[#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 \(p\).

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Definitions and notation

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.

Remark 1. The assertion about positive proportion includes infinitude and predicts an explicit density depending on a.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Artin's primitive root conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Artin's primitive root conjecture, record artins-primitive-root-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Artin's primitive root conjecture.” TheoremDB. P18. Problem statement; statement text SHA-256 d941f242a176968e586c580ef7751c6ea35ba58aaae27d3c5e685554ad62530b. https://theoremdb.org/statement/?ref=P18

BibTeX

@misc{theoremdb-problem-d941f242a176968e586c580ef7751c6ea35ba58aaae27d3c5e685554ad62530b,
  title = {{Artin's primitive root conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 d941f242a176968e586c580ef7751c6ea35ba58aaae27d3c5e685554ad62530b},
  url = {https://theoremdb.org/statement/?ref=P18}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P18] Artin's primitive root conjecture

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

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Computational notes

2See also

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2Problem setup

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Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance