Erdős Problem #1139 — open problem in number theory

Erdős Problem #1139

Let 1≤ u_1 < u_2 < ⋯ be the sequence of integers with at most 2 prime factors. Is it true that limsup_k → ∞ u_k+1-u_k/log k=∞?

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-1139,
  title        = {Erdős Problem #1139},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1139}},
  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

Let $1 \leq u_{1} < u_{2} < \dots$ be the sequence of integers with at most $2$ prime factors. Is it true that $k \to \infty lim sup \frac{u _{k + 1} - u _{k}}{lo g k} = \infty ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1139». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1139 :
    answer(sorry) ↔
      letI u := Nat.nth (fun n ↦ 0 < n ∧ Ω n ≤ 2)
      atTop.limsup (fun k : ℕ ↦ (((u (k + 1) : ℝ) - (u k : ℝ)) / Real.log (↑k + 1) : EReal)) = ⊤

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/1139. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1139

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.