Erdős Problem #891 — open problem in number theory

Erdős Problem #891

Let 2=p_1 < p_2 < ⋯ be the primes and k≥ 2. Is it true that, for all sufficiently large n, there must exist an integer in [n,n+p_1⋯ p_k) with >k many prime factors?

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-891,
  title        = {Erdős Problem #891},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-891}},
  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 $2 = p_{1} < p_{2} < \dots$ be the primes and $k \geq 2$ . Is it true that, for all sufficiently large $n$ , there must exist an integer in $[n, n + p_{1} \dots p_{k})$ with $> k$ many prime factors?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«891». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_891 :
    answer(sorry) ↔
      ∀ k ≥ 2, ∀ᶠ n in atTop,
      ∃ m ∈ Ico n (n + ∏ i ∈ range k, i.nth Nat.Prime), k < ω m

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/891. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_891.variants.case_k_2 — This is unknown even for k=2 - that is, is it true that in every interval of 6 (sufficiently large) consecutive integers there must exist one with at least 3…

References

erdosproblems.com/891
[Po18] Pólya, Georg, Zur arithmetischen {U}ntersuchung der {P}olynome. Math. Z. (1918), 143--148.
[Wikipedia] https://en.wikipedia.org/wiki/Dickson%27s_conjecture

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.