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Algebraic number theory and heights

Lehmer’s Conjecture on Mahler Measure

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Can a nonzero, noncyclotomic integer polynomial have Mahler measure arbitrarily close to one?

∃λ>1∀P∈ℤ[x]∖{0},M(P)=1orM(P)≥λ
Known results and sources
A reciprocal polynomial’s roots around the unit circle with one pair just outside and inside, emphasizing the unresolved gap above Mahler measure one.
Lehmer’s conjecture asks whether noncyclotomic integer polynomials stay a fixed distance above Mahler measure one.

Research problem

Exact mathematical statement

For a nonzero integer polynomial

P(x)=a0∏j=1d(x-αj),P(x)=a_0\prod_{j=1}^d(x-\alpha_j),

define

M(P)=|a0|∏j=1dmax{1,|αj|}.M(P)=|a_0|\prod_{j=1}^d\max\{1,|\alpha_j|\}.

Lehmer’s conjecture asks whether there is an absolute constant λ>1\lambda>1 such that every P∈ℤ[x]P\in\mathbb Z[x] satisfies either M(P)=1M(P)=1 or M(P)≥λM(P)\ge\lambda.

Problem infographic

Problem at a glance

A problem-first explainer showing polynomial roots, the product defining Mahler measure, the value-one cyclotomic case, and the open universal-gap question.
Mahler measure multiplies only the root magnitudes outside the unit circle; the conjecture asks for a universal separation above one.

Current mathematical picture

Where work on Lehmer’s Conjecture on Mahler Measure stands

Open conjecture

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

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A single-level Chebyshev row-difference determinant gain by itself reaches the leading quadratic scale needed for a contradiction. Convert the surviving balanced or ultra-nonnormal structure into a contradiction without assuming the still-unverified inertia-coinvariant certificate or a missing regulator, normalizer, torsion-norm, or scale-feasibility theorem.

Route status · Narrowed route
Main reductionDegree-spectrum constraint

The source reports a full degree-spectrum inequality bounding the selected reciprocal orbit products by one half of log Mahler measure.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeIndependently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Lehmer’s Conjecture on Mahler Measure in numbers

4.2kretained lines of mathematical investigation4,160 in the current working snapshot
Argument development
3,608 · 87%
Explored or eliminated routes
127 · 3%
Computational analysis
49 · 1%
Open obligations
135 · 3%
Definitions and setup
241 · 6%
7selected mapped statements1routes investigated4open questions4contribution-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 byDerrick Henry LehmerProposed1933Status checkedAug 14, 2026
Source recordedAug 14, 2026

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

12 selected steps

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

12 selected steps

Scroll horizontally to explore the route

Working route overview for Lehmer’s Conjecture on Mahler MeasureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Is there a universal gap above Mahler measure one? — Depends on missing premiseIs there a universal gapabove Mahler measure one?Current reduction — Depends on missing premiseCurrent reductionDegree-spectrum constraint — Depends on missing premiseDegree-spectrum constraintReciprocal-unit reduction — Depends on missing premiseReciprocal-unit reductionClosing target — Depends on missing premiseClosing targetExact gap question — Depends on missing premiseExact gap questionExact orbit-product compression — Depends on missing premiseExact orbit-productcompressionSingle-level Chebyshev row-difference determinant route — stoppedSingle-level Chebyshevrow-difference determinantrouteIndependently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants. — OpenIndependently verify orfalsify the Section 44inertia-coinvariant…Prove a relative-unit or regulator obstruction incompatible with the source’s saturated high-rank, low-height configuration. — OpenProve a relative-unit orregulator obstructionincompatible…Establish whether the confluent determinant route can contribute at the leading quadratic logarithmic scale. — OpenEstablish whether theconfluent determinant routecan…Balanced and ultra-nonnormal frontier — OpenBalanced and ultra-nonnormalfrontier
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

The following claim is rejected or insufficient in the recorded route: A single-level Chebyshev row-difference determinant gain by itself reaches the leading quadratic scale needed for a contradiction. Convert the surviving balanced or ultra-nonnormal structure into a contradiction without assuming the still-unverified inertia-coinvariant certificate or a missing regulator, normalizer, torsion-norm, or scale-feasibility theorem.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Independently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants.Suggested move: Write the exact free coinvariant lattice, residue-color group, induced Frobenius action, and every divisibility exponent in a standalone audit.
Ready to work on
02
Prove a relative-unit or regulator obstruction incompatible with the source’s saturated high-rank, low-height configuration.Suggested move: Test successive-minima and relative regulator lower bounds against the exact signature and norm-one unit-rank ledger without upgrading conditional hypotheses.
Ready to work on
03
Balanced and ultra-nonnormal frontier

Four narrow balanced tasks remain, while the ultra-nonnormal branch is also open.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Establish whether the confluent determinant route can contribute at the leading quadratic logarithmic scale.Suggested move: Complete the prerequisite scale-feasibility calculation, tracking independent directions, normalization weights, derivative losses, and the total order of the gain.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The general reciprocal Mahler-measure gap remains a longstanding unsolved problem; bounded computations and nonreciprocal lower bounds do not resolve it.

[4][2]
External progress

What the literature has established

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

  1. Computational resultA bounded search found no tested polynomial with measure below Lehmer’s degree-ten example; this is computational evidence, not a proof of the conjecture.[3]
  2. Authoritative summarySmyth’s nonreciprocal lower bound reduces the unresolved gap question to the reciprocal case.[2]
  3. Historical sourceLehmer published the small-Mahler-measure problem and the degree-ten polynomial that still supplies the least known value above one.[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusLehmer’s Conjecture on Mahler Measure
Solved special caseNonreciprocal Mahler-measure gap

Nonreciprocal irreducible noncyclotomic integer polynomials have Smyth’s fixed gap above one.

[2]
Related problemSmall Salem numbers

Salem numbers and small reciprocal polynomials provide the central unresolved test class.

[4]

Formal and computational footholds

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

  • computation · source linked; not reproduced by ProofAtlasMossinghoff bounded searches for small Mahler measure

    Published bounded searches and enumerations; they do not establish a universal gap.

    [3]

Formalization opportunities

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

  • Formalization targetA statement-aligned formal library treatment of Mahler measure and the exact universal-gap statement was not located in this bounded pass.

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. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

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.

5 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction3 of 73
  • lemma3 of 73
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementIs there a universal gap above Mahler measure one?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact gap questionintermediate
  • retained route statementReciprocal-unit reductionintermediate
  • retained route statementExact orbit-product compressionintermediate
  • retained route statementDegree-spectrum constraintintermediate
  • Recorded relationshipThe source 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 source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureSingle-level Chebyshev row-difference determinant routereported failure
  • Research targetIndependently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants.open
  • Research targetProve a relative-unit or regulator obstruction incompatible with the source’s saturated high-rank, low-height configuration.open
  • Research targetEstablish whether the confluent determinant route can contribute at the leading quadratic logarithmic scale.open
  • Research targetInertia-coinvariant audit gatesuperseded
  • Research targetBalanced and ultra-nonnormal frontieropen
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: A single-level Chebyshev row-difference determinant gain by itself reaches the leading quadratic scale needed for a contradiction. Convert the surviving balanced or ultra-nonnormal structure into a contradiction without assuming the still-unverified inertia-coinvariant certificate or a missing regulator, normalizer, torsion-norm, or scale-feasibility theorem.
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 bridgeIndependently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants.

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 pointIndependently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants.

Lehmer’s Conjecture on Mahler Measure · ready to start

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Research contextPrepared context for any AI agent

Can a nonzero, noncyclotomic integer polynomial have Mahler measure arbitrarily close to one?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Factorization of certain cyclotomic functionsoriginal source · D. H. Lehmer · Annals of Mathematics / JSTOR · 1933 · DOI 10.2307/1968172 · accessed Aug 14, 2026
  2. 2
    What Is... Lehmer’s Number?survey or monograph · Eriko Hironaka · Notices of the American Mathematical Society · 2009 · accessed Aug 14, 2026
  3. 3
    Polynomials with small Mahler measurepeer reviewed result · Michael J. Mossinghoff · American Mathematical Society · 1998 · accessed Aug 14, 2026
  4. 4
    Around the Unit Circle: Mahler Measure, Integer Matrices and Roots of Unitysurvey or monograph · James McKee, Chris Smyth · Springer · 2021 · DOI 10.1007/978-3-030-80031-4 · accessed Aug 14, 2026

Important qualifications

  • Selective publisher and society-source snapshot, not an exhaustive literature or computational-record review.
  • Historical priority beyond the original published record was not independently adjudicated.

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