CHAPTER X. 



NUMBER OF SOLUTIONS OF QUADRATIC CONGRUENCES IN 



H UNKNOWNS. 



For n ^ 4, report was made in Ch. VIII on the papers by Libri, 29 Schone- 

 mann, 31 Frattini, 75 Lipschitz, 77 Dickson, 95 Tengbergen, 107 and L. Aubry, 110 

 and to many papers proving merely the existence of solutions. See also 

 Hermite, 21 Lebesgue, 63 and Pepin 80 of Ch. VIII, Vol. I of this History. 



V. A. Lebesgue 1 noted that F = Zc^x* = (mod p), where p is a prime 

 2h + 1, may be reduced by multiplication of the variables by constants to 

 a form 

 (1) y\ + - + y} = n(z\ + + ) (mod p), 



where n = 1 if p = 4g 1, and n is a quadratic non-residue of p if 

 p = 4g -f i. Let Nl, N k , N' k denote the number of sets of solutions of 



2/i + + yl = a (mod p), 



according as a = 0, a is a quadratic residue or non-residue of p. In view 

 of his 2 general theorem, the number of sets of solutions of (1) is 



according as n = I or a quadratic non-residue of p. Also, if P is the 

 number of solutions of F = and TT the number of F ax 2 = 0, the 

 number for F = a is (IT P )/(p 1). It is proved that, if A; is odd, 



Nl = p*- 1 , N k = p*- 1 + t, N', = p*- 1 -t, t = (- i)<p-c*-y-' ; 



while, if k is even, 



Nl = p*~i + (p - 1)1, N k = N' k = p*- 1 - Z, I = (- i)(p-i>/"*/p(*/-i. 



Lebesgue 3 gave a simpler proof of the last results and also found the 

 number of sets of solutions prime to p. 



C. Jordan 4 proved by induction from n = I and n = mton = l + m 

 that, if ai 2 n ^ 0, a,ix\ + + a 2n zL ^ ( m d P)> where p is an 

 odd prime, has p 2n-1 p n ~ l v sets of solutions if k 4 s (mod p), and 

 p2n-i _|_ (p _ pn-i),, sets if A: = 0, where 



are Legendre symbols. Also, aix\ + + a Zn +iX 2 2n+ i = k (mod p) has 

 p 2n + pV sets of solutions. As a corollary, there are (p l)/2 variations 



1 Jour, de Math., 2, 1837, 266-275. 



2 Vol. I, pp. 224-5 of this History. 



Jour, de Math., 12, 1847, 467^71. 



4 Comptes Rendus Paris, 62, 1866, 687-90; TraitS des substitutions, 1870, 156-61 (with a 

 misprint of sign in the theorem on p. 610). 



325 



