consider 



/ = P*(diX\ + - + a m x 2 m + b lz XiX 2 + ) = c (mod P x ), 



where at least one coefficient GI, , a ro , &i 2 , is not divisible by the 

 prime P. First, let P > 2. By means of a linear transformation, we may 

 remove the terms XiX 2 , etc., not squares. The problem is reduced to 



A,x\ + + A p x z p + P'(B iy \ + + Bjff) + - s d (mod P"). 



The number of sets of solutions, in which o?i, , x p are not all divisible by 

 P, is P r U, where r = (ju l)(w 1) + r& p, n = p + q + , and U 

 is the number of sets of solutions of A\x\ + + A p x z p = d (mod P), 

 given above. 4 For solutions in which x\, , x p are divisible by P, we can 

 remove a power of P and are led to the preceding case. 



For P = 2, we can transform / linearly into 2 a S a + 2^ + , where 

 each 2 P is of one of the four types S p = x\y\ + + x P y P , 



S + Az> S + Az 2 + AiZ 2 S + w 2 + uv + v z 



p 



where A and AI are odd integers, A ^ 7, and p may be zero. The number 

 of solutions is found by treating these four cases in turn. 



T. Pepin 9 proved Jordan's 4 results by expressing the number of solu- 

 tions in terms of the number for the congruence in which the number of 

 unknowns is less by two. 



1 Comptes Rendus Paris, 62, 1866, 868-72. 



8 Trans. Phil. Soc. London, 157, 1867, 286-7, 18; Coll. Math. Papers, I, 492-4. 



7 Trait< des substitutions, 1870, 198. 



8 Jour, de Math., (2), 17, 1872, 368-402. Comptes Rendus Paris, 74, 1872, 1093. 

 Nouv. Ann. Math., (2), 10, 1871, 227-234. 



