[#P2526] An ideal Prouhet-Tarry-Escott solution of size eleven

Problem. Do there exist disjoint sets \(A,B\subset\mathbb Z\), each of size \(11\), such that \(\sum_{a\in A}a^k=\sum_{b\in B}b^k\) for every \(1\le k\le10\)?

An illustration accompanying the statement of An ideal Prouhet-Tarry-Escott solution of size eleven. — An illustration accompanying the statement. The statement alone defines the problem.

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1Context

The problem is an explicit Diophantine construction with standard affine normalizations. Bounded-height sweeps and modular exclusions retain their value even when they miss a solution.

2Remarks

Remark 1. Equal power sums through degree ten make this an ideal Prouhet-Tarry-Escott solution.

Remark 2. A common translation, a common nonzero scaling, or exchanging the two sets preserves the equations.

3What counts as a solution

Supply two disjoint 11-element integer sets and verify all ten power-sum equalities exactly, or prove that no such pair exists.

1Status

Saved packet · July 24, 2026

What counts as a solution

Saved packet status (Integral ideal solutions of size eleven remain unknown). The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500.^[1][3]

1Packet records

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Record	Kind	Assessment
The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500.^[1][3] Relevance to this problem For An ideal Prouhet-Tarry-Escott solution of size eleven, record ipte11-claim-currently-open (“Integral ideal solutions of size eleven remain unknown”) records a bound, answer, status fact, or structural consequence. The record states: The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope the published existence status of an integral ideal Prouhet-Tarry-Escott solution of size 11, checked on 2026-07-24 Sources Reference 1, Reference 3 Details No pair of eleven-element integer sets is supplied in the candidate record. Its displayed pair has twelve elements on each side and serves as a neighboring known case. Coppersmith, Mossinghoff, Scheinerman, and VanderKam state that integral ideal PTE solutions are known for sizes at most ten and for size twelve. Their 2024 paper reports no new integral solution at sizes nine through sixteen. For odd size eleven, symmetry means \(B=-A\). Section 4.1 and Table 2 give an exhaustive search for primitive symmetric ideal solutions of height at most \(3500\), where height is the largest absolute coordinate. The table records zero solutions. Their search extends the height-2000 computation of Borwein, Lisoněk, and Percival. This is a bounded result inside the symmetric family. It leaves asymmetric solutions and symmetric solutions of greater height untouched. Common translation and nonzero scaling preserve the equations, so a future witness should state its affine normalization. For a symmetric witness, a centered representative is automatic; dividing by the common gcd gives a primitive representative. The literature search checked the 2003 search paper, the 2024 Mathematics of Computation paper and its tables, and later PTE work available through July 24, 2026. It found no claimed integer witness of size eleven and no proof of general nonexistence. The mathematical status is open. Open full record →
Exact integer arithmetic verifies eleven equal moments, distinct disjoint sets, centering, primitive gcd one, and the first unequal moment.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Exact integer arithmetic verifies eleven equal moments, distinct disjoint sets, centering, primitive gcd one, and the first unequal moment.^[1] Relevance to this problem For An ideal Prouhet-Tarry-Escott solution of size eleven, record ipte11-artifact-size12-control-check (“The supplied size-twelve control witness is valid and primitive”) supplies evidence or a replay used to check the packet. The record states: Exact integer arithmetic verifies eleven equal moments, distinct disjoint sets, centering, primitive gcd one, and the first unequal moment. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the size-twelve symmetric ideal PTE witness displayed in the candidate record, including moments 1 through 12 and affine-normalization checks Sources Reference 1 Run `python3 check.py` Runtime CPython 3 standard library Details The candidate's control pair is \[ A=\{\pm22,\pm61,\pm86,\pm127,\pm140,\pm151\}, \] \[ B=\{\pm35,\pm47,\pm94,\pm121,\pm146,\pm148\}. \] The replay checks that both sides contain twelve distinct integers and are disjoint. Their sums vanish, and the gcd of all twenty-four coordinates is one, so this representative is centered and primitive. Direct summation gives equal powers for \(1\leq k\leq11\). At \(k=12\), the difference \(\sum_Aa^{12}-\sum_Bb^{12}\) is \[ 809283534716112648192000. \] The two monic root polynomials differ by the nonzero constant \[ -67440294559676054016000. \] This certifies a nontrivial ideal solution of size twelve. It supplies a control for the definitions and gives no size-eleven witness. The four-line output has SHA-256 digest `0617ec2157d54c020ce602649b765768866531635ca4c54ea3484e0ebebe2fb6`. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. The 2023 search literature reports ideal integer solutions for sizes through ten and for size twelve, with size eleven still unresolved.

Newton identities imply that the monic polynomials \(\prod_{a\in A}(x-a)\) and \(\prod_{b\in B}(x-b)\) differ by a nonzero constant. Record factorization ranges, congruence filters, and the normalization used in every search.
The symmetric restriction \(B=-A\) turns the odd power sums into the main equations. Searches confined to this attractive family cannot exclude an asymmetric solution.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. A size-twelve ideal solution is \(A=\{\pm22,\pm61,\pm86,\pm127,\pm140,\pm151\}\) and \(B=\{\pm35,\pm47,\pm94,\pm121,\pm146,\pm148\}\).

Computational notes

Independent integer arithmetic verified that the displayed size-twelve sets have equal power sums for every exponent from 1 through 11. At exponent 12, the first sum minus the second is 809283534716112648192000, confirming the expected first unequal moment.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemAn ideal Prouhet-Tarry-Escott solution of size eleven

All 1 recorded relations between these records and the problem

The supplied size-twelve control witness is valid and primitive informs Integral ideal solutions of size eleven remain unknown

2See also

A bounded three-cubes search for 114diophantine equations
Prime exceptions to connectivity of the Markoff graphdiophantine equations
Square-class collisions in the Pell-Lucas sequencediophantine equations

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“An ideal Prouhet-Tarry-Escott solution of size eleven.” TheoremDB. P2526. Problem statement; statement text SHA-256 fde5950caf94e528848d360fbcf7c8570e2d2ab8046a19596e5ffeb7c298755a. https://theoremdb.org/statement/?ref=P2526

BibTeX

@misc{theoremdb-problem-fde5950caf94e528848d360fbcf7c8570e2d2ab8046a19596e5ffeb7c298755a,
  title = {{An ideal Prouhet-Tarry-Escott solution of size eleven}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 fde5950caf94e528848d360fbcf7c8570e2d2ab8046a19596e5ffeb7c298755a},
  url = {https://theoremdb.org/statement/?ref=P2526}
}

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This problem includes 2 records joined by 1 typed links, sourced from doi.org^[1], current as of July 24, 2026.

1Lean verification

1References

Packet source. Don Coppersmith, Michael Mossinghoff, Danny Scheinerman, and Jeffrey VanderKam, “Ideal solutions in the Prouhet–Tarry–Escott problem”. Mathematics of Computation 93(349) (2023), 2473-2501. DOI 10.1090/mcom/3917. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:2304.11254, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at The paper reports extensive searches for ideal solutions and identifies size eleven as an unresolved integer case.Also cited at Don Coppersmith, Michael J. Mossinghoff, Danny Scheinerman, and Jeffrey M. VanderKam, Ideal solutions in the Prouhet-Tarry-Escott problem, Mathematics of Computation 93 (2024), 2473-2501: Introduction, pages 2474-2475; Section 4.1 size n=11 result, page 2486; Table 2, page 2487. Peter Borwein, Petr Lisoněk, and Colin Percival, Computational investigations of the Prouhet-Tarry-Escott problem, Mathematics of Computation 72 (2003), 2063-2070: Table 2 and size-11 discussion, page 2069.Also cited at Mathematics of Computation 93 (2024), pages 2474-2475 and 2486-2487.Also cited at Coppersmith, Mossinghoff, Scheinerman, and VanderKam, Mathematics of Computation 93 (2024), equation (5) on page 2475; executable reproduction in this record.Source used to formulate or check the problem record.For An ideal Prouhet-Tarry-Escott solution of size eleven: The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500.Source named by the research packet.
Srikanth Raghavendran and Veena Narayanan, “The Prouhet Tarry Escott Problem: A Review,” Mathematics 7(3) (2019), article 227. DOI 10.3390/math7030227. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at sections and tables on known ideal integer Prouhet-Tarry-Escott solutions and the open size-11 case.For An ideal Prouhet-Tarry-Escott solution of size eleven, this source directly records size 11 as missing in the 2019 historical review; later sources are required for current search limits.
Peter Borwein, Petr Lisoněk, and Colin Percival, “Computational investigations of the Prouhet-Tarry-Escott problem,” Mathematics of Computation 72(244) (2003), 2063-2070. DOI 10.1090/S0025-5718-02-01504-1. Table 2 and Section 2.6, especially p. 2069. ↗website · reference source · checked 2026-07-24Source use: citation only.For An ideal Prouhet-Tarry-Escott solution of size eleven, this source reports the computational status and known ideal Prouhet-Tarry-Escott solutions near size eleven.Also cited at Mathematics of Computation 72 (2003), Table 2 and Section 2.6, page 2069.

Canonical missing-size target in the ideal Prouhet-Tarry-Escott problem.

[#P2526] An ideal Prouhet-Tarry-Escott solution of size eleven

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Computational notes

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Background

Definitions

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