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Level A · Machine-checkable Hard Number theory P-erdos-261

Erdős Problem #261

Do all positive integers n have the required property?

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-erdos-261,
  title        = {Erdős Problem #261},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-261}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Disputed
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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

erdos_261.parts.ii. Do all positive integers have the required property?

erdos_261.parts.iii. Is there a rational number such that has at least representations by pairwise distinct positive integers ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«261» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_261.parts.ii : answer(sorry) ↔ ∀ n > 0, Erdos261Prop n
theorem erdos_261.parts.iii : answer(sorry) ↔ ∃ x : ℚ,
    𝔠 ≤ #{a : ℕ → ℕ | Erdos261InfiniteRepresentation x a}

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 — check erdosproblems.com/261. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/261
  • [BoLo90] Borwein, Peter and Loring, Terry A., Some questions of Erdős and Graham on numbers of the form . Math. Comp. (1990), 377--394.
  • [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102--109.
  • [TUZ20] Tengely, Szabolcs and Ulas, Maciej and Zygadlo, Jakub, On a Diophantine equation of Erdős and Graham. J. Number Theory (2020), 445--459.

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.