Let $n>1$ be an non square positive integer (you can have it prime, if you wish), does there exist a prime $p>2$ such that $n$ generates the multiplicative group of $\mathbb F_p$? It sounds true, but I could not find an immediate proof for that... maybe using some reciprocity law? Not sure.
The general question in a strong form is the contents of Artin's Conjecture:
Every integer which is neither a perfect square nor equal to $−1$ is a primitive root modulo infinitely many primes.
This remains unproved.