Erdős Problem #463 — open problem in number theory

Erdős Problem #463

Is there a function f with f(n)→∞ as n→∞ such that, for all large n, there is a composite number m such that n + f(n) < m < n + p(m) Here p(m) is the least prime factor of m.

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-463,
  title        = {Erdős Problem #463},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-463}},
  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

Is there a function $f$ with $f (n) \to \infty$ as $n \to \infty$ such that, for all large $n$ , there is a composite number $m$ such that $n + f (n) < m < n + p (m)$ Here $p (m)$ is the least prime factor of $m$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«463». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_463 : answer(sorry) ↔ ∃ (f : ℕ → ℕ) (_ : Tendsto f atTop atTop),
    ∀ᶠ n in atTop,
      ∃ m, m.Composite ∧
        n + f n < m ∧ m < n + m.minFac

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/463. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/463

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.