Erdős Problem #683 — open problem in number theory

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].

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-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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_683 : answer(sorry) ↔
    ∃ 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.