Artin's primitive root conjecture

Let $a$ be an integer that is not a square number and not $-1$. A prime $p$ is $a$-rooted if $a$ generates the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$, equivalently if

$$\{a^i \bmod p \mid i\in\mathbb{Z}\}=\{1,\ldots,p-1\}.$$

Artin's primitive root conjecture. The set of $a$-rooted primes is infinite.

This is a famous conjecture concerning primitive roots modulo primes. It remains open in general, although important conditional and partial results are known.