Artin's conjecture on primitive roots

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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. However, it is proven that there cannot be more than two primes for which Artin's conjecture fails. In October 2026, OpenAI released a catalogue of manuscripts produced by an internal model, including a paper (029) claiming an unconditional proof of the existence part of Artin's conjecture on primitive roots, establishing a density lower bound of c a...

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