ProofAtlas

Diophantine equations and exponential arithmetic

Beal Conjecture

Collaboration beta

Can pairwise-coprime positive bases satisfy A^x+B^y=C^z when all three integer exponents exceed two?

Ax+By=Cz, x,y,z>2⇒gcd(A,B,C)>1
Beal Prize
Known results and sources
Three perfect-power towers illustrate A to x plus B to y equals C to z, with integer exponents greater than two and a question about a prime factor shared by the positive bases.
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

For positive integers A, B, C and integers x, y, z greater than two, does

Ax+By=CzA^x+B^y=C^z

always imply

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

The universal question remains open. In an equation solution, a prime dividing two bases also divides the third, so a hypothetical counterexample has pairwise-coprime bases. The full question includes bases equal to one; their later exclusion uses the stated external Catalan–Mihăilescu theorem.

Reducing each exponent to four or an odd prime leaves infinitely many signatures and must preserve the original placements and signs. The source reports closure of every placement with two original exponents divisible by four and of the reducible prescribed septic cases for the positive primitive ordered (3,5,7) signature. Irreducibility, however, does not exclude the surviving solvable or nonsolvable sectors.

For the solvable sector, the source reports six locally viable fields and abstract genus-three isogenies. The explicit quartic identification with its primitive parameter, arithmetic rank bounds, and complete point argument remain separate tasks. Stronger inherited finite-logarithm thresholds also remain conditional on missing exceptional-case work. None of these sector results proves the full conjecture.

Problem infographic

Problem at a glance

The equation A^x+B^y=C^z has positive integer bases A, B, C, including 1, and integer exponents x, y, z greater than 2. Beal predicts a prime dividing all three bases. A counterexample would satisfy the equation with gcd(A,B,C)=1, which also gives pairwise coprimality for an equation solution. The full conjecture remains unproved.
Beal asks whether every perfect-power solution with exponents above two shares a prime factor among its bases. Reduction to four and odd primes leaves infinitely many signatures.

Current mathematical picture

Where work on Beal Conjecture stands

Open conjecture

The full Beal conjecture remains open. The new source downgrades inherited finite-logarithm thresholds to conditional claims, reports the two-fourth-power and prescribed reducible-septic exclusions, and separates the abstract Fano isogenies from the missing explicit arithmetic model. Global labeled relations and uniform surviving exponent families remain unresolved.

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 retains finite-logarithm determinant identities on admissible invertible nodes, while uniform simultaneous-minor nonvanishing and complete exceptional-case verification remain open. Conditional on the correctly scoped two-minor target, the claimed consequence is t>=N^2-2N+4; s>=4 is used only at h=N-2.

Evidence posture · Source-reported route statement · dependencies incomplete
Completed special caseNecessary irreducibility of the prescribed septic

For a positive primitive ordered solution a^3+b^5=c^7, the source reports that the prescribed septic is irreducible: all 3+4,2+5 and 1+6 factor partitions are excluded. The exact point/rank and formal-log arguments are attachment-dependent source claims. Irreducibility is not a contradiction and does not settle this signature.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeUniform primitive contradiction for all exponent families

Find a uniform primitive contradiction across all surviving exponent triples greater than two.

Task status · Ready to work on
Latest mathematical updateThe Beal revision clarifies source-reported reductions, conditional finite-logarithm bounds, the missing Fano model map, and separate open arithmetic and exponent-family questions.

The Beal revision clarifies source-reported reductions, conditional finite-logarithm bounds, the missing Fano model map, and separate open arithmetic and exponent-family questions.

Retained source record

Work mapped so far

Beal Conjecture in numbers

4.8kretained lines of mathematical investigation1,190 in the current working snapshot
Argument development
4,077 · 85%
Explored or eliminated routes
115 · 2%
Computational analysis
134 · 3%
Open obligations
191 · 4%
Definitions and setup
297 · 6%
12selected mapped statements2routes investigated20open questions20contribution-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 recordedSep 8, 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

22 selected steps

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

22 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 primitive-family map — Depends on missing premiseCurrent primitive-family mapMissing explicit model and arithmetic transport — Depends on missing premiseMissing explicit model andarithmetic transportPrimitive signature reduction — Depends on missing premisePrimitive signaturereductionSix locally viable solvable fields — Depends on missing premiseSix locally viable solvablefieldsAbstract Fano kernel and middle variety — Depends on missing premiseAbstract Fano kernel andmiddle varietyClosing target — Depends on missing premiseClosing targetConditional finite-logarithm thresholds — Depends on missing premiseConditional finite-logarithmthresholdsExact Beal statement — Depends on missing premiseExact Beal statementNecessary irreducibility of the prescribed septic — Depends on missing premiseNecessary irreducibility ofthe prescribed septicSource-reported closure of two fourth-divisible exponents — Depends on missing premiseSource-reported closure oftwo fourth-divisibleexponentsRepresentation-dimension contradiction for maximal Kummer images — stoppedRepresentation-dimensioncontradiction for maximalKummer…Blind extension of finite-logarithm quotient strips — stoppedBlind extension offinite-logarithm quotientstripsFANO-QUARTIC-ID-v34 — OpenFANO-QUARTIC-ID-v34Complete the arithmetic degree-eight descent after the model bridge. — OpenComplete the arithmeticdegree-eight descent afterthe…Uniform primitive contradiction for all exponent families — OpenUniform primitivecontradiction for allexponent…Exclude synchronized silent and hidden-power blocks — OpenExclude synchronized silentand hidden-power blocksSelected-prime circular-unit global index — OpenSelected-prime circular-unitglobal indexOff-diagonal or wild-symbol escape — OpenOff-diagonal or wild-symbolescapePrime-N finite-logarithm node capacity — OpenPrime-N finite-logarithmnode capacityRepeated-odd valuation and class/unit gap — OpenRepeated-odd valuation andclass/unit gap
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

Stronger finite-logarithm strip thresholds remain conditional because complete exceptional-case programs are missing; the entry audit and two recovered first-minor zeros do not restore the lost full scan. The current source calls for exact reconstruction of the original remainder and the correctly scoped uniform two-minor theorem.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

20 featured tasks
01
FANO-QUARTIC-ID-v34

Construct the selected-orbit resolvent, three adjoints, explicit nonconstant quartic map and rational t-function identifying the model with the Fano quotient.

Suggested move: Verify the exact model identity and primitive-parameter transport before transferring Q4 arithmetic.
Ready to work on
02
Uniform finite-logarithm two-minor theorem

Prove the uniform two-minor target over its complete admissible domain, with the corrected boundary consequences for t.

Suggested move: Prevent simultaneous consecutive-minor zeros using a complete structural argument and exceptional-node accounting.
Ready to work on
03
Uniform primitive contradiction for all exponent families

Find a uniform primitive contradiction across all surviving exponent triples greater than two.

Suggested move: Preserve the original primitive generators in mixed one-quartic, qualified repeated-prime and distinct all-odd families.
Ready to work on
04
Arithmetic full image and incidence-norm application

Prove the arithmetic full image needed for the incidence-norm application, using the model bridge or a direct argument on the abstract curve.

Suggested move: Distinguish the generic finite-module implication from the actual arithmetic representation and its required field data.
Ready to work on
05
Complete points and primitive t-filter

Complete the local/global point argument and apply the primitive t-filter after a verified rank upper bound.

Suggested move: Retain every rational-prime valuation condition and prove the point enumeration or exclusion complete in the actual solvable quotient.
Ready to work on
06
Compatible genus-three rank upper bound

After transporting the rank lower bound two, prove a compatible genus-three rank upper bound using both degree-eight descents.

Suggested move: Compute the two Selmer dimensions with the actual fields, units, class terms and local images; seek sum two after the model/rank bridge.
Ready to work on
07
Primitive points on the solvable Fano quotient

Prove that no point on C_F over D=Q(sqrt(-7)) has rational t in (0,1) satisfying all 3,5,7 primitive valuation conditions.

Suggested move: Use the abstract quotient and its isogeny to the inherited threefold while retaining the all-prime t-filter.
Ready to work on
08
Bad primes, units, 2-class obstruction and local images

Compute the actual bad primes, supported square classes, unit contribution, 2-class obstruction and local images required by the degree-eight descent.

Suggested move: Construct the exact arithmetic data after the required field/image inputs, retaining the class term rather than treating ideal powers as element powers.
Ready to work on
09
Degree-seven torsion point field and dual/flag embeddings

Construct the defining degree-seven torsion point field over D and the required dual and flag embeddings.

Suggested move: Give exact field/isogeny data and verify the embeddings used by the arithmetic descent.
Ready to work on
10
Complete the arithmetic degree-eight descent after the model bridge.Suggested move: After the explicit map is established, determine the actual point-line and flag fields, bad places, units and 2-class groups, and Selmer local images. Transport the rank lower bound two, prove a compatible upper bound, compute all relevant points and apply the original primitive filter.
Ready to work on
11
Rebuild conditional finite-logarithm exceptional cases

Reconstruct complete exceptional-case programs for the stronger finite-logarithm bounds from the original family and symbolic divisor.

Suggested move: Prove the degree bound and reproduce every effective coefficient, content prime, denominator exception and leading-degree drop; test the entire original remainder.
Ready to work on
12
Prime-N finite-logarithm node capacity

Complete the sharp capacity and full-node accounting for the prime-N finite-logarithm order interface.

Suggested move: Audit every admissible node and exceptional multiplier with N prime and the exact-element/unramified/unit hypotheses retained.
Ready to work on
13
Distinguished septic generator and primitive labels

Exclude the remaining irreducible septic configurations using the distinguished generator and every primitive valuation label, allowing the nonsolvable S7 sector.

Suggested move: Derive an arithmetic incompatibility that survives the source's 49/81 and 3/11 countermodels to label-free reasoning.
Ready to work on
14
Off-diagonal or wild-symbol escape

Find an equation-specific off-diagonal class, independent orbit, restrictive class relation or independently fixed wild symbol escaping the ordinary-ray and same-channel barriers.

Suggested move: Retain the exact twist, selected orbit, target-prime power and wild-symbol normalization when deriving a new incompatible global relation.
Ready to work on
15
Repeated-odd valuation and class/unit gap

Force a primitive cyclotomic divisor whose valuation is not divisible by the other exponent, or eliminate the surviving cyclotomic unit/class alternatives.

Suggested move: Use the original signed factorization and coprimality, separating the ordinary-prime order statement from the missing valuation restriction.
Ready to work on
16
Selected-prime circular-unit global index

Determine the global circular-unit index required by the selected-prime argument without inferring it from local silence.

Suggested move: Bind the relevant class/unit object and prove the global index implication under its exact field and selected-prime hypotheses.
Ready to work on
17
Marked sharp three-point support input

Establish the exact marked, descended, good-reduction three-point specialization input needed for the sharper cover-support set.

Suggested move: Bind the external theorem's complete hypotheses and prove the marked unramified-descent application, retaining the safe finite set until then.
Ready to work on
18
Exclude synchronized silent and hidden-power blocks

Exclude every remaining synchronization of silent or exceptional prime blocks in the exact radical branch while retaining hidden nth-power weights.

Suggested move: Use the actual Frobenius action, target order and weighted archimedean constraints to prove a global incompatibility for the remaining blocks.
Ready to work on
19
Global labeled Kummer relation

Find a nonzero arithmetic relation for the original labeled radical generators that is incompatible with the allowed full Kummer image.

Suggested move: Specify its field, group action, class, local/global map and nonvanishing; retain the primitive generator labels.
Ready to work on
20
Uniform equation-specific relation for the surviving radical frameworks

Find a uniform equation-specific primitive contradiction in the surviving radical and mixed-exponent frameworks. A proposed labeled arithmetic relation must be incompatible with the allowed full Kummer image; fixed-signature finiteness or full image alone does not close the full target.

Suggested move: Specify the relation's field, group action, class, local/global map and nonvanishing. Retain the original additive-grid labels, radical parameters and primitivity; a norm, standard Steinberg or another universal multiplicative identity alone is insufficient.
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.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

The Beal revision clarifies source-reported reductions, conditional finite-logarithm bounds, the missing Fano model map, and separate open arithmetic and exponent-family questions.The Beal revision clarifies source-reported reductions, conditional finite-logarithm bounds, the missing Fano model map, and separate open arithmetic and exponent-family questions.

Changed the research frontierLater mathematical revision

Source revision · date not established

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

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.

8 standing statements4 proposed statements20 open questions2 narrowed routes2 conditional results3 completed special cases
Statements by mathematical role12 selected mapped statements
  • lemma5 of 125
  • reduction5 of 125
  • special case1 of 121
  • theorem candidate1 of 121
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.32 displayed rows · 2 routes included
  • retained route statementAbstract Fano kernel and middle varietyintermediate
  • retained route statementConsecutive-minor reformulationintermediate
  • retained route statementExact Beal statementintermediate
  • retained route statementPrimitive signature reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementCan pairwise-coprime bases solve A^x+B^y=C^z when every exponent exceeds two?
  • retained route statementMissing explicit model and arithmetic transportconditional
  • retained route statementConditional finite-logarithm thresholdsconditional
  • retained route statementCurrent primitive-family mapintermediate
  • retained route statementNecessary irreducibility of the prescribed septicspecial case
  • retained route statementSix locally viable solvable fieldsspecial case
  • retained route statementSource-reported closure of two fourth-divisible exponentsspecial case
  • 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
  • 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 relationshipThe current family reduction uses the source-reported closure of placements with two fourth-divisible exponents. This leaves the expressly retained exponent families.supports · reported by source
  • Recorded relationshipNecessary irreducibility is one stated component of the current positive primitive (3,5,7) route. Its solvable and nonsolvable sectors remain unresolved, as does the wider exponent union.supports · reported by source
  • Recorded relationshipThe six-field classification is stated for the remaining solvable irreducible prescribed septic sector. Neither irreducibility nor local realizability supplies a contradiction.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 conditional finite-logarithm interface is one source-reported route component. Its missing exceptional-case verification is an unresolved premise and does not establish the full conjecture.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Useful failureBlind extension of finite-logarithm quotient stripsreported failure
  • Useful failureRepresentation-dimension contradiction for maximal Kummer imagesreported failure
  • Research targetComplete the arithmetic degree-eight descent after the model bridge.open
  • Research targetFANO-QUARTIC-ID-v34open
  • Research targetUniform finite-logarithm two-minor theoremopen
  • Research targetRebuild conditional finite-logarithm exceptional casesopen
  • Research targetGlobal labeled Kummer relationopen
  • Research targetUniform equation-specific relation for the surviving radical frameworksopen
  • Research targetUniform primitive contradiction for all exponent familiesopen
  • 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 stripsStronger finite-logarithm strip thresholds remain conditional because complete exceptional-case programs are missing; the entry audit and two recovered first-minor zeros do not restore the lost full scan. The current source calls for exact reconstruction of the original remainder and the correctly scoped uniform two-minor 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 bridgeFind a uniform primitive contradiction across all surviving exponent triples greater than two.

Constructing the Fano-to-quartic map and its primitive parameter is a concrete task within the solvable ordered (3,5,7) sector. The nonsolvable sector, other exponent families and conditional finite-logarithm bounds remain separate challenges.

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.

  • Do not replace the full conjecture by ordered (3,5,7).
  • A fixed-signature finiteness result or full residual image alone does not complete the task.

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Prepared starting pointFANO-QUARTIC-ID-v34

Beal Conjecture · ready to start

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Research contextPrepared context for any AI agent

Can pairwise-coprime positive bases satisfy A^x+B^y=C^z when all three integer 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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