# P18: Artin's primitive root conjecture

- ID: `P18`
- Reference: `artins-primitive-root-conjecture`
- Page: https://theoremdb.org/statements/P18
- Record maturity: Reviewed problem with recorded work

## 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 \(p\).

### Context

The conjecture predicts how often a fixed integer generates the multiplicative group modulo a varying prime.

### Problem setup

- **Definition (The residue class of a).** The residue class of a is a primitive root modulo p when its multiplicative order is p - 1.
- **Definition (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.** The conjecture predicts how often a fixed integer generates the multiplicative group modulo a varying prime.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper on 2026-07-22.
- Hooley proved the conjecture conditionally on a generalized Riemann hypothesis. Unconditional results cover weaker alternatives and families.

- Recorded example: The integer 2 is a primitive root modulo 3, 5, 11, and 13.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- Prime searches can estimate densities for fixed a over finite ranges without proving the asymptotic claim.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `artins-primitive-root-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2502.19601
   - Also cited at abstract and main conditional and unconditional results
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2502.19601, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.