Artin's Conjecture

Conjecture 2.5 (Emil Artin)   Suppose $a\in\mathbb{Z}$ is not $-1$ or a perfect square. Then there are infinitely many primes $p$ such that $a$ is a primitive root modulo $p$ .

There is no single integer $a$ such that Artin's conjecture is known to be true. For any given $a$ , Pieter [#!pieter:artin!#] proved that there are infinitely many $p$ such that the order of $a$ is divisible by the largest prime factor of $p-1$ . Hooley [#!hooley:artin!#] proved that something called the Generalized Riemann Hypothesis implies Conjecture 2.5.14.

Remark 2.5   Artin conjectured more precisely that if $N(x,a)$ is the number of primes $p\leq x$ such that $a$ is a primitive root modulo $p$ , then $N(x,a)$ is asymptotic to $C(a)\pi(x)$ , where $C(a)$ is a positive constant that depends only on $a$ and $\pi(x)$ is the number of primes up to $x$ .