Diophantine equations and arithmetic curves

Equal sums of two fifth powers

Collaboration beta

Can two different unordered pairs of positive integers have the same sum of fifth powers? The source excludes nine pair-sum ratios at every height and corrects a proposed stronger claim about rational points, while the unrestricted equation remains unresolved.

\forall a, b, c, d \in ℤ_{> 0} : a^{5} + b^{5} = c^{5} + d^{5} \Rightarrow {{a, b}} = {{c, d}}

Known results and sources

Illustration: Can genuinely different unordered pairs of positive integers have the same sum of fifth powers?

Research problem

Exact mathematical statement

Prove that for all positive integers $a,b,c,d$ ,

a^5+b^5=c^5+d^5\quad\Longrightarrow\quad \{\{a,b\}\}=\{\{c,d\}\}.

The braces denote unordered multisets. Repeated entries within either pair are permitted. Swapping entries or exchanging sides does not create a different representation; signed zero-sum examples are outside the target.

For a hypothetical nontrivial equality, arrange $a+b<c+d$ , set $a+b=mZ$ , $c+d=nZ$ , $\gcd(m,n)=1$ , and $X=b-a,\ Y=d-c$ . The equivalent positive rational-point target is the absence of points on

mX^4-nY^4+2m^3X^2Z^2-2n^3Y^2Z^2+\frac{m^5-n^5}{5}Z^4=0

with $m,n\in\mathbb Z_{>0}$ , $m<n$ , and

Z>0,\qquad |X|<mZ,\qquad |Y|<nZ.

Allow $X=0$ or $Y=0$ . Reconstruct the entries as $(mZ\mp X)/2$ and $(nZ\mp Y)/2$ , check positivity, and clear denominators. Global primitiveness does not imply coprimality within each pair.

Problem infographic

Problem at a glance

Illustration: The positive-integer fifth-power equation is paired with the conjectured unordered-multiset conclusion and examples explaining repeats and swaps. — Illustration: Repeated entries are permitted; swapping a pair is the same representation. The existence of a genuinely different positive pair remains unresolved.

Current mathematical picture

Where work on Equal sums of two fifth powers stands

Open problem

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

Useful failureUniversal emptiness of admissible projective fibers or quotients

The point has Z=0 and cannot reconstruct positive entries. It refutes neither the original target nor any of the nine fixed exclusions. Classify affine or positive lifts, or split parameter families with exhaustive scope and distinct positive-region obstructions.

Route status · Narrowed route

Main reductionA lift needs three globally compatible squares

The source's conic descent requires three quadratic forms at the same (S,T) to share one signed squarefree class d_sf across all places. Resultant support justifies finite class enumeration for a fixed fiber. Separate local norm-conic solutions are weaker; the general Hilbert-symbol rows require nonzero leading coefficient and discriminant, and the u*d abbreviation additionally requires u≠0.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.Task status · Ready to work on

Work mapped so far

Equal sums of two fifth powers in numbers

1.1kretained lines of mathematical investigation1,096 in the current working snapshot

Argument development: 630 · 57%
Explored or eliminated routes: 134 · 12%
Computational analysis: 186 · 17%
Open obligations: 76 · 7%
Definitions and setup: 70 · 6%

9selected mapped statements2routes 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.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.

Suggested move: Suggested next step: Formulate a theorem that tolerates the fourth-power boundary family, prove the exceptional parameter classification is exhaustive, and address the positive region in every resulting class.

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

14 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

2 recorded

Narrowed routeUniversal emptiness of admissible projective fibers or quotients

Route status · Narrowed route

Narrowed routeSearching for a real-only obstruction after the size filter

The intervals of real fifth-power sums for fixed pair sums m and n overlap exactly under the strict size inequality; the construction does not imply rational differences. Finite-place and global rational compatibility, simultaneous squares, and positive-lift arithmetic remain possible obstructions.

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.Suggested move: Suggested next step: Formulate a theorem that tolerates the fourth-power boundary family, prove the exceptional parameter classification is exhaustive, and address the positive region in every resulting class.

Ready to work on

Establish a universal incompatibility for the three simultaneous square conditions at one shared parameter pair and one global squareclass.Suggested move: Suggested next step: Derive squareclass support, character rows, projective-chart coverage, and exceptional valuations symbolically for a stated parameter family; preserve all three equations and positivity rather than separate local norm tests.

Ready to work on

Determine rigorous arithmetic status for the smallest surviving fibers (81,131) and (101,131) without inferring it from starting data.Suggested move: Suggested next step: Construct their exact rational coefficients and bounded conic seeds, then produce a complete all-height obstruction or an exactly verified rational lift with all signs, square conditions, and strict positivity checked.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

The existence of positive integers a, b, c, d with a⁵ + b⁵ = c⁵ + d⁵ and {a,b} ≠ {c,d} remains unresolved in the selected sources. Gaussian-integer examples and finite searches do not settle this domain.

[1][2][3]

External progress

What the literature has established

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

2015-11-21
PreprintCampbell and Zujev constructed fifth-power identities involving Gaussian integers. These do not supply the sought nontrivial positive-integer identity.[2]
1998
Peer reviewedEkl reported computational searches for equal sums of like powers and explicitly left solvability of the [5.2.2] case as a question.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusEqual sums of two fifth powers

Related problemGaussian-integer fifth-power identities

Allowing Gaussian integers changes the domain; identities there do not answer whether two distinct unordered pairs of positive integers have equal fifth-power sums.

[2]

Related problemEqual sums of like powers [k.m.n]

The fifth-power two-against-two problem is one specific exponent and term-count case in Ekl's wider computational study.

[1]

Formal and computational footholds

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

formal statement · statement onlyLean Taxicab(5,2,2) existence statement
Merged PR #2333 contains an open existence statement for positive fifth-power taxicab representations with proof placeholders. It is a formal problem statement, not a solution. Exact alignment of its IsTaxicabFor definition was not established in this collection.
[3]
computation · source linked; not reproduced by ProofAtlasEkl's equal-sums search study
The paper reports finite computational searches. No search bound or global nonexistence conclusion is independently certified in this collection.
[1]

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 questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma4 of 94
equivalence1 of 91
computational claim1 of 91

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 a sum of two positive fifth powers have two different unordered representations?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact positive-region quartic reductionintermediate
retained route statementNecessary reduced-ratio numerator boundintermediate
retained route statementNine fixed parameter pairs excluded at every heightintermediate
retained route statementRational points at infinity classifiedintermediate
retained route statementThe strict size filter settles real feasibilityintermediate
retained route statementA lift needs three globally compatible squaresintermediate
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
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 failureUniversal emptiness of admissible projective fibers or quotientsreported failure
Useful failureSearching for a real-only obstruction after the size filterreported failure
Research targetProve a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.open
Research targetEstablish a universal incompatibility for the three simultaneous square conditions at one shared parameter pair and one global squareclass.open
Research targetDetermine rigorous arithmetic status for the smallest surviving fibers (81,131) and (101,131) without inferring it from starting data.open
Narrowed routeUniversal emptiness of admissible projective fibers or quotientsThe point has Z=0 and cannot reconstruct positive entries. It refutes neither the original target nor any of the nine fixed exclusions. Classify affine or positive lifts, or split parameter families with exhaustive scope and distinct positive-region obstructions.
Narrowed routeSearching for a real-only obstruction after the size filterThe intervals of real fifth-power sums for fixed pair sums m and n overlap exactly under the strict size inequality; the construction does not imply rational differences. Finite-place and global rational compatibility, simultaneous squares, and positive-lift arithmetic remain possible obstructions.

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 a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.

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.

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Prepared starting pointProve a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.

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Sources and references3 cited works · next context review by Dec 7, 2026

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

1
New results in equal sums of like powerspeer reviewed result · Randy L. Ekl · Mathematics of Computation 67(223), 1309–1315 · 1998 · DOI 10.1090/S0025-5718-98-00979-X · accessed Sep 7, 2026
2
Gaussian integer solutions for the fifth power taxicab number problempreprint · Geoffrey B. Campbell, Aleksander Zujev · arXiv · 2015-11-21 · ARXIV 1511.07424 · accessed Sep 7, 2026
3
Taxicab conjecture statements, merged pull request #2333formalization · google-deepmind/formal-conjectures · 2026-03-17 · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.
Direct access to Ekl's AMS PDF failed; the bounded historical claims use search-indexed text from that exact primary publisher URL. No precise search cutoff is asserted.
Repository issue prose was not used as mathematical authority. The formalization entry refers to the actual proposed Lean declarations and their unresolved proof placeholders.

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