Analytic number theory · primitive roots · Kummer extensions

Artin’s Primitive-Root Conjecture

Collaboration beta

Choose an integer that is neither a square nor one of the excluded trivial cases. Artin’s conjecture predicts that it generates every nonzero residue modulo a positive proportion of primes, with a precise density. The source controls some ranges and isolates a tensor-sum bottleneck, but reports no proof of the conjecture.

N_{a} (x) : = # {p \leq x : {ord}_{p} (a) = p - 1} \sim δ (a) Li (x),

Known results and sources

Artin’s conjecture predicts that an admissible fixed integer is a primitive root for a precise positive proportion of primes; the retained source does not prove the prediction.

Research problem

Exact mathematical statement

Fix an integer $a$ with $|a|>1$ , $a\ne-1$ , and $a$ not a square. For primes $p\nmid a$ , let $\operatorname{ord}_p(a)$ be the multiplicative order of $a$ modulo $p$ . Artin’s Primitive-Root Conjecture predicts

N_a(x):=\#\{p\le x: \operatorname{ord}_p(a)=p-1\}\sim\delta(a)\operatorname{Li}(x),

where the source writes

\delta(a)=\sum_{n\ge1}\frac{\mu(n)}{[\mathbb Q(\zeta_n,a^{1/n}): \mathbb Q]}>0.

The retained source explicitly says that the conjecture is not proved and that no literature-status search was performed; its claims preserve the source’s own workstream and standard-input labels.

Problem infographic

Problem at a glance

Problem-first infographic for Artin’s Primitive-Root Conjecture showing powers of a fixed admissible integer modulo a prime, the full-cycle primitive-root condition, the predicted positive prime density, the source-reported controlled lower and upper q ranges, and the unresolved varying-radical middle range. — The conjecture asks how often a fixed admissible integer generates every nonzero residue modulo a prime; the source narrows the analytic tail to a Möbius-weighted tensor cancellation problem but reports the conjecture open.

Current mathematical picture

Where work on Artin’s Primitive-Root Conjecture stands

Open conjecture

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Complete residue-symbol or character symmetrization is recorded as an exact route elimination: after full summation, the dense family collapses to an affine transform of the original two-valued Kummer trace and supplies no new averaging dimension. High factorial moments, straightforward norm amplification, short Fourier truncation, and naive diagonal second moments are likewise marked eliminated or circular in their stated forms. Prove cancellation in the aggregate Möbius-weighted pure-tensor target, or in an equivalent finite-rank formulation, without taking absolute values term by term. The first unverified conversion must preserve the coefficients across growing-degree and growing-conductor twists, and any mechanism must survive the one-large-factor rank-one obstruction.

Route status · Narrowed route

Main reductionCurrent reduction

Finite Chebotarev and inclusion-exclusion reduce the conjecture to vanishing rough nonprimitive mass. A moving-cutoff detector then rewrites that mass, up to a negligible error in the source, as a full Möbius-weighted sum of centered pure Kummer tensor traces over rough squarefree indices. For fixed small-prime cutoff, the finite sieve is compressed into finitely many fixed twists, leaving aggregate cancellation across growing degree and conductor as the analytic bottleneck.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeBilinearize the upper rank-one tensor blocks.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Artin’s Primitive-Root Conjecture in numbers

2.2kretained lines of mathematical investigation2,161 in the current working snapshot

Argument development: 1,881 · 87%
Explored or eliminated routes: 37 · 2%
Computational analysis: 4 · 0%
Open obligations: 79 · 4%
Definitions and setup: 160 · 7%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byEmil ArtinProposed1927Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Bilinearize the upper rank-one tensor blocks.

Suggested move: For fixed small-prime cutoff and dyadic q above the one-quarter-power scale, derive a Type-I/II or dispersion estimate that improves on Brun–Titchmarsh while retaining the fixed sieve and centered Kummer sign.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Bilinearize the upper rank-one tensor blocks.Suggested move: For fixed small-prime cutoff and dyadic q above the one-quarter-power scale, derive a Type-I/II or dispersion estimate that improves on Brun–Titchmarsh while retaining the fixed sieve and centered Kummer sign.

Ready to work on

Build an aggregate finite-rank explicit formula.Suggested move: Convert the finite fixed-twist prime sums to a full von Mangoldt or contour expression with prime powers, ramification, zeros, and possible poles accounted for while preserving every Möbius and rank weight.

Ready to work on

Verify savings on the rank-one obstruction.Suggested move: Stress-test any aggregate cancellation mechanism on the one-large-factor rank-one term and its prime-residual subcase, rather than obtaining savings only from composite tensor ranks.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

The original unconditional conjecture remains open: no single eligible fixed integer is known unconditionally to be a primitive root modulo infinitely many primes. Hooley proved the corrected density statement under generalized Riemann hypotheses, and Heath-Brown proved a strong unconditional three-integer disjunction, but neither result supplies an unconditional solution for a specified integer.

[5][6][2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
PreprintFan and Pollack established a uniform Artin–Hooley asymptotic and a uniform least-prime bound under GRH; these remain conditional results in a preprint.[6]
2024
Peer reviewedHazra, Murty, and Sivaraman improved unconditional lower bounds for a two-variable Artin variant while explicitly retaining the original conjecture as open.[5]
1986
Peer reviewedHeath-Brown proved an unconditional disjunction showing, in particular, that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes, without identifying which one.[3][5]
1967
Peer reviewedHooley proved the corrected Artin density formula assuming generalized Riemann hypotheses for the relevant Dedekind zeta functions.[2][1]

Mathematical neighborhood

Related results and reusable starting points

Current focusArtin's primitive root conjecture

Dependency or reductionGRH for relevant Dedekind zeta functions

Effective control of splitting primes in Kummer extensions under generalized Riemann hypotheses yields the corrected Artin density formula.

[2][1]

Related problemTwo-variable Artin conjecture

The two-variable variant asks about primes for which one rational number lies in the subgroup generated by another; its unconditional and conditional results do not solve the fixed-integer conjecture.

[5]

Related problemArtin's constant and correction factor

Artin's constant is the generic density factor, while arithmetic entanglement can require a rational correction factor depending on the integer.

[1][9]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedmathlib primitive-roots-of-unity and Kummer-extension support
mathlib defines primitive roots of unity and formalizes substantial Kummer-extension splitting and irreducibility results. The cited modules do not state Artin's prime-density conjecture or Hooley's theorem.
[7][8]
dataset · source linked; not reproduced by ProofAtlasOEIS A005596: Artin's constant
OEIS provides a maintained decimal expansion, formulas, references, and sample programs for Artin's constant. The linked digits were not independently reproduced in this collection run.
[9]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA formal statement relating an integer's image in ZMod p to generation of the multiplicative group for varying prime p.
Formalization targetFormal natural-density and asymptotic-counting infrastructure for sets of primes.
Formalization targetEffective Chebotarev density estimates for the tower of Kummer extensions used in Hooley's argument.
Formalization targetA formal treatment of the required generalized Riemann hypotheses and their error-term consequences.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction1 of 91
lemma7 of 97

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementArtin’s Primitive-Root Conjecture
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact centered-incidence identityintermediate
retained route statementFinite sieve reduces to a rough tailintermediate
retained route statementLogarithmic small-q rangeintermediate
retained route statementUpper residual-prime band is negligibleintermediate
retained route statementPure-tensor Möbius targetintermediate
retained route statementFixed twists expose the first open conversionintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetBilinearize the upper rank-one tensor blocks.open
Research targetBuild an aggregate finite-rank explicit formula.open
Research targetVerify savings on the rank-one obstruction.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Complete residue-symbol or character symmetrization is recorded as an exact route elimination: after full summation, the dense family collapses to an affine transform of the original two-valued Kummer trace and supplies no new averaging dimension. High factorial moments, straightforward norm amplification, short Fourier truncation, and naive diagonal second moments are likewise marked eliminated or circular in their stated forms. Prove cancellation in the aggregate Möbius-weighted pure-tensor target, or in an equivalent finite-rank formulation, without taking absolute values term by term. The first unverified conversion must preserve the coefficients across growing-degree and growing-conductor twists, and any mechanism must survive the one-large-factor rank-one obstruction.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeBilinearize the upper rank-one tensor blocks.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointBilinearize the upper rank-one tensor blocks.

Artin’s Primitive-Root Conjecture · ready to start

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Sources and references10 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
The correction factor in Artin's primitive root conjecturepeer reviewed result · Peter Stevenhagen · Journal de Théorie des Nombres de Bordeaux · 2003 · accessed Aug 7, 2026
2
On Artin's conjecturepeer reviewed result · Christopher Hooley · Journal für die reine und angewandte Mathematik · 1967 · DOI 10.1515/crll.1967.225.209 · MR MR0207630 · accessed Aug 7, 2026
3
Artin's conjecture for primitive rootspeer reviewed result · D. R. Heath-Brown · The Quarterly Journal of Mathematics · 1986-03-01 · DOI 10.1093/qmath/37.1.27 · accessed Aug 7, 2026
4
Artin's Primitive Root Conjecture – A Surveysurvey or monograph · Pieter Moree · Integers · 2012-11-30 · ARXIV math/0412262 · DOI 10.1515/integers-2012-0043 · accessed Aug 7, 2026
5
A note on the two variable Artin's conjecturepeer reviewed result · S. G. Hazra, M. Ram Murty, J. Sivaraman · Journal of Number Theory · 2024 · DOI 10.1016/j.jnt.2024.03.008 · accessed Aug 7, 2026
6
Counting primes with a given primitive root, uniformlypreprint · Steve Fan, Paul Pollack · arXiv · 2025-05-08 · ARXIV 2505.05601 · accessed Aug 7, 2026
7
Mathlib.RingTheory.RootsOfUnity.PrimitiveRootsformalization · mathlib community · accessed Aug 7, 2026
8
Mathlib.FieldTheory.KummerExtensionformalization · mathlib community · accessed Aug 7, 2026
9
A005596: Decimal expansion of Artin's constantsoftware or dataset · OEIS Foundation · OEIS A005596 · accessed Aug 7, 2026
10
Artin's conjecture on primitive rootsencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

The search covered the original fixed-integer conjecture, its standard conditional proof, one central unconditional disjunction, recent related work, and visible formal or computational footholds; it was not an exhaustive survey of Artin-type generalizations.
A scoped search found no end-to-end formalization of the prime-density conjecture. The listed mathlib modules are partial infrastructure, and the search does not establish that no fuller formalization exists.
The OEIS computation was linked but not independently reproduced by ProofAtlas in this collection run.

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