Artin's primitive root conjecture

Prove Artin's primitive root conjecture: Establish that for any integer a that is neither a perfect square nor equal to −1, there exist infinitely many primes p such that a is a primitive root modulo p (equivalently, ord_p(a) = p − 1), without assuming the generalized Riemann hypothesis.

Background

The paper’s Sample problem 1 discusses counting primes p for which ord_p(2) > ord_p(3) and relates this to Artin’s framework about primitive roots. The authors note the classic Artin’s primitive root conjecture, which asks for integers a with infinitely many primes p such that a is a primitive root mod p (i.e., ord_p(a) = p − 1).

They explain that Hooley (1967) proved the conjecture under a generalized Riemann hypothesis (GRH) for Dedekind zeta functions of number fields, but an unconditional proof remains unknown. As part of the problem design, the paper introduces a truncated order ord_{p, x}(a) to avoid reliance on error-term bounds that currently require GRH, emphasizing the conjecture’s open status.

References

Artin's conjecture is open. For our purposes, the relevant fact is that the conjecture has been solved on the assumption of a generalization of the Riemann hypothesis (for zeta functions of number fields).

— FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI  (2411.04872 - Glazer et al., 2024) in Appendix, Section 'Sample problem 1 — high difficulty', Background

This problem is tightly connected to the still-open Artin’s conjecture on the existence of primitive roots modulo infinitely many prime numbers.

— Unclustered BWTs of any Length over Non-Binary Alphabets  (2508.20879 - Fici et al., 28 Aug 2025) in Section 6 (Special case related to Artin’s conjecture)