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Level A · Machine-checkable Hard Number theory P-erdos-390

Erdős Problem #390

Does there exists a constant c such that f n - 2 n ~ c (n / log 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. 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-390,
  title        = {Erdős Problem #390},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-390}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Disputed
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«390». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_390 :
    answer(sorry) ↔ ∃ c,
      (fun n => f n - 2 * n : ℕ → ℝ) ~[atTop] (fun n => c * n / log (n : ℝ))

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/390. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/390
  • [EGS82] Erdős, P., R. K. Guy, and J. L. Selfridge. "Another Property of 239 and some related questions." Congr. Numer. 34 (1982): 243-257.

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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.