Artin's conjecture on primitive roots
in numbers theory, a given integer a which is not a perfect square and not −1 is a primitive root modulo infinitely many primes p
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 2026. In fact, there is no single value of a for which Artin's conjecture is proved.
Formulation
Let a be an integer that is not a square number and not −1. Write a = a0b2 with a0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then the conjecture states
S(a) has a positive asymptotic density inside the set of primes.
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