Erdős Problem #450 — open problem in number theory

Erdős Problem #450

How large must y=y(ε,n) be such that the number of integers in (x,x+y) with a divisor in (n,2n) is at most ε y? The bound is required for every x and every window length at least y, and y(ε,n) is the least such threshold (or ∞ if there is none).

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-450,
  title        = {Erdős Problem #450},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-450}},
  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

How large must $y = y (ϵ, n)$ be such that the number of integers in $(x, x + y)$ with a divisor in $(n, 2 n)$ is at most $ϵy$ ?

The bound is required for every $x$ and every window length at least $y$ , and $y (ϵ, n)$ is the least such threshold (or $\infty$ if there is none). A linear scale $y \leq C (ϵ) n$ is known to suffice for fixed $ϵ$ and all large $n$ (see erdos_450.linear_scale_suffices).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«450». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_450 (ε : ℝ) (hε : 0 < ε) (n : ℕ) : windowThreshold ε n = answer(sorry)

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/450. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/450

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.