Chap. VII] BiNOMIAL CONGRUENCES. 215 
F. J. Studnicka^^^ treated at length the solution in integers x, y (y<h) 
of hx-{-l = y^, discussed by Leibniz in 1716. 
L. Gegenbauer^^^ gave a new derivation of the equations of Berger^^^ 
leading to asymptotic expressions for the number of solutions of x^=Z). 
A. Tonelli^^^ gave a method of solving x^=c (mod p), when p is a prime 
4/i+l and some quadratic non-residue g' of p is known. Set p = 2'y-{-l, 
where y is odd. By Euler's criterion, the power 72^"^ of c and g are con- 
gruent to +1, —1. Set €0 = or 1, according as the power 72^"^ of c is 
congruent to +1 or —1. Then 
For s^3, set ei = or 1 according as the square root of the left member is 
= -f-lor-l. Then 
Proceeding similarly, we ultimately get 
g2eycy=-^l (mod p), e = €o+2€i+ . . . +2'-\_2- 
Thus a;= ±^'^c^^+^^/2 (mod p). Then Z^=c (mod p^) has the root 
X=x^'-'c(^^-'^^"'+i^/2 (mod p^). 
G. B. Mathews'^^ (p. 53) treated the cases in which x^^a (mod p) is 
solvable by formulas. Cf. Legendre.-^^^ 
S. Dickstein^^^ noted that H. Wronski^^^ gave the solution 
rM -l(7r-l) 
y==hK+{-iy+'+Mi, = /i+(-1)'+'A| ^, ttJ +Mj 
of 2"— a?/"=0 (mod M) with (iV^)^ in place of K, and gave, as the condition 
for solvability, 
a(lV^)2"-l=0(modM). 
But there may be solutions when the last condition is satisfied by no 
integer A;. 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 y, z 
in a parameter K, subject to the condition aK" — 1 = (mod M). 
M. F. J. Mann"^'^ proved that, if n=2^XV. . ., where X, m, • • • are dis- 
tinct odd primes, the number of solutions of x^= 1 (mod n) is GGiGi . . . 
QiQi . . . , where G= 1 if n or p is odd, otherwise G is the g. c. d. of 2p and 2^~'^, 
and where Gi, Gi,.., gi, g2,. . are the g. c. d.'s of p with X"~\ m''~\- • •> 
X — l,jLi— 1,. . ., respectively. 
A. Tonelli^^^ gave an explicit formula for the roots of x^=c (mod p^), 
"iCasopis, Prag, 18, 1889, 97; cf. Fortschritte Math., 1889, 30. 
"^Denkschriften Ak. Wiss. Wien (Math.), 57, 1890, 520. 
"'Gottingen Nachrichten, 1891, 344-6. 
I'^BuU. Internat. de I'Acad. Sc. de Cracovie, 1892, 372 (64-65); Berichte Krakauer Ak. Wiss., 
26, 1893, 155-9. 
"^aMath. Quest. Educ. Times, 56, 1892, 24-7. 
"^AttiR. Accad. Lincei, Rendiconti, (5), 1, 1892, 116-120. 
