224 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. VIII
L. Poinsot10 gave the proof due to Crelle.9
J. A. Grunert11 proceeded by induction from n — l to n, making use of the first part of Lagrange's proof. D. A. da Silva12 gave a proof.
NUMBER OF ROOTS OP HIGHER CONGRUENCES. G. Libri16 found that/(z, y, . . .)=0 (mod m) has
1 * 4 hr1 2kirf . . . 2kirf\ — S £ . . A S cos — -+i sin — -f mx~av~c U-o ™ m J
sets of solutions such that a^x^b, c^y^d, ____ The total number of sets of solutions is
155... l+e«2£+co.^+ . . . +cos 2(»-
V. A. Lebesgue17 proved that if p is a prime we obtain as follows the residue modulo p of the number Sk of sets of solutions of F(XI, . . ., sn)=0 (mod p), in which each xt- is chosen from 0, 1, . . ., p — 1, and F is a polynomial with integral coefficients. Let SA be the sum of the coefficients of the terms Axf. . .xk° of the expansion of Fp~l in which each of the exponents a, . . ., g is a multiple >0 of p-1. Then Sk= (-1) *+1 SA (mod p).
Henceforth, let p = /im-j-l. First, let F=zm-a. In F*"1 the coefficient of x"«»-i-»> is (P;1) ( - a)n= an (mod p) . The exponent of x will be a multiple >0 of p — 1 only when n = k(p — l)/d, for A; = 0, 1,. . ., d— 1, where dis the g. c. d. of m and p - 1. Thus ^sSa*^"1^ (mod p), while evidently /Si<p. According as a{p~1)/d= 1 or not, we get Si = d or 0.
Next, let F=-xm-aym-b. Set c=ai/m+Z>. In (xn-c)p-1 we omit the terms in which the exponent of x is not a multiple >0 of p — 1 and also the xm(P-i) not containing y. Since the arithmetical coefficient is =1 as in the first case, we get
chyjn(p-l-h) i C2h^n(p—l-2h) \ i^m-Dh^mk
In this, we replace ckh by those terms of (aym+b)hh in which the exponents are multiples >0 of p — 1, viz.,
Set 7/ = l, and sum for k = l, . . ., m — 1; we get — S2 (mod p). It is shown otherwise that S2 is a multiple < mp of m.
To these two cases is reduced the solution of
(1) F-aiXJm+. . .+ajfc£jtms=ci (mod p
"Jour, de MathSmatiques, 10, 1845, 12-15.
"Klttgel's Math. Worterbuch, 5, 1831, 1069-71.
"Proprietaries . . . Congruencias binomias, Lisbon, 1854. Cf. C. Alasia, Rivista di fisica, mat.
e ec. nat., 4, 1903, p. 9. "M6m. divers Savants Ac. Sc. de 1'Institut de France (Math.), 5, 1838, 32 (read 1825). Jour.