Erdős (Erdos) Problem #709: status in 2026

Erdős Problem #709

Let f(n) be minimal such that, for any A=a_1,…,a_n⊆ [2,∞)∩ℕ of size n, in any interval I of f(n)max(A) consecutive integers there exist distinct x_1,…,x_n∈ I such that a_i| x_i. Obtain good bounds for f(n), or even an asymptotic formula.

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 #709 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_709 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #709 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_709`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 3 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. 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-709,
  title        = {Erdős Problem #709},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-709}},
  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

Let $f (n)$ be minimal such that, for any $A = {a_{1}, \dots, a_{n}} \subseteq [2, \infty) \cap N$ of size $n$ , in any interval $I$ of $f (n) max (A)$ consecutive integers there exist distinct $x_{1}, \dots, x_{n} \in I$ such that $a_{i} ∣ x_{i}$ .

Obtain good bounds for $f (n)$ , or even an asymptotic formula.

We read "good bounds" as determining the order of magnitude of $f (n)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«709».

theorem erdos_709 : f =Θ[atTop] erdos_709.answer

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/709. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/709
[ErSu59] P. Erdős and J. Surányi, Bemerkungen zu einer Aufgabe eines mathematischen Wettbewerbs, Mat. Lapok 10 (1959), 39–48.
[Er92c] P. Erdős, Some of my forgotten problems in number theory, Hardy–Ramanujan J. 15 (1992), 34–50.
[vD26] W. van Doorn, On the length of an interval that contains distinct multiples of the first $n$ positive integers, Integers (2026), #A7.

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.