Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Number theory P-erdos-462

Erdős Problem #462

Let p(n) denote the least prime factor of n. Is there a constant C>0 such that Σ_x≤ n≤ x+C√(x)(log x)^2p(n)/n≫ 1 for all sufficiently large x?

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.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-erdos-462,
  title        = {Erdős Problem #462},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-462}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

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

The problem

The question

Let denote the least prime factor of . Is there a constant such that for all sufficiently large ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«462». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_462 : answer(sorry) ↔
    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ᶠ x : ℕ in atTop,
      c ≤ ∑ n ∈ Finset.Icc x
        ⌊(x : ℝ) + C * Real.sqrt x * (Real.log x) ^ 2⌋₊, (n.minFac : ℝ) / n

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/462. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/462

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.