Artin's conjecture on primitive roots

Statement

Infinitude version

Suppose $a$ is an integer that is not equal to $-1$ and is not a perfect square, i.e., $a$ is not the square of an integer. Then, there exist infinitely many primes $p$ such that $a$ is a primitive root modulo $p$.

Density version

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Relation with other conjectures and known facts

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

External links

• A survey of Artin's conjecture and the developments related to it (PDF)