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

External links

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