# P2650: A bounded three-cubes search for 114

- ID: `P2650`
- Reference: `three-cubes-114-height-1e20`
- Page: https://theoremdb.org/statements/P2650
- Record maturity: Reviewed problem with recorded work

## Problem

Do integers \(x,y,z\) with \(\max(|x|,|y|,|z|)\le10^{20}\) satisfy \(x^3+y^3+z^3=114\)?

### Remarks

- **Remark.** Permutations do not change the height bound.
- **Remark.** All three variables may be negative, zero, or positive.

### What counts as a solution

- Give a solution within the height bound, or provide a complete exact certificate excluding every integer triple in the box.

## Status

No verified representation of 114 is known, and the published computations do not certify the full height box. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Proposition 1 (The height-10^20 question remains open).** No verified representation of 114 is known, and the published computations do not certify the full height box.

Define the height by
\[
H(x,y,z)=\max\{|x|,|y|,|z|\}.
\]
The target asks for an integer triple satisfying
\[
x^3+y^3+z^3=114,\qquad H(x,y,z)\leq10^{20}.
\]
Permutations count as the same representation for search purposes, although the equation itself uses ordered variables.

Booker and Sutherland listed 114 among the seven cases below 1,000 that remained unresolved in 2021. Their September 2019 computation included 114 and covered every solution with
\[
\min\{|x|,|y|,|z|\}\leq10^{17}.
\]
Their later computation reached a larger rectangle in the search parameters, rather than the full height box. Grantham and Walsh's arXiv preprint, submitted on 22 November 2022 as version 1, reports that their substantial effort on \(k=114\) remained unsuccessful. Thus the current source record supplies neither a solution of height at most \(10^{20}\) nor a complete exclusion certificate.

The fully covered prefix leaves a precise unresolved part of the requested box:
\[
10^{17}<\min\{|x|,|y|,|z|\}
\leq H(x,y,z)\leq10^{20}.
\]
Some of this region lies inside the later parameter search recorded separately.

### Background and intake notes

A height cap turns an unbounded equation into a precise range-certification task whose swept intervals carry lasting value.

- Original intake status: Booker and Sutherland reported 114 among the unresolved admissible integers in 2021. Later solution and search-bound status remain unverified, while the height-10^20 box is a separate exact claim.
- Use the factorization x^3+y^3=(x+y)(x^2-xy+y^2) and lattice sieving for each z. Publish disjoint z intervals and residue filters with hashes.
- Trap: a symmetric box loop scales cubically and duplicates permutations. Congruence sieves can exclude candidates only when every allowed residue class is retained.

- Recorded example: Since 114 is congruent to 6 modulo 9, the usual 4-or-5 modulo 9 obstruction does not apply.

### Other known results

- **Computation 1** (reproduced): The 2019 Charity Engine run searched all solutions with minimum absolute coordinate at most 10^17 and reported none for 114. [2](#reference-2)
- **Computation 2** (reproduced): The 2020 Charity Engine run searched the stated (d,z) rectangle for 114 and reported no solution. [2](#reference-2)
- **Computation 3** (reproduced): Cubes modulo nine force each coordinate into one of the residue classes 2, 5, or 8. [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python program checks the cube identity, the height cap, and the necessary residues using arbitrary-precision integers.

### Computational notes

- An exact two-pointer sweep checked every x,y,z with absolute value at most 3000. It made 36012001 ordered two-sum comparisons after symmetry reduction and found no solution. Cubic-residue tests for every modulus at most 100 found no global obstruction. This is local evidence only and does not reproduce the published large-scale search.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `three-cubes-114-height-1e20`, 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>Jon Grantham and P. G. Walsh, Representing integers as a sum of three cubes, arXiv:2211.12149v1 (2022). Version 1, closing sentence of Section 2 https://arxiv.org/abs/2211.12149
   - Also cited at Grantham and Walsh, Representing integers as a sum of three cubes, arXiv:2211.12149v1, submitted 22 November 2022, closing sentence of Section 2
   - preprint; reference source; arXiv:2211.12149v1; checked 2026-07-25
   - Source use: citation_only
   - Reports that the authors' search for a representation of 114 was unsuccessful.
2. <a id="reference-2"></a>Andrew R. Booker and Andrew V. Sutherland, On a question of Mordell, Proceedings of the National Academy of Sciences 118(11) (2021). Booker and Sutherland, Sections 1 and 5.B, especially the algorithm coverage statement on pages 1-2, the September 2019 parameters on page 10, and Remark 5.1 on page 11 https://doi.org/10.1073/pnas.2022377118
   - Also cited at Booker and Sutherland, On a question of Mordell, PNAS 118(11), 2021, Sections 1 and 5
   - Also cited at Booker and Sutherland, Section 5.B and Remark 5.1, page 11
   - Also cited at Booker and Sutherland, Section 3, Lemma 3.2; exact residue replay in tc114-artifact-exact-verifier
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Describes the complete search through minimum coordinate 10^17 and the modular tests used for the 114 audit.
   - Source named by the research packet.