CHAP, vii] BINOMIAL CONGRUENCES. 215 F. J. Studnicka191 treated at length the solution in integers x, y of bx+l=y2, discussed by Leibniz in 1716. L. Gegenbauer192 gave a new derivation of the equations of Berger188 leading to asymptotic expressions for the number of solutions of x2z=D. A. Tonelli193 gave a method of solving z2=c (mod p), when p is a prime 4/i+l and some quadratic non-residue g of p is known. Set p = 2*7+1, where 7 is odd. By Euler's criterion, the power 72s"1 of c and g are congruent to +1, 1. Set e0 = 0 or 1, according as the power y2*~2 of c is congruent to +1 or 1. Then For s^3, set x = 0 or 1 according as the square root of the left member is = +1 or -1. Then Proceeding similarly, we ultimately get 02TCT= + ! (mod p)9 e = e0+2l + . . . Thus x= =fc0<rc(-H-i>/2 (mod p). Then X2=c (mod px) has the root X=Xpx->x-2pX~1+1)/2 (mod px). G. B. Mathews77 (p. 53) treated the cases in which x2=a (mod p) is solvable by formulas. Cf . Legendre.156 S. Dickstein194 noted that H. Wronski169 gave the solution -|CT-D T] +MJ of zn~aynz=Q (mod M) with (I*/1)2 hi place of K, and gave, as the condition for solvability, a(lY1)2n~l=0(modM'). But there may be solutions when the last condition is satisfied by no integer k. This is due to the fact that the value assigned to y imposes a limitation, which may be avoided by using the same expressions for yt z in a parameter K, subject to the condition aKn 1=0 (mod M). M. F. J. Mann194a proved that, if n=2AXV . .., where X, /*, ... are distinct odd primes, the number of solutions of xp=l(mod ri) is GGiG^. . . giQ2 . . . , where G = 1 if n or p is odd, otherwise G is the g. c. d. of 2p and 2k~1y and where GI, Cr2,. ., Qi, 82, > are the g. c. d.'s of p with Xa-1, M6"1,- > X 1,/z 1,..., respectively. A. Tonelli195 gave an explicit formula for the roots of x2=c (mod px), 181Casopis, Prag, 18, 1889, 97; cf. Fortschritte Math., 1889, 30. 192Denkschriften Ak. Wiss. Wien (Math.), 57, 1890, 520. 1MG6ttingen Nachrichten, 1891, 344-6. "4BuU. Intermit, de I'Acad. Sc. de Cracovie, 1892, 372 (64-65); Berichte Krakauer Ak. Wiss., 26, 1893, 155-9. 194°Math. Quest. Educ. Times, 56, 1892, 24r-7. 185Atti R. Accad. Lincei, Rendiconti, (5), 1, 1892, 116-120.