Erdős Problem #377 — open problem in number theory

Erdős Problem #377

Is there some absolute constant C > 0 such that Σ_p ≤ n 1_pnmid 2n choose n1/p ≤ C for all 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-377,
  title        = {Erdős Problem #377},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-377}},
  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 some absolute constant $C > 0$ such that $p \leq n \sum 1_{p ∤ (n 2 n)} \frac{1}{p} \leq C$ for all $n$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«377». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_377 : answer(sorry) ↔
    ∃ C > (0 : ℝ), ∀ (n : ℕ), sumInvPrimesNotDvdCentralBinom n ≤ 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/377. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/377

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.