Artin's conjecture on primitive roots
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This article is about the conjecture of Emil Artin on primitive roots. For the conjecture of Artin on L-functions, see Artin L-function.
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of 2024. In fact, there is no single value of a for which Artin's conjecture is proved.