224 History of the Theory of Numbers. [Chap.viii 
'J 
L. Poinsot^" gave the proof due to Crelle.' ■ 
J. A. Gninert^^ proceeded by induction from n — 1 to n, making use of 
the first part of Lagrange's proof. 
D. A. da Silva^' gave a proof. 
Number of Roots of Higher Congruences. 
G. Libri^" found that /(a:, ?/, . . .) = (mod m) has 
I 
1 b d 
-22 
Wl x=o v=t 
;--! 2A-7rf . . 2Uj\ 
.ALi COS \-x sm \ 
Lfc=o Tnn m \ 
sets of solutions such that a^x^6, c^?/^d, .... The total number of 
sets of solutions is 
1 ^ ^ r, , 27r/, 47r/ , , Am-X)'KJ\ 
— 2 2 . . . O +COS — ^+cos — ^+ . . . +COS 2^^ '-^ V 
7^1=0 v=o y m m m \ 
V. A. Lebesgue^^ proved that if p is a prime we obtain as follows the 
residue modulo p of the number S>k of sets of solutions of F{xx, . . ., x„) = 
(mod p), in which each x, is chosen from 0, 1,. . ., p — 1, and F is a poly- 
nomial with integral coefficients. Let 2A be the sum of the coefficients of 
the terms Ax^ ■ ■ x/ of the expansion of F^~'^ in wliich each of the exponents 
a, . . . , ^ is a multiple > of p - 1 . Then Sk= ( - 1) *"^^ 2 A (mod p) . 
Henceforth, let p = hm+l. First, let F = x"'—a. In F^~^ the coefficient 
of a;""'?-!-"^ is (p-^)(-a)"=a" (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 c? is the 
g. c. d. of m and p-L Thus 51=20*^^"^^'''^ (mod p), while evidently Si<p. 
According as a'-^~^^^'^=l or not, we get Si=d or 0. 
Next, let F = x"'-ay"'-h. Set c = ay"'-\-h. In (x"'-c)p-^ we omit the 
terms in which the exponent of x is not a multiple >0 of p — 1 and also the 
^rn(p-i) jjq|. containing y. Since the arithmetical coefficient is =1 as in the 
first case, we get 
In this, we replace c'''' by those terms of (ay"* +6)'''' in which the exponents 
are multiples >0 of p — 1, viz., 
SGB^"^") 
kh-lhUh 
Set 2/ = 1, and sum for A* = 1, . . . , w — 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) ^ = 01X1"*+. . .-\-a^k"=a (mod p = hm-\-l). 
"Jour, de Math^matiques, 10, 1845, 12-15. 
"Klugel'8 Math. Worterbuch, 5, 1831, 1069-71. 
"Proprietades . . .Congruencias binomias, Lisbon, 1854. Cf. C. Alasia, Rivista di fisica, mat. 
e sc. nat., 4, 1903, p. 9. 
"Mdm. divers Savants Ac. Sc. de I'Institut de France (Math.), 5, 1838, 32 (read 1825). Jour. 
fur Math., 9, 1832, 54. To be considered in vol. n. 
"Jour, de Math., 2, 1837, 253-292. Cf. vol. 3, 113; vol. 4, 366. 
