History Of The Theory Of Numbers - I

CHAP, vii] BINOMIAL CONGRUENCES. 209
p an odd prime is the g. c. d. of t and $(pn); the same holds for modulus 2pn. He found the number of roots of ofer (mod m), m arbitrary. By using S0(0 =6, if t ranges over the divisors of 6, he proved (pp. 25-26) the known result that the number of roots of xn= 1 (mod p) is the g. c. d. 5 of n andp— 1. The product of the roots of the latter is congruent to (—I)34"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. Arndt163 used indices to find the number of roots of z3=a.
A.L. Crelle164gave an exposition of knownresults on binomial congruences.
L. Poinsot166 considered the direct solution of xn=A (mod p), where p is a prime and n is a divisor of p — l=nw (to which the contrary case reduces). Let the necessary condition Am^ 1 be satisfied. Hence we may replace A by Al+mk and obtain the root x==A€ if l+mk = ne is solvable for integers k, e, which is the case if m and n are relatively prime [cf. Gauss167]. The fact that we obtain a single root z== Ae is explained by the remark that it is a root common to xn=A and xmz=l, which have a single common root when n is prime to m. Next, let n and m be not relatively prune. 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 Ae/, where l.+m/k=net. The number of roots of ccn==l (mod N) is found (pp. 87-101).
C, F. Arndt166 proved that z'=l (mod 2n), 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 t and 2n~2. The sum of the kth powers of the roots of x'= 1 (mod p) is divisible by the prime p if k 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 ofel (mod p) is divisible by the prime p, while their product is s +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 — — (—I)7", the m integers are the roots of rcm=l (mod p).
A. Cauchy167 stated that if 7=pxgM..., where p, qt... are m distinct primes, and if n is an odd prime, xn== 1 (mod J) has nm distinct roots, including primitive roots, i. e., numbers belonging to the exponent n. [But #3=1 (mod 5) has a single root.]
Cauchy168 later restricted p, g,... to be prunes ==1 (mod n). Then xn=l (mod px) has a primitive root rx, and #n==l (mod <?M) has a primitive root r2, so that of = 1 (mod I) has a primitive root, viz., an integer =7^ (mod px) and ==r2 (mod <p), etc.; but no primitive root if p, g,.. . are not all s 1 (mod n).
W3Von den Kubischen Resten, Torgau, 1842, 12 pp.
1MJour. fur Math., 28, 1844, 111-154.
"'Jour, de MathSmatiques, (1), 10, 1845, 77-87.
1MArchiv Math. Phys., 6, 1845, 380, 396-9.
187Comptes Rendus Paris, 24, 1847, 996; Oeuvres, (1), 10, 299.
lfl8Comptes Rendus Paris, 25, 1847, 37; Oeuvres, (1), 10, 331.