Erdős Problem #886 — open problem in number theory

Erdős Problem #886

Let ε>0. Is it true that, for all large n, the number of divisors of n in (n^1/2,n^1/2+n^1/2-ε) is O_ε(1)? Erdős attributes this conjecture to Ruzsa.

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-886,
  title        = {Erdős Problem #886},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-886}},
  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

Let $ϵ > 0$ . Is it true that, for all large $n$ , the number of divisors of $n$ in $(n^{1/2}, n^{1/2} + n^{1/2 - ϵ})$ is $O_{ϵ} (1)$ ?

Erdős attributes this conjecture to Ruzsa.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«886». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_886 :
    answer(sorry) ↔ ∀ ε > 0, ∃ K : ℕ, ∀ᶠ n in atTop, (Erdos886Divisors n ε 1).card ≤ K

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/886. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/886
[ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.

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.