Pollock's (tetrahedral numbers) conjecture

Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most 5 tetrahedral numbers.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-pollocks-conjecture,
  title        = {Pollock's (tetrahedral numbers) conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/pollocks-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Fpollocks-conjecture.json)](https://cairn-commons.com/problems/pollocks-conjecture)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

pollock_tetrahedral. Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most $5$ tetrahedral numbers.

pollock_tetrahedral.salzer_levine. Salzer–Levine strengthening (as stated on Wikipedia/OEIS): there are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$ .

Every positive integer is the sum of at most 5 tetrahedral numbers.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.PollocksConjecture (2 statements).

theorem pollock_tetrahedral (N : ℕ) :
    ∃ f : Fin 5 → ℕ, N = ∑ i, tetrahedral (f i)

theorem pollock_tetrahedral.salzer_levine :
    IsGreatest NotSumOfFourTetrahedral 343867

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
A797
L. E. Dickson, History of the Theory of Numbers, Vol. II: Diophantine Analysis, Dover (2005), pp. 22–23
Frederick Pollock, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant, Abstracts of the Papers Communicated to the Royal Society of London 5 (1850), 922–924
H. E. Salzer and N. Levine, Table of integers not exceeding 100000 that are not expressible as the sum of four tetrahedral numbers, Math. Comp. 12 (1958), 141–144
MathWorld: Pollock's Conjecture

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.