[#P18] Artin's primitive root conjecture
Problem. If \(a\in\mathbb{Z}\) is neither \(-1\) nor a perfect square, then \(a\) is a primitive root modulo \(p\) for a positive proportion of primes \(p\).
1Context
The conjecture predicts how often a fixed integer generates the multiplicative group modulo a varying prime.
2Problem setup
Definition 1 (The residue class of a). The residue class of a is a primitive root modulo p when its multiplicative order is p - 1.
Definition 2 (The assertion about positive proportion includes infinitude and predicts an explicit density depending on a). The assertion about positive proportion includes infinitude and predicts an explicit density depending on a.
Remark 1. The conjecture predicts how often a fixed integer generates the multiplicative group modulo a varying prime.
3What counts as a solution
- Prove the predicted positive density for every qualifying integer a, or give a qualifying a and prove that the relevant prime set has zero density or fails to have the asserted density.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Goldmakher, Martin, and Péringuey prove their refinements under GRH and weaker versions unconditionally; Artin's predicted positive density remains open unconditionally. Exact unresolved remainder: Prove the predicted positive density for every qualifying integer a without GRH, or give a qualifying a for which the asserted density fails.[1]
1Records
Notes and companion material
Original intake status. The cited 2025 paper states that Artin's predicted positive proportion remains open without the generalized Riemann hypothesis. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Hooley proved the conjecture conditionally on a generalized Riemann hypothesis. Unconditional results cover weaker alternatives and families.
Recorded example 1. The integer 2 is a primitive root modulo 3, 5, 11, and 13.
Computational notes
- Prime searches can estimate densities for fixed a over finite ranges without proving the asymptotic claim.
2See also
How to cite
TheoremDB contributors, “Artin's primitive root conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/artins-primitive-root-conjectureThis page as plain text: artins-primitive-root-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. Leo Goldmakher, Greg Martin, and Paul Péringuey, “Refinements of Artin's primitive root conjecture”. arXiv:2502.19601 (2025). Leo Goldmakher, Greg Martin, and Paul Péringuey, arXiv:2502.19601, abstract. ↗preprint · primary source · arXiv:2502.19601, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2025 paper states that Artin's predicted positive proportion remains open without the generalized Riemann hypothesis. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and main conditional and unconditional results.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States the unconditional gap and proves conditional refinements plus weaker unconditional variants.Source named by the research packet.
An original CC0 restatement prepared by TheoremDB maintainers.