^li Chap. VII] BiNOMIAL CONGRUENCES. 213 
L. Sancery" (pp. 17-23) employed the modulus M = p'' or 2^', where p 
is an odd prime. Let a belong to the exponent n modulo M. Let A be the 
g. c. d. of m and 4>{M)/n. Set A=AiA2 where Ai = pi*'p2*' • • • > and p,- 
is a prime dividing both A and n, and p/< is the power of Pi dividing A. 
Let b be any divisor of Ag. Then a:"'=a (mod M) has 0(nAi5)/0(n) roots 
belonging to the exponent nAjS ; the power aAi5 of such a root is congruent 
to a, where a can be found by means of a linear congruence. Given a 
number belonging to the exponent nAi5, we can find Ai5 roots of the con- 
gruence. 
C. G. Reuschle^^^" tabulated the roots of /=0 (mod p), where p = wX+l 
and X are primes and / is the maximum irreducible algebraic prime factor 
of a^ — 1; also the roots of 
T^Hc^O, r/Hc^^O, TyHc^sO, rf^'n-\-d=Q, 
for c<13, d= —1 to —26, d=+2 to +21, and for various cubic and 
quartic congruences. 
A. Kunerth's method for ^^=c (mod h) will be given in Vol. 2, Ch. XII. 
E. Lucas^^^^ treated a;^+l=0 (mod p"), where p is a prime >2, for use 
in the question of the number of satins. Given a^+l=0 (mod p), set 
{a+i)"' = A+Bi, ^B=l (mod p'"). 
Then A^ is a root x of the proposed congruence. 
B. Stankewitsch^^^ proved that if x^^q (mod p) is solvable, p being an 
odd prime, the positive root <p/2 is =B/A (mod p), where 
1-2 i 
A=Si_,+qSi.3+q%_s+ ...+q^ S^, B = Si+qSi_2+ ■ ■ • +q^ 
where i = {p — l)/2 and Sk denotes the sum of the products of 1, 2, . . ., i 
taken k at a time. Let n be a divisor of p — 1. Let F{x) be the g. c. d. 
modulo p of x" — ! and Il(x''^"- — 1), where a ranges over the distinct 
prime factors of n. Call f{x) the quotient of x'' — l by i^(a;). Then the 
roots of f{x) = (mod p) are the primitive roots of x"= 1 (mod p) . [Cf . 
Cauchy.14] 
N. V. Bougaief^^^ noted that if p = 8n+5 is a prime and if x^=q 
(mod p) is solvable, it has the root g(p+3)/8 ^j. (pzl)! g(p+3)/8 according as 
q2n+i^-^ or -1. If p = 2^Z+l, I odd, and q'=l, it has the root x=q^'+'^/\ 
[Legendre.^^®] 
T. Pepin^^^ treated x^= 2 by tables of indices. 
P. Gazzaniga^^^ gave a generalization of Gauss' lemma (the case n = 8 = 2, 
1820 Tafeln Complexer Primzahlen . . . , Berlin, 1875. Errata, Cunningham."* 
^^^^ G6om6trie des tigsus, Assoc, fran^., 40, 1911, 83-6; French transl. of his Italian paper in 
I'Ingegnere Civile, 1880, Turm. 
is'Moscow Math. Soc, 10, 1882-3, I, 112 (in Russian). 
^<^Ibid., p. 103. 
"»Atti Accad. Pont. Nuovi Lincei, 38, 1884-5, 201. 
"»Atti Reale Istituto Veneto, (6), 4, 1885-6, 1271-9. 
