Artin's Conjecture

Conjecture 3.1 (Emil Artin)   If $a\in\mathbb{Z}$ is not $-1$ or a perfect square, then the number $N(x,a)$ of primes $p\leq x$ such that $a$ is a primitive root modulo $p$ is asymptotic to $C(a)\pi(x)$, where $C(a)$ is a constant that depends only on $a$. In particular, there are infinitely many primes $p$ such that $a$ is a primitive root modulo $p$.

Nobody has proved this conjecture for even a single choice of $a$. There are partial results, e.g., that there are infinitely many $p$ such that the order of $a$ is divisible by the largest prime factor of $p-1$. (See, e.g., Moree, Pieter, A note on Artin's conjecture.)