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Level A · Machine-checkable Hard Number theory P-erdos-18

Erdős Problem #18

Conjecture 1. Are there infinitely many practical numbers m such that h(m) < (log log m)^O(1)? More precisely: does there exist a constant C > 0 such that for infinitely many practical numbers m, we have h(m) < (log log m)^C?

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-18,
  title        = {Erdős Problem #18},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-18}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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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

erdos_18a. Conjecture 1. Are there infinitely many practical numbers such that ?

More precisely: does there exist a constant such that for infinitely many practical numbers , we have ?

erdos_18b. Conjecture 2. Is it true that ? That is, for all , is for sufficiently large ?

erdos_18c. Conjecture 3. Or perhaps even ?

Erdős offered \$250 for a proof or disproof.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«18» (3 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_18a : answer(sorry) ↔
    ∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧
      (practicalH m : ℝ) < (log (log m)) ^ C
theorem erdos_18b : answer(sorry) ↔
    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (n : ℝ) ^ ε
theorem erdos_18c : answer(sorry) ↔
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (log n) ^ C

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/18. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/18
  • [ErGr80] Erdős, P. and Graham, R. L. (1980). Old and New Problems and Results in Combinatorial Number

Theory. Monographies de L'Enseignement Mathématique, 28. Université de Genève. (See the sections on Egyptian fractions or practical numbers).

  • [Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.

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.