# Heath-Brown theorem on Artin's conjecture

From Number

## Statement

This result, proved by Heath-Brown, states that Artin's conjecture on primitive roots holds for all except at most two exceptional values. The explicit formulation is below.

### Infinitude version

There is a set of size at most such that the following holds. If is an integer that is not equal to and is not a perfect square, and if , there are infinitely many prime numbers such that is a primitive root modulo .

### Density version

*Fill this in later*

## Remarks

The result is not constructive in terms of the exceptions. In other words, no explicit information about the possible exceptions arises from the result. Thus, there is *no* particular number for which we can deduce that there are infinitely many primes for which is a primitive root modulo .