# Artin's conjecture on primitive roots

## Statement

### Infinitude version

Suppose  is an integer that is not equal to  and is not a perfect square, i.e.,  is not the square of an integer. Then, there exist infinitely many primes  such that  is a primitive root modulo .

### Density version

Fill this in later

## Relation with other conjectures and known facts

Name of conjecture/fact Statement Conditional to ...
Hooley's theorem Artin's conjecture holds for all  (special cases of) generalized Riemann hypothesis
Gupta-Ram Murty theorem Artin's conjecture holds for infinitely many  Unconditional
Heath-Brown theorem on Artin's conjecture Artin's conjecture holds for all but two exceptional values of . However, no explicit information about the explicit values of  Unconditional