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Level A · Machine-checkable Hard Number theory P-erdos-11

Erdős Problem #11

Is every odd n > 1 the sum of a squarefree number and a power of 2?

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-11,
  title        = {Erdős Problem #11},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-11}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Verified
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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

Is every odd the sum of a squarefree number and a power of 2?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«11».

theorem erdos_11 (n : ℕ) (hn : Odd n) (hn' : 1 < n) :
    ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l

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/11. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_11.variants.not_four_dvd — Erdős often asked this under the weaker assumption that n > 1 is not divisible by 4.
  • erdos_11.variants.two_pow_two — Is every odd n > 1 the sum of a squarefree number and two powers of 2?

References

erdosproblems.com/11

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.