Chap. VII] BiNOMIAL CONGRUENCES. 209 
p an odd prime is the g. c. d. of t and (/)(p") ; the same holds for modulus 2p". 
He found the number of roots of x'^=r (mod m), m arbitrary. By using 
S</>(0 =5, if i ranges over the divisors of 5, he proved (pp. 25-26) the known 
result that the number of roots of x"= 1 (mod p) is the g. c. d. 5 of n andp — 1. 
The product of the roots of the latter is congruent to ( — 1)*"^^; the sum of 
the roots is divisible by p; the sum of the squares of the roots is divisible 
bypif 6>2. 
P. F. Arndt^^^ used indices to find the number of roots of x^ = a. 
A. L. Crelle^^^gave an exposition of known results on binomial congruences. 
L. Poinsot^^^ considered the direct solution of x"=A (mod p), where p 
is a prime and n is a divisor of p — l=nm (to which the contrary case 
reduces). Let the necessary condition ^""=1 be satisfied. Hence we may 
replace A by A^+"''^ and obtain the root rc=A^ if l-\-mk = ne is solvable for 
integers k, e, which is the case if m and n are relatively prime [cf. Gauss^^^]. 
The fact that we obtain a single root x=A^ is explained by the remark that 
it is a root common to a:"=A and x"'=l, which have a single common root 
when n is prime to m. Next, let n and m be not relatively prime. Then 
there is no root A' if A belongs to the exponent m modulo p. But if A 
belongs to a smaller exponent m' and if m' is prime to n, there exists as 
before a root A", where l-\-m'k = ne'. The number of roots of a;"=l 
(mod N) is found (pp. 87-101). 
C. F. Arndt^^^ proved that x'=l (mod 2"), n>2, has the single root 1 if 
t is odd; while for t even the number of roots is double the g. c. d. of i and 
2n-2^ The sum of the A:th powers of the roots of x'= 1 (mod p) is divisible 
by the prime p if A: is not a multiple of t. By means of Newton's identities 
it is shown that the sum, sum of products by twos, threes, etc., of the roots 
of x*= 1 (mod p) is divisible by the prime p, while their product is = + 1 or 
— 1 according as the number of roots is odd or even. If the sum, sum of 
products by twos, threes, etc., of m integers is divisible by the prime p, 
while their product is =—( — 1)'", the m integers are the roots of x'"=l 
(mod p). 
A. Cauchy^" stated that if I = p\'' . . ., where p, q,... are m distinct 
primes, and if n is an odd prime, x"= 1 (mod 7) has rf distinct roots, includ- 
ing primitive roots, i. e., numbers belonging to the exponent n. [But 
x^= 1 (mod 5) has a single root.] 
Cauchy^^^ later restricted p, q,. . . to be primes =1 (mod n). Then 
a:"=l (mod p^) has a primitive root ri, and rc^^l (mod q") has a primitive 
root 7-2, so that x''^ 1 (mod /) has a primitive root, viz., an integer =ri (mod 
p*") and =r2 (mod q"), etc.; but no primitive root ii p, q,. . . are not all =1 
(mod n). 
i«3Von den Kubischen Resten, Torgau, 1842, 12 pp. 
»*Jour. fiir Math., 28, 1844, 111-154. 
»«Jour. de Math^matiques, (1), 10, 1845, 77-87. 
""Archiv Math. Phys., 6, 1845, 380, 396-9. 
"'Comptes Rendus Paris, 24, 1847, 996; Oeuvres, (1), 10, 299. 
"sComptes Rendus Paris, 25, 1847, 37; Oeuvres, (1), 10, 331. 
