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

- ID: `P2526`
- Reference: `ideal-pte-size-11`
- Page: https://theoremdb.org/statements/P2526
- Record maturity: Reviewed problem with recorded work

## 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 \(1\le k\le10\)?

### Remarks

- **Remark.** Equal power sums through degree ten make this an ideal Prouhet-Tarry-Escott solution.
- **Remark.** A common translation, a common nonzero scaling, or exchanging the two sets preserves the equations.

### What 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.

## Status

The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500. [1](#reference-1) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- 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.

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

### Runnable artifacts

- **Artifact 1** (reproduced): Exact integer arithmetic verifies eleven equal moments, distinct disjoint sets, centering, primitive gcd one, and the first unequal moment. [1](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `ideal-pte-size-11`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. The paper reports extensive searches for ideal solutions and identifies size eleven as an unresolved integer case. https://arxiv.org/abs/2304.11254
   - 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
   - preprint; reference source; arXiv:2304.11254, checked 2026-08-01; checked 2026-07-24
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Srikanth Raghavendran and Veena Narayanan, “The Prouhet Tarry Escott Problem: A Review,” Mathematics 7(3) (2019), article 227. DOI 10.3390/math7030227. sections and tables on known ideal integer Prouhet-Tarry-Escott solutions and the open size-11 case https://doi.org/10.3390/math7030227
   - journal_article; secondary source; checked 2026-08-01
   - Source use: original_summary
   - 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.
3. <a id="reference-3"></a>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 https://www.daemonology.net/papers/pte.pdf
   - Also cited at Mathematics of Computation 72 (2003), Table 2 and Section 2.6, page 2069
   - website; reference source; checked 2026-07-24
   - Source 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.