Erdős Problem #1201 — open problem in number theory

Erdős Problem #1201

Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)⋯(n+k))>n^1-ε is at least 1-η (where P(m) is the greatest prime divisor of m)?

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-1201,
  title        = {Erdős Problem #1201},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1201}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is it true that for every $ϵ, η > 0$ there exists a $k$ such that the density of $n$ for which $P (n (n + 1) \dots (n + k)) > n^{1 - ϵ}$ is at least $1 - η$ (where $P (m)$ is the greatest prime divisor of $m$ )?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1201». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1201 :
    answer(sorry) ↔
      ∀ ε > 0, ∀ η > 0, ∃ k : ℕ,
        atTop.liminf (fun x : ℕ ↦
          (((count (· ∈ Erdos1201Set ε k) x : ℝ) / (x : ℝ)) : EReal)) ≥ (1 - η : 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/1201. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1201

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.