# P10: Sum of three cubes problem

- ID: `P10`
- Reference: `sum-of-three-cubes-problem`
- Page: https://theoremdb.org/statements/P10
- Record maturity: Reviewed problem with recorded work

## Problem

The Diophantine equation \(x^3+y^3+z^3=k\) is solvable in integers \(x,y,z\) for each \(k\in\mathbb{Z}\) satisfying \(k\not\equiv\pm4\pmod 9\).

### Context

A simple congruence gives the only known general obstruction, while finding or excluding representations remains difficult.

### Problem setup

- **Definition (Cubes modulo 9 are congruent only to 0, 1, or -1, so integers congruent to 4 or 5 modulo 9 are impossible).** Cubes modulo 9 are congruent only to 0, 1, or -1, so integers congruent to 4 or 5 modulo 9 are impossible.
- **Definition (The variables may be negative and need not be distinct).** The variables may be negative and need not be distinct.
- **Remark.** A simple congruence gives the only known general obstruction, while finding or excluding representations remains difficult.

### What counts as a solution

- Prove representability for every integer outside the two obstructed residue classes, or give an admissible integer and prove that it has no representation by three integer cubes.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Booker and Sutherland found explicit representations for 33 and 42, completing every admissible positive k<=100. Universal representability outside residues 4 and 5 modulo 9 remains open. Exact unresolved remainder: Prove representation for every admissible integer, or prove one admissible integer is not representable. [1](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Booker and Sutherland found explicit representations for 33 and 42, completing every admissible positive k<=100. Universal representability outside residues 4 and 5 modulo 9 remains open. Exact unresolved remainder: Prove representation for every admissible integer, or prove one admissible integer is not representable.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Booker and Sutherland found explicit representations for 33 and 42, completing every admissible positive k<=100. Universal representability outside residues 4 and 5 modulo 9 remains open.

Exact unresolved remainder: Prove representation for every admissible integer, or prove one admissible integer is not representable.

### Background and intake notes

- Original intake status: The cited research paper treats universal representability outside the modulo-9 obstruction as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper on 2026-07-22.
- Solutions can involve variables vastly larger than k. Check current computational tables before claiming a first representation of a particular integer.

- Recorded example: 6 = 2^3 + (-1)^3 + (-1)^3.

### Open directions

- **Route 1** (reported): Prove representability for every integer outside the two obstructed residue classes, or give an admissible integer and prove that it has no representation by three integer cubes. [1](#reference-1)

### Computational notes

- A search bounded by |x|, |y|, and |z| cannot prove that an unsolved target has no representation beyond that bound.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `sum-of-three-cubes-problem`, 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>Victor Y. Wang, “Sums of cubes and the Ratios Conjectures”. arXiv:2108.03398 (2021). Victor Y. Wang, arXiv:2108.03398, abstract and introduction https://arxiv.org/abs/2108.03398
   - Also cited at Abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2108.03398, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited research paper treats universal representability outside the modulo-9 obstruction as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Packet-linked current theoretical framing.
   - Source named by the research packet.
2. <a id="reference-2"></a>Andrew V. Sutherland, Sums of cubes, project results page, joint work with Andrew R. Booker. math.mit.edu checked 2026-08-01. Displayed identities https://math.mit.edu/~drew/sumsofcubes.html
   - website; primary source; checked 2026-08-01
   - Source use: original_summary
   - Primary project record for computed representations.
3. <a id="reference-3"></a>Edward Dunne, 42, AMS Beyond Reviews, 11 September 2019. Representation of 42 and status through 100 https://blogs.ams.org/beyondreviews/2019/09/11/42/
   - website; secondary source; checked 2026-08-01
   - Source use: original_summary
   - Explains completion of admissible cases through 100.