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Level A · Machine-checkable Hard Number theory P-erdos-413

Erdős Problem #413

Are there infinitely many barriers for ω?

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-413,
  title        = {Erdős Problem #413},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-413}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The question

erdos_413.parts.i. Are there infinitely many barriers for ω?

erdos_413.parts.ii. Does there exist some ε > 0 such that there are infinitely many ε-barriers for ω?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«413» (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_413.parts.i :
    answer(sorry) ↔ { n | IsBarrier (fun m => ω m) n }.Infinite
theorem erdos_413.parts.ii :
    answer(sorry) ↔
        (∃ ε > (0 : ℝ), { n | IsBarrier (fun n => ε * ω n) n }.Infinite)

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/413. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_413.variants.bigOmega — Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity.

References

Erdős called a natural number n a barrier for ω, the number of distinct prime divisors, if m + ω(m) ≤ n for all m < n. He believed there should be infinitely many such barriers, and even posed a relaxed variant asking whether there is some ε > 0 for which infinitely many n satisfy m + ε · ω(m) ≤ n for every m < n.

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.