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Level A · Machine-checkable Hard Number theory P-erdos-940

Erdős Problem #940

Let r ≥ 3. Is it true that the set of integers which are the sum of at most r r-powerful numbers has density 0?

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-940,
  title        = {Erdős Problem #940},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-940}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Disputed
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Refuted
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On the literature board
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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

Let . Is it true that the set of integers which are the sum of at most -powerful numbers has density ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«940». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_940 :
    answer(sorry) ↔ ∀ r ≥ 3,
      {n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • 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 — check erdosproblems.com/940. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_940.variants.three_cubes — Is it true that the set of integers which are the sum of at most three cubes has density 0?
  • erdos_940.variants.large_integers — Let r ≥ 3. Are there infinitely many integers which are not the sum of at most r-many r-powerful numbers?

References

  • erdosproblems.com/940
  • [BaBr94] Baker, R. C. and Brüdern, J., _On sums of two squarefull numbers_. Math. Proc. Cambridge Philos. Soc. (1994), 1-5.
  • [He88] Heath-Brown, D. R., _Ternary quadratic forms and sums of three square-full numbers_. (1988), 137-163.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.