Erdős Problem #385 — open problem in number theory

Erdős Problem #385

Let F(n) := maxm + p(m) | textrmm < n composite where p(m) is the least prime divisor of m. Is it true that F(n)>n for all sufficiently large n?

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.

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Cite

@misc{cairn-erdos-385,
  title        = {Erdős Problem #385},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-385}},
  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

erdos_385.parts.i. Let $F(n) := \max\{m + p(m) \mid \textrm{$ m < n $composite}\}\}$ where $p (m)$ is the least prime divisor of $m$ . Is it true that $F (n) > n$ for all sufficiently large $n$ ?

erdos_385.parts.ii. Let $F(n) := \max\{m + p(m) \mid \textrm{$ m < n $composite}\}\}$ where $p (m)$ is the least prime divisor of $m$ . Does $F (n) - n \to \infty$ as $n \to \infty$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«385» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_385.parts.i : answer(sorry) ↔ ∀ᶠ n in atTop, n < F n

theorem erdos_385.parts.ii : answer(sorry) ↔ atTop.Tendsto (fun n ↦ F n - n) atTop

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/385. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_385.variants.lb — A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least n+(1-o(1))√(n)

References

erdosproblems.com/385

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.