ProofAtlas

Number theory · equal sums of powers · arithmetic geometry

Prouhet–Tarry–Escott Problem

Collaboration beta

Do ideal equal-sums-of-like-powers identities exist at every size, including the missing size 11?

∀n≥2, ∃A,B∈Multisetn(ℤ), A≠B: ∑a∈Aaj=∑b∈Bbj(1≤j≤n-1)
Known results and sources
A text-free mathematical cover representing Prouhet–Tarry–Escott Problem.
Do ideal equal-sums-of-like-powers identities exist at every size, including the missing size 11?

Research problem

Exact mathematical statement

Two multisets A={a1,…,an}A=\{a_1,…,a_n\} and B={b1,…,bn}B=\{b_1,…,b_n\} form an ideal Prouhet–Tarry–Escott solution when

∑i=1naij=∑i=1nbij(1≤j≤n-1).\sum_{i=1}^n a_i^j=\sum_{i=1}^n b_i^j\qquad(1\le j\le n-1).

The full problem asks whether a nontrivial integral ideal solution exists for every size nn, or exactly which sizes admit one. The source reports size 11 as the first missing case in the standard table.

Problem infographic

Problem at a glance

A text-free scientific explainer showing the objects, constraints, and unresolved route for Prouhet–Tarry–Escott Problem.
The packet centers size 11, which it reports as the first missing case in the standard table. It describes a six-anchor contraction and two secondary arithmetic-geometric routes, but intake did not execute the supplied scripts or certificates, and no rational point or complete split is supplied.

Current mathematical picture

Where work on Prouhet–Tarry–Escott Problem stands

Open problem

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

Useful failureRepresentation-theoretic all-degree existence as written

The current work retains an explicit subordinate counterexample to the load-bearing proposition. Use the six-anchor contraction and exact factor-pattern geometry, with rational verification of every candidate.

Route status · Narrowed route
Main reductionCurrent reduction

The primary symmetric size-11 route reduces construction to complete rational splitting of a quintic determined by six positive anchors.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeFind a rational point where the six-anchor contraction quintic splits completely.Task status · Ready to work on

Work mapped so far

Prouhet–Tarry–Escott Problem in numbers

932retained lines of mathematical investigation932 in the current working snapshot
Argument development
771 · 83%
Explored or eliminated routes
32 · 3%
Computational analysis
52 · 6%
Open obligations
36 · 4%
Definitions and setup
41 · 4%
6selected mapped statements2routes investigated6open questions6contribution-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 checkedAug 28, 2026
Source recordedAug 28, 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 Prouhet–Tarry–Escott ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Equal-size integer multisets should match all power sums up to the ideal degree at every size—or reveal a precise exceptional pattern. — Depends on missing premiseEqual-size integer multisetsshould match all power sumsup…Constant polynomial difference — Depends on missing premiseConstant polynomialdifferenceCurrent reduction — Depends on missing premiseCurrent reductionIdeal power sums — Depends on missing premiseIdeal power sumsClosing target — Depends on missing premiseClosing targetSix-anchor contraction and near-miss — Depends on missing premiseSix-anchor contraction andnear-missRepresentation-theoretic all-degree existence as written — stoppedRepresentation-theoreticall-degree existence aswrittenUnretained large-resultant route — stoppedUnretained large-resultantrouteFind a rational point where the six-anchor contraction quintic splits completely. — OpenFind a rational point wherethe six-anchor contractionquintic…Close the fixed-quintic rational split fiber. — OpenClose the fixed-quinticrational split fiber.Address nonsymmetric size 11 and the full all-degree mechanism. — OpenAddress nonsymmetric size 11and the full all-degreemechanism.Size 11 target — OpenSize 11 targetSecondary square routes — OpenSecondary square routesNonsymmetric size 11 — OpenNonsymmetric size 11
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-theoretic all-degree existence as written

The current work retains an explicit subordinate counterexample to the load-bearing proposition. Use the six-anchor contraction and exact factor-pattern geometry, with rational verification of every candidate.

Route status · Narrowed route
Narrowed routeUnretained large-resultant route

The current work demotes them to legacy until their defining artifacts are regenerated. Regenerate exact artifacts before promoting any legacy computation.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks
01
Find a rational point where the six-anchor contraction quintic splits completely.Suggested move: Parameterize the near-miss factorization stratum of the six-anchor contraction and search for a rational complete split.
Ready to work on
02
Close the fixed-quintic rational split fiber.Suggested move: Perform a Prym descent or exploit the denominator-prime constraints on the fixed-quintic collision curve.
Ready to work on
03
Size 11 target

The immediate target is a disjoint integral size-11 degree-10 pair.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Secondary square routes

The source reports an open fixed-quintic second-fiber route and an open nine-plus-two route with three equations and a square-discriminant condition.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Nonsymmetric size 11

Failure of the symmetric branch would leave the general nonsymmetric problem open.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
06
Address nonsymmetric size 11 and the full all-degree mechanism.Suggested move: Develop the nonsymmetric size-11 branch or a valid all-degree propagation theorem.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 28, 2026
Current statusOpen problem

Ideal integer solutions are known at selected sizes and have been searched extensively, but the cited sources do not give a construction for every size or an exact characterization of exceptional sizes.

[1][2]
External progress

What the literature has established

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

  1. PreprintCoppersmith, Mossinghoff, Scheinerman, and VanderKam extended searches for ideal solutions through larger sizes without a universal construction.[2]
  2. Peer reviewedA peer-reviewed review records Wright's all-sizes ideal-solution conjecture as open and surveys known ideal and non-ideal constructions.[1]
2 cited sources0 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Formalization opportunities

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

  • Formalization targetA formalization needs disjoint integer multisets, equality of every power sum through degree n-1, nontriviality modulo affine equivalence, and quantification over every size.
  • Formalization targetSearch bounds and isolated ideal solutions require exact independently reproduced certificates before being used as computation evidence.

Detailed research inventory

Claims, milestones, and routes in the current map

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

4 standing statements2 proposed statements6 open questions2 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction1 of 61
  • lemma2 of 62
  • equivalence2 of 62
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included
  • retained route statementEqual-size integer multisets should match all power sums up to the ideal degree at every size—or reveal a precise exceptional pattern.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementIdeal power sumsintermediate
  • retained route statementConstant polynomial differenceintermediate
  • retained route statementSix-anchor contraction and near-missintermediate
  • 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
  • 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-theoretic all-degree existence as writtenreported failure
  • Useful failureUnretained large-resultant routereported failure
  • Research targetFind a rational point where the six-anchor contraction quintic splits completely.open
  • Research targetClose the fixed-quintic rational split fiber.open
  • Research targetAddress nonsymmetric size 11 and the full all-degree mechanism.open
  • Research targetSize 11 targetopen
  • Research targetSecondary square routesopen
  • Research targetNonsymmetric size 11open
  • ComputationTwo independent exact certificate families reportedly verify the m=6 Hurwitz identities and the finite-field forcing data.The exact finite identities constrain the route but do not provide a rational point on the complete-splitting locus. Intake did not execute the scripts. · reported unreproduced
  • Narrowed routeRepresentation-theoretic all-degree existence as writtenThe current work retains an explicit subordinate counterexample to the load-bearing proposition. Use the six-anchor contraction and exact factor-pattern geometry, with rational verification of every candidate.
  • Narrowed routeUnretained large-resultant routeThe current work demotes them to legacy until their defining artifacts are regenerated. Regenerate exact artifacts before promoting any legacy computation.
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 rational point where the six-anchor contraction quintic splits completely.

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

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointFind a rational point where the six-anchor contraction quintic splits completely.

Prouhet–Tarry–Escott Problem · 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

Do ideal equal-sums-of-like-powers identities exist at every size, including the missing size 11?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
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A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

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Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Nov 28, 2026

The mathematical context was checked on Aug 28, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The Prouhet Tarry Escott Problem: A Reviewsurvey or monograph · V. N. Srinivasa Rao, S. V. Bhagavan · Mathematics · 2019 · DOI 10.3390/math7030227 · accessed Aug 28, 2026
  2. 2
    Ideal Solutions in the Prouhet-Tarry-Escott Problempreprint · Don Coppersmith, Michael J. Mossinghoff, Danny Scheinerman, Jeffrey M. VanderKam · arXiv · 2023-04-21 · ARXIV 2304.11254 · accessed Aug 28, 2026

Important qualifications

  • This was a bounded primary-source and publisher-record search, not an exhaustive literature, priority, citation, rights, or authorship review.
  • Open status means that the cited source states or studies the problem as a conjecture or open problem and the bounded search found no statement-aligned primary resolution; it does not prove that no later claim exists.
  • Recent preprints are recorded only with their stated preprint posture and are not treated as peer-reviewed or independently verified.
  • No submitted attachment, submitted URL, packet-reported computation, or model output was treated as independent external authority.
  • No statement-aligned formalization, certificate, or independently reproduced computation was established by this search.

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