216 History of the Theory of Numbers. [Chap, vii 
when p is an odd prime, and a quadratic non-residue ^ of p is known. 
Set p = 2'a+l, where s^ 1 and a is odd. Then y = ap^~^ is odd, and 
{}>(p^) = 2'y. Tonelli's earlier work for modulus p now holds for modulus 
p^ and we get x= ± g'^c^'^'^^^^. If s = 1, then e = and the root is that given 
by Lagrange if X = l. If s = 2, whence p = 4a+l = 8Z+5, the expression 
for X is given a form free of e = cq: 
x= ± (c»+3)V^+^^/^ y = ap^-\ 
A. Tonelli^^^ expressed the root x in a form free of e for every s: 
7+1 
where the v's are given by the recursion formula 
v.-H = c''-S?.t^'\ . .t^--,%+k (/i = 2, 3,. . .). 
Here k is an existing integer such that A;-|-l is a quadratic residue of p, 
and A: — 1 a non-residue. Thus, if s = 3, 
7+1 
x=^{c^''+ky {{c-'+ky^C+kl^'c 2 , 
where we may take A: = —2 if a is not divisible by 3, but A; = —4 if a is divi- 
sible by 3, while neither a nor 4a +1 are di\'isible by 5. 
N. Amici^^ proved that a;^*=6 (mod 2"), h odd, k^v — 2, is solvable only 
when h is of the form 2^'"^^/i+l and then has 2^+^ roots, as shown by use of 
indices. For (x'")^ = 5, the same condition on b is necessary ; thus it remains 
to solve x'"=j8 (mod 2') when m is odd. If i3 = 8A;+l or 8A:+3, it has an 
index to the base 8/z + 3 and we get an unique root. If /3 = 8/j — 3 or 8A: — 1, 
then x'"= — j3 has a root a by the preceding case, and —a is a root of the 
proposed congruence. 
Jos. Mayer^^^ found the number of roots of x^=a (mod p''), for the 
primes 2, 3, p = 6m='= 1. If fli, Go,. . . are residues of nth powers modulo p, 
and if g is the g. c. d. of n and p — 1, then 0102- . . = -f-l or —1 (modp), 
according as p' = (p — l)/g is odd or even. If p' is even, we can pair the 
numbers belonging to the exponent p' so that the sum of a pair is or p; 
hence there exists a residue of an nth power = — 1 (mod p) ; but none if 
p' is odd. 
K. Zsigmondy^^ obtained by the use of abelian groups known theorems 
on the number, product and sum of the roots of x*= 1 (mod m). 
G. Speckmann^^^ considered x^=a (mod p), where p is an odd prime. 
Set P=(p — 1)/2. When they exist, the roots may be designated P — k, 
P-\-l-\-k, whose sum is p. The successive differences of P^, (P-|-l)^ 
(P+2)2,. . . arep, p+2, p+4, . . .. Thesumof 2 = s+l termsof 2,4, 6, . . . is 
s'^+Ss+2 = z^+z. Adding to the latter the remainder r obtained by di\'iding 
P^ by p, we must get pn-{-a. Hence in pn-\-a—r we give to n the values 
i»«Atti R. Accad. Lincei, Rendiconti, (5), 2, 1893, 259-265. 
"'Ueber nte Potenzreate und binomische Congruenzen dritten Grades, Progr., Freising, 1895. 
>»^\rcliiv Math. Phys., (2), 14, 1896. 445-8; 15, 1897, 335-6. 
