Erdős (Erdos) Problem #683: status in 2026

Erdős Problem #683

Let P(n, k) be the largest prime factor of C(n, k). There exists c > 0 such that P(n, k) ≥ min(n - k + 1, k^1 + c) for all 0 < k ≤ n/2. Erdős stated this for 1 ≤ k ≤ n with the bound min(n-k+1, k^1+c) [Er79d].

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-09-04).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #683 as open (status updated 4 September 2025).
Formal Conjectures marks the Lean statement erdos_683 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #683 on erdosproblems.com	open 2025-09-04	erdosproblems.com
Formal Conjectures: status of `erdos_683`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 2 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.

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

Cite

@misc{cairn-erdos-683,
  title        = {Erdős Problem #683},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-683}},
  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-683.json)](https://cairn-commons.com/problems/erdos-683)

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

Let $P (n, k)$ be the largest prime factor of $(k n)$ . There exists $c > 0$ such that $P (n, k) \geq min (n - k + 1, k^{1 + c})$ for all $0 < k \leq n /2$ .

Erdős stated this for $1 \leq k \leq n$ with the bound $min (n - k + 1, k^{1 + c})$ [Er79d]. The minimum is needed even for $k \leq n /2$ : at $n = 2 k$ every prime factor of $(k 2 k)$ is at most $2 k$ , so $P (n, k) \geq k^{1 + c}$ fails for large $k$ . The range $k \leq n /2$ is natural (cf. #961 and the discussion).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«683».

theorem erdos_683 :
    ∃ c > (0 : ℝ), ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 →
      (P n k : ℝ) ≥ min (↑(n - k + 1) : ℝ) ((k : ℝ) ^ (1 + c))

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/683. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_683.variant.exp_sqrt — Standard heuristics suggest that P(n, k) > min(n - k + 1, e^c√(k)) for some constant c > 0.

References

erdosproblems.com/683
[Er34] Erdős, Paul, A Theorem of Sylvester and Schur. J. London Math. Soc. (1934), 282--288.
[Er55d] Erdős, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.
[Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.

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.