CHAP, vii] BINOMIAL CONGRUENCES. 213 L. Sancery61 (pp. 17-23) employed the modulus M =p* or 2p", 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 </>(M)/n. Set A=A!A2 where A1=p1eip2e*. . -, and p% is a prime dividing both A and n, and p'< is the power of pt dividhig A. Let 5 be any divisor of A2. Then xm^a (mod M) has (j>(n^f)/<t>(n) roots belonging to the exponent nA^; the power aA^ 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 nA^, we can find AX6 roots of the congruence. C. G. Reuschle182a tabulated the roots of /=sO (mod p), where p=m\+l and X are prunes and / is the maximum irreducible algebraic prime factor of ax — 1; also the roots of for c<13, d= — 1 to —26, d=+2 to +21, and for various cubic and quartic congruences. A. Kunerth's method for y2=c (mod 6) will be given in Vol. 2r Ch. XII. E. Lucas1826 treated #2+l=0 (mod p*"), where p is a prune >2, for use hi the question of the number of satins. Given a2+l=0 (mod p), set (a+i)m = A +Bi, /3JB= 1 (mod pm) . Then Aft is a root x of the proposed congruence. B. Stankewitsch183 proved that if x2z=q (mod p) is solvable, p being an odd prune, the positive root <p/2 is =B/A (mod p), where t-2 * A =£i_1+2£i_3+22Si-5+ . • • +q 2 Si, B-S,+0SM+ . . . +g2, where i = (p — 1)/2 and SA denotes the sum of the products of 1, 2,. . ., t taken k at a tune. Let n be a divisor of p — L Let F(x) be the g. c. d. modulo p of xn — 1 and II(a;n/a— 1), where a ranges over the distinct prime factors of n. Call f(x) the quotient of xn— I by ^(z). Then the roots of /(oO—O (mod p) are the primitive roots of zn==l (mod p). [Cf. Cauchy.14] N. V. Bougaief184 noted that if p = 8n+5 is a prime and if x2z=q (mod p) is solvable, it has the root g(p+3)/8 or (^)! q(p+3}/8 according as g2^1-! or -1. If p = 2xZ+l, I odd, and gfs&l, it has the root x=gc/+1)/2. [Legendre.156] T. Pepin185 treated £3==2 by tables of indices. P. Gazzaniga186 gave a generalization of Gauss' lemma (the case n =6 = 2, i82a Xafeln Complexer Primzahlen. . ., Berlin, 1875. Errata, Cunningham.131 mfc G6om6trie des tissue, Asaoc. frang., 40, 1911, 83-6; French transl. of his Italian paper in 1'Ingegnere Civile, 1880, Turin. 1MMoscow Math. Soc., 10, 1882-3, I, 112 (in Russian). wlbid., p. 103. 1MAtti Accad. Pont. Nuovi Lincei, 38, 1884-5, 201. 18BAtti Reale Istituto Veneto, (6), 4, 1885-6, 1271-9.