Let $m$ be a number in the list $2,4,p^k$ or $2p^k$ for some $k\geq 1$ Then we know that $m$ has a primitive root, but finding one can be a rather challenging problem (theoretically and computationally).

Gauss conjectured that the number $10$ is a primitive root for infinitely many primes $p$ Much later, in $1927$ Emil Artin made the following conjecture:

However, up to now, nobody has been able to show that a single integer $n$ is a primitive root for infinitely many primes. It can be shown that the number $3$ is a primitive root for every Fermat prime but, unfortunately, the existence of infinitely many Fermat primes is far from obvious, and in fact it is quite dubious (only five Fermat primes are known!).