Artin's conjecture on primitive roots

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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

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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 (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

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