Heath-Brown theorem on Artin's conjecture

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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

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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 .