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Level A · Machine-checkable Hard Number theory P-pollocks-conjecture

Pollock's (tetrahedral numbers) conjecture

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

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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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 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

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

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

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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.