# Artin's conjecture on primitive roots

From Number

## Contents

## 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 |