Erdős (Erdos) Problem #382: status in 2026

Erdős Problem #382

Let u ≤ v be such that the largest prime dividing Π_u ≤ m ≤ v m appears with exponent at least 2. Is it true that v - u = v^o(1)? We read this as: for every ε > 0, every such pair with v large enough satisfies v - u ≤ v^ε.

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 #382 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #382 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.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-erdos-382,
  title        = {Erdős Problem #382},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-382}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Ferdos-382.json)](https://cairn-commons.com/problems/erdos-382)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

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_382.parts.i. Let $u \leq v$ be such that the largest prime dividing $\prod_{u \leq m \leq v} m$ appears with exponent at least $2$ . Is it true that $v - u = v^{o (1)}$ ?

We read this as: for every $ϵ > 0$ , every such pair with $v$ large enough satisfies $v - u \leq v^{ϵ}$ . Cambie observed that this question reduces to old conjectures on gaps between primes; for example, it follows from Cramér's conjecture. See also Erdős Problems 380 and 383.

erdos_382.parts.ii. Let $u \leq v$ be such that the largest prime dividing $\prod_{u \leq m \leq v} m$ appears with exponent at least $2$ . Can $v - u$ be arbitrarily large?

That is, for every $k$ , is there such a pair with $v - u \geq k$ ? Cambie gives a heuristic that suggests the answer is yes.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«382» (2 statements).

theorem erdos_382.parts.i :
    ∀ ε > (0 : ℝ), ∀ᶠ v : ℕ in atTop, ∀ u : ℕ, LargestPrimeRepeated u v →
      ((v - u : ℕ) : ℝ) ≤ (v : ℝ) ^ ε

theorem erdos_382.parts.ii :
    ∀ k : ℕ, ∃ u v : ℕ, LargestPrimeRepeated u v ∧ k ≤ v - u

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/382. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/382
[ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathematique (1980).

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.