Erdős Problem #1142 — open problem in number theory

Erdős Problem #1142

Are there infinitely many n > 2 such that n - 2^k is prime for all k ≥ 1 with 2^k < n? The only known such n are 4, 7, 15, 21, 45, 75, 105 (OEIS A039669).

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-1142,
  title        = {Erdős Problem #1142},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1142}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Are there infinitely many $n > 2$ such that $n - 2^{k}$ is prime for all $k \geq 1$ with $2^{k} < n$ ?

The only known such $n$ are $4, 7, 15, 21, 45, 75, 105$ (OEIS A039669).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1142». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1142 :
    answer(sorry) ↔ Infinite { n | Erdos1142Prop n }

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/1142. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1142
A039669
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).
[MiWe69] Mientka, W. E. and Weitzenkamp, R. C., On f-plentiful numbers, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.

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.