On Artin's conjecture.

The problem of determining the prime numbers p for which a given number a is a primitive root, modulo JP, is mentioned, for the partieular case a — 10, by Gauss in the section of the Disquisitiones Arithmeticae that is devoted to the periodie deeimal expansions of fraetions with denominator p. Several writers in the nineteenth Century subsequently alluded to the problem, but, since their results were for the most part of an inconclusive nature, we are content to single out from their work the interesting theorem that 2 is a primitive root, modulo p, if p be of the form 4g + l, where q is a prime. The early work, however, was eonfined almost entirely to special cases, it not being until 1927 that the problem was formulated definitely in a general sense. In the latter year the late Emil Artin enunciated the celebrated hypothesis, now usually known äs Artin's conjecture, that for any given non-zero integer a other than l, —l, or a perfect square there exist infinitely many primes p for which is a primitive root, modulo p. Furthermore, letting Na(#) denote the number of such primes up to the limit #, he was led to conjecture an asymptotic formula of the form