ProofAtlas

Diophantine equations and exponential arithmetic

Beal Conjecture

Collaboration beta

Can three pairwise-coprime positive integers satisfy a sum of perfect powers when all three exponents exceed two?

Ax+By=Cz, x,y,z>2⇒gcd(A,B,C)>1
Beal Prize
Known results and sources
Three towers of perfect powers balance in the equation A to the x plus B to the y equals C to the z, while the three bases are tested for a common prime factor.
Beal asks whether every positive-integer perfect-power solution with exponents above two must share a prime factor among its bases.

Research problem

Exact mathematical statement

Let A,B,C,x,y,zA,B,C,x,y,z be positive integers satisfying

Ax+By=Cz,x,y,z>2.A^x+B^y=C^z, \qquad x,y,z>2.

The Beal Conjecture asserts that

gcd(A,B,C)>1.\gcd(A,B,C)>1.

Equivalently, there is no solution with A,B,CA,B,C pairwise coprime.

Problem infographic

Problem at a glance

A problem-first diagram showing positive integer bases and exponents in A to x plus B to y equals C to z, primitive factor wheels, and the open common-prime-factor question.
A hypothetical counterexample would have pairwise-coprime bases; the universal question remains open despite reductions of the exponent signatures.

Current mathematical picture

Where work on Beal Conjecture stands

Open conjecture

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

Useful failureRepresentation-dimension contradiction for maximal Kummer images

The source explicitly states that maximal Kummer image is not contradictory by representation dimension alone. A global arithmetic relation specific to the binomial Kummer generators, using local fingerprints and Frobenius data, remains viable.

Route status · Narrowed route
Main reductionConsecutive-minor reformulation

The source reports that r consecutive minor vanishings are equivalent to interpolation divisibility and r forward-difference equations.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve the uniform two-minor nonvanishing target for every admissible N and h.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

Beal Conjecture in numbers

3.6kretained lines of mathematical investigation3,628 in the current working snapshot
Argument development
3,256 · 90%
Explored or eliminated routes
65 · 2%
Computational analysis
43 · 1%
Open obligations
85 · 2%
Definitions and setup
179 · 5%
8selected mapped statements2routes 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 byAndrew BealStatus checkedAug 15, 2026
Source recordedAug 15, 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Beal 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 pairwise-coprime bases solve A^x+B^y=C^z when every exponent exceeds two? — Depends on missing premiseCan pairwise-coprime basessolve A^x+B^y=C^z when everyexponent…Consecutive-minor reformulation — Depends on missing premiseConsecutive-minorreformulationCurrent reduction — Depends on missing premiseCurrent reductionPrimitive signature reduction — Depends on missing premisePrimitive signaturereductionClosing target — Depends on missing premiseClosing targetExact Beal statement — Depends on missing premiseExact Beal statementPacket-reproduced finite-log bound — Depends on missing premisePacket-reproduced finite-logboundReported strip elimination — Depends on missing premiseReported strip eliminationRepresentation-dimension contradiction for maximal Kummer images — stoppedRepresentation-dimensioncontradiction for maximalKummer…Blind extension of finite-logarithm quotient strips — stoppedBlind extension offinite-logarithm quotientstripsProve the uniform two-minor nonvanishing target for every admissible N and h. — OpenProve the uniform two-minornonvanishing target forevery…Independently regenerate every source-reported strip certificate for m=2,...,17, especially the missing raw ledger for m=13,...,17. — OpenIndependently regenerateevery source-reported stripcertificate…Find a global arithmetic relation that closes the surviving one- or two-radical and mixed-exponent Kummer branches. — OpenFind a global arithmeticrelation that closes thesurviving…Global Kummer relation — OpenGlobal Kummer relation
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

2 recorded
Narrowed routeRepresentation-dimension contradiction for maximal Kummer images

The source explicitly states that maximal Kummer image is not contradictory by representation dimension alone. A global arithmetic relation specific to the binomial Kummer generators, using local fingerprints and Frobenius data, remains viable.

Route status · Narrowed route
Narrowed routeBlind extension of finite-logarithm quotient strips

The source directs future work not to resume with blind strip extension and instead lists structural routes for consecutive minors. Hypergeometric recurrences, paired reflection, subresultants, determinant identities, and an exceptional-zero classification remain source-supported structural directions.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove the uniform two-minor nonvanishing target for every admissible N and h.Suggested move: Derive a recurrence or determinant identity that prevents simultaneous consecutive minor zeros, starting with the exact consecutive-difference formulation.
Ready to work on
02
Independently regenerate every source-reported strip certificate for m=2,...,17, especially the missing raw ledger for m=13,...,17.Suggested move: Write a complete exact generator, retain the per-position log and certificates, and replay them independently before citing the stronger frontier.
Ready to work on
03
Global Kummer relation

A global arithmetic relation for the special binomial Kummer generators remains the full-conjecture bottleneck.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Find a global arithmetic relation that closes the surviving one- or two-radical and mixed-exponent Kummer branches.Suggested move: Combine local target-prime constraints, selected-prime harmonic fingerprints, and Frobenius collisions without inferring a contradiction from representation dimension alone.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The official AMS records continue to administer and list the Beal Prize, with award procedures requiring a refereed proof or counterexample. No qualifying resolution is identified in the official sources collected here.

[2][3]
External progress

What the literature has established

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

  1. Authoritative summaryAMS Council records describe the million-dollar prize procedures and the exact prize conjecture.[2]
  2. Historical sourceThe AMS Notices article published the Beal Conjecture and its original prize problem.[1]
3 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBeal Conjecture
Related problemFermat's Last Theorem

The original AMS article presents Beal as a generalization in the setting of unequal exponents.

[1]

Formalization opportunities

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

  • Formalization targetA formal library covering the primitive exponent-signature reductions and every external generalized-Fermat input used by a proposed proof.
  • Formalization targetIndependent exact replay of any computational certificate before it can support a formal theorem.

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

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 statements3 proposed statements4 open questions2 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction3 of 83
  • lemma3 of 83
  • computational claim1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included
  • retained route statementCan pairwise-coprime bases solve A^x+B^y=C^z when every exponent exceeds two?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact Beal statementintermediate
  • retained route statementPrimitive signature reductionintermediate
  • retained route statementPacket-reproduced finite-log boundintermediate
  • retained route statementReported strip eliminationintermediate
  • retained route statementConsecutive-minor reformulationintermediate
  • 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 failureRepresentation-dimension contradiction for maximal Kummer imagesreported failure
  • Useful failureBlind extension of finite-logarithm quotient stripsreported failure
  • Research targetProve the uniform two-minor nonvanishing target for every admissible N and h.open
  • Research targetIndependently regenerate every source-reported strip certificate for m=2,...,17, especially the missing raw ledger for m=13,...,17.open
  • Research targetFind a global arithmetic relation that closes the surviving one- or two-radical and mixed-exponent Kummer branches.open
  • Research targetGlobal Kummer relationopen
  • ComputationThe source reports exact elimination of quotient strips m=2,...,17, but labels the resulting bounds [PC] because a complete generator and the raw certificate ledger for m=13,...,17 were not retained. This computation was not independently reproduced for this intake.Reported-unreproduced only: conditional on independent regeneration, the source's candidate frontier is h >= (N+37)/2 with the displayed G_17^sharp(N) lower bound for t. · reported unreproduced
  • Narrowed routeRepresentation-dimension contradiction for maximal Kummer imagesThe source explicitly states that maximal Kummer image is not contradictory by representation dimension alone. A global arithmetic relation specific to the binomial Kummer generators, using local fingerprints and Frobenius data, remains viable.
  • Narrowed routeBlind extension of finite-logarithm quotient stripsThe source directs future work not to resume with blind strip extension and instead lists structural routes for consecutive minors. Hypergeometric recurrences, paired reflection, subresultants, determinant identities, and an exceptional-zero classification remain source-supported structural directions.
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 bridgeProve the uniform two-minor nonvanishing target for every admissible N and h.

2 approaches have 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 pointProve the uniform two-minor nonvanishing target for every admissible N and h.

Beal Conjecture · ready to start

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Research contextPrepared context for any AI agent

Can three pairwise-coprime positive integers satisfy a sum of perfect powers when all three exponents exceed two?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problemoriginal source · R. Daniel Mauldin · American Mathematical Society · 1997-12 · accessed Aug 15, 2026
  2. 2
    Beal Prize procedures in the January 2014 AMS Council minutesauthoritative webpage · American Mathematical Society · 2014-01 · accessed Aug 15, 2026
  3. 3
    Report of the Treasurerauthoritative webpage · American Mathematical Society · 2023-09 · accessed Aug 15, 2026

Important qualifications

  • Current status and recognition are based only on official American Mathematical Society records.
  • No submitted-packet URL was fetched and no source-package attachment was executed or rendered.
  • Prize availability is contextual recognition, not mathematical evidence of openness or resolution by itself.

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