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Number theory · powerful integers · Pell equations · primitive divisors

Erdős–Mollin–Walsh Conjecture

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Can three consecutive positive integers all have every prime factor repeated? The packet says no proof is known, but turns any hypothetical example into a tightly constrained Pell-sequence repair problem.

∄n∈ℤ>0: n, n+1, n+2are all powerful
Erdős Problem #364
Known results and sources
Problem-first card for the open Erdős–Mollin–Walsh conjecture, showing three consecutive integer tiles and asking whether all three can be powerful.
Can three consecutive positive integers all be powerful? The conjecture remains open.

Research problem

Exact mathematical statement

There do not exist three consecutive positive powerful integers. Here a positive integer is powerful when every prime dividing it has exponent at least two.

∄n∈ℤ>0:n, n+1, n+2are all powerful.\nexists n\in\mathbb{Z}_{>0}: \quad n,\ n+1,\ n+2\text{ are all powerful}.

The submitted source reports reductions, packet-scoped proofs, and a certified case study, but no completed proof of the conjecture.

Problem infographic

Problem at a glance

Three-panel deterministic explainer of the Erdős–Mollin–Walsh conjecture: the consecutive-powerful target, the source-reported Pell repair reduction, and three open bridges that stop before a contradiction.
The packet reports a rigid Pell and defect-closure reduction, while lift reciprocity, closure nontermination, and the fundamental branch remain open.

Current mathematical picture

Where work on Erdős–Mollin–Walsh Conjecture stands

Open conjecture

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

Useful failureParity-only primitive-divisor and ordinary genus arguments

The source explicitly warns that ordinary quadratic characters see only parity and thus do not attack the powerful threshold. Valuation-sensitive lift reciprocity, certified nontermination of the repair closure, terminal-system methods, and genuinely coupled class-field or Q-curve invariants remain proposed routes.

Route status · Narrowed route
Main reductionLargest-prime layer

The current work reports layer isolation, a powerful nonsquare normalized prime atom, and endpoint, middle, and exceptional P=3 displacement laws.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment.Task status · Ready to work on

Work mapped so far

Erdős–Mollin–Walsh Conjecture in numbers

1kretained lines of mathematical investigation1,034 in the current working snapshot
Argument development
835 · 81%
Explored or eliminated routes
40 · 4%
Computational analysis
30 · 3%
Open obligations
38 · 4%
Definitions and setup
91 · 9%
8selected 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 byPaul Erdős, R. A. Mollin and P. G. Walsh (independent rediscovery)Proposed1976Status checkedAug 29, 2026
Source recordedAug 29, 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

13 selected steps

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

13 selected steps

Scroll horizontally to explore the route

Working route overview for Erdős–Mollin–Walsh ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can three consecutive positive integers all be powerful? — Depends on missing premiseCan three consecutivepositive integers all bepowerful?Current reduction — Depends on missing premiseCurrent reductionDeterministic closure criterion — Depends on missing premiseDeterministic closurecriterionExact Pell interface — Depends on missing premiseExact Pell interfaceLargest-prime layer — Depends on missing premiseLargest-prime layerLeast-index repair structure — Depends on missing premiseLeast-index repair structureClosing target — Depends on missing premiseClosing targetProper-cut residue — Depends on missing premiseProper-cut residueParity-only primitive-divisor and ordinary genus arguments — stoppedParity-onlyprimitive-divisor andordinary…Construct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment. — OpenConstruct avaluation-sensitivereciprocity…Prove that deterministic defect closure cannot reach a finite fixed point by forcing a new simple prime or a strictly increasing rank-graph invariant. — OpenProve that deterministicdefect closure cannot reacha…Exclude the independent fundamental branch q=1, D divides v, with powerful fundamental trace u. — OpenExclude the independentfundamental branch q=1, Ddivides…Valuation-sensitive lift reciprocity — OpenValuation-sensitive liftreciprocity
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 routeParity-only primitive-divisor and ordinary genus arguments

The source explicitly warns that ordinary quadratic characters see only parity and thus do not attack the powerful threshold. Valuation-sensitive lift reciprocity, certified nontermination of the repair closure, terminal-system methods, and genuinely coupled class-field or Q-curve invariants remain proposed routes.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Construct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment.Suggested move: For one endpoint atom F_{Pd}, derive a product identity from a cyclotomic logarithmic derivative, resultant, Kummer pairing, or ray-class norm-residue symbol.
Ready to work on
02
Prove that deterministic defect closure cannot reach a finite fixed point by forcing a new simple prime or a strictly increasing rank-graph invariant.Suggested move: Extend the certificate logic into a proof-DAG closure engine that records exact ranks and valuations without treating probable primes or cofactors as certificates.
Ready to work on
03
Valuation-sensitive lift reciprocity

The lead open obligation is a valuation-sensitive reciprocity or product law contradicting the reported nonzero rank moments when every primitive divisor repeats.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Exclude the independent fundamental branch q=1, D divides v, with powerful fundamental trace u.Suggested move: Test real ring-class, continued-fraction, endpoint-kernel, or coupled Q-curve invariants specifically on the k=1 branch without importing nonfundamental assumptions.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 29, 2026
Current statusOpen conjecture

The general conjecture remains open: no proof or counterexample is recorded in the maintained problem list. Known structured special-case exclusions do not resolve the universal assertion that no three consecutive positive integers are all powerful, meaning divisible by p squared for every prime p dividing them.

[1][2]
External progress

What the literature has established

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

  1. Authoritative summaryThe maintained problem list records published exclusions for two structured cube-centered families, not a proof of the general conjecture.[2]
  2. Authoritative summaryMollin and Walsh independently rediscovered the conjecture; Reinhart records their characterization in terms of fundamental units.[1][2]
  3. Authoritative summaryThe source history attributes the no-triple conjecture to Erdős's work on consecutive integers.[1][2]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusErdős–Mollin–Walsh conjecture
Equivalent formulationFundamental-unit characterization

Mollin and Walsh characterized counterexamples through fundamental units of real quadratic fields; Reinhart states the corresponding equivalent obstruction and its unresolved cases.

[1]
Logical consequenceabc conjecture

The abc conjecture would imply that only finitely many triples of consecutive powerful numbers exist. This conditional finiteness statement does not imply that there are none.

[2]

Formalization opportunities

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

  • Formalization targetAn exact machine-checked statement defining powerful positive integers and the universal nonexistence assertion.
  • Formalization targetA checked dependency chain for any proposed general proof or counterexample; no such formal artifact was identified in this bounded collection.

Detailed research inventory

Claims, milestones, and routes in the current map

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

6 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction3 of 83
  • lemma2 of 82
  • equivalence2 of 82
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementCan three consecutive positive integers all be powerful?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact Pell interfaceintermediate
  • retained route statementLeast-index repair structureintermediate
  • retained route statementDeterministic closure criterionintermediate
  • retained route statementProper-cut residueintermediate
  • retained route statementLargest-prime layerintermediate
  • 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
  • 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 failureParity-only primitive-divisor and ordinary genus argumentsreported failure
  • Research targetConstruct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment.open
  • Research targetProve that deterministic defect closure cannot reach a finite fixed point by forcing a new simple prime or a strictly increasing rank-graph invariant.open
  • Research targetExclude the independent fundamental branch q=1, D divides v, with powerful fundamental trace u.open
  • Research targetValuation-sensitive lift reciprocityopen
  • Narrowed routeParity-only primitive-divisor and ordinary genus argumentsThe source explicitly warns that ordinary quadratic characters see only parity and thus do not attack the powerful threshold. Valuation-sensitive lift reciprocity, certified nontermination of the repair closure, terminal-system methods, and genuinely coupled class-field or Q-curve invariants remain proposed routes.
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 bridgeConstruct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment.

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 pointConstruct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment.

Erdős–Mollin–Walsh Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Can three consecutive positive integers all have every prime factor repeated? The current work says no proof is known, but turns any hypothetical example into a tightly constrained Pell-sequence repair problem.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references2 cited works · next context review by Nov 29, 2026

The mathematical context was checked on Aug 29, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A counterexample to the Pellian equation conjecture of Mordellpeer reviewed result · Andreas Reinhart · Acta Arithmetica · 2024-06-12 · DOI 10.4064/aa240214-3-4 · accessed Aug 29, 2026
  2. 2
    Erdős Problem #364maintained problem list · Erdős Problems · accessed Aug 29, 2026

Important qualifications

  • This was a bounded authoritative-source review, not an exhaustive priority or citation search.
  • The maintained problem list reports a computational search bound through OEIS, but no proof certificate or independently reproducible exhaustive artifact was verified in this collection.
  • No exhaustive formalization search was completed; an empty formalizations list does not establish that none exist.
  • No source-package attachment was executed or rendered, and no URL copied from the submitted packet was fetched.
  • Metadata has no proof, review, acceptance, credit, publication, or deployment effect.

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