[#P10] Sum of three cubes problem

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 \(k\not\equiv\pm4\pmod 9\).

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1Context

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

2Remarks

Remark 1. Cubes modulo 9 are congruent only to 0, 1, or -1, so integers congruent to 4 or 5 modulo 9 are impossible.

Remark 2. The variables may be negative and need not be distinct.

3What 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.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (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.^[1][2][3]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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][2][3] Relevance to this problem For Sum of three cubes problem, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3 Details 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Sum of three cubes problem, record sum-of-three-cubes-problem-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

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 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Solutions can involve variables vastly larger than k. Check current computational tables before claiming a first representation of a particular integer.

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

Computational notes

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

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Sum of three cubes problem.” TheoremDB. P10. Problem statement; statement text SHA-256 86ffc901b7e5b73410f811cb3ee3b2629fbb2240462b7321836d41d8dcfc1b37. https://theoremdb.org/statement/?ref=P10

BibTeX

@misc{theoremdb-problem-86ffc901b7e5b73410f811cb3ee3b2629fbb2240462b7321836d41d8dcfc1b37,
  title = {{Sum of three cubes problem}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 86ffc901b7e5b73410f811cb3ee3b2629fbb2240462b7321836d41d8dcfc1b37},
  url = {https://theoremdb.org/statement/?ref=P10}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Victor Y. Wang, “Sums of cubes and the Ratios Conjectures”. arXiv:2108.03398 (2021). Victor Y. Wang, arXiv:2108.03398, abstract and introduction. ↗preprint · primary source · arXiv:2108.03398, checked 2026-07-31 · checked 2026-07-31Source 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.Also cited at Abstract and introduction.Also cited at Editorial research route recorded 2026-07-31.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.
Andrew V. Sutherland, Sums of cubes, project results page, joint work with Andrew R. Booker. math.mit.edu checked 2026-08-01. Displayed identities. ↗website · primary source · checked 2026-08-01Source use: original summary.Primary project record for computed representations.
Edward Dunne, 42, AMS Beyond Reviews, 11 September 2019. Representation of 42 and status through 100. ↗website · secondary source · checked 2026-08-01Source use: original summary.Explains completion of admissible cases through 100.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P10] Sum of three cubes problem

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

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Computational notes

2See also

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1References

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1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance