Erdős (Erdos) Problem #929: status in 2026

Erdős Problem #929

Let k≥ 2 be large and let S(k) be the minimal x such that there is a positive density set of n where n+1,n+2,…,n+k are all divisible by primes ≤ x. Estimate S(k) - in particular, is it true that S(k)≥ k^1-o(1)? A problem of Erdős [Er76d, p. 26]. This is the first part.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #929 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #929 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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.

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Cite

@misc{cairn-erdos-929,
  title        = {Erdős Problem #929},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-929}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_929.parts.i. Let $k \geq 2$ be large and let $S (k)$ be the minimal $x$ such that there is a positive density set of $n$ where $n + 1, n + 2, \dots, n + k$ are all divisible by primes $\leq x$ .

Estimate $S (k)$ - in particular, is it true that $S (k) \geq k^{1 - o (1)}$ ?

A problem of Erdős [Er76d, p. 26]. This is the first part. We read "estimate" as $S (k) = Θ (answer)$ as $k \to \infty$ .

erdos_929.parts.ii. Let $k \geq 2$ be large and let $S (k)$ be the minimal $x$ such that there is a positive density set of $n$ where $n + 1, n + 2, \dots, n + k$ are all divisible by primes $\leq x$ .

Estimate $S (k)$ - in particular, is it true that $S (k) \geq k^{1 - o (1)}$ ?

This is the second part. We read $o (1)$ as an arbitrary $ε > 0$ for all large $k$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«929» (2 statements).

theorem erdos_929.parts.i : (fun k ↦ (S k : ℝ)) =Θ[atTop] erdos_929.parts.i.answer

theorem erdos_929.parts.ii :
    ∀ ε : ℝ, 0 < ε → ∀ᶠ k : ℕ in atTop, (k : ℝ) ^ (1 - ε) ≤ (S k : ℝ)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/929. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/929
[Er76d] Erdős, P., *Problems and results on number theoretic properties of consecutive integers and related questions*. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. PDF

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.