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,
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.