Erdős (Erdos) Problem #18: status in 2026

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?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #18 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #18 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	3 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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 2026-11-13}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_18a. Conjecture 1. Are there infinitely many practical numbers $m$ such that $h (m) < (lo g lo g m)^{O (1)}$ ?

More precisely: does there exist a constant $C > 0$ such that for infinitely many practical numbers $m$ , we have $h (m) < (lo g lo g m)^{C}$ ?

erdos_18b. Conjecture 2. Is it true that $h (n!) < n^{o (1)}$ ? That is, for all $ε > 0$ , is $h (n!) < n^{ε}$ for sufficiently large $n$ ?

erdos_18c. Conjecture 3. Or perhaps even $h (n!) < (lo g n)^{O (1)}$ ?

Erdős offered \$250 for a proof or disproof.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«18» (3 statements).

theorem erdos_18a :
    ∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧
      (practicalH m : ℝ) < (log (log m)) ^ C

theorem erdos_18b :
    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (n : ℝ) ^ ε

theorem erdos_18c :
    ∃ 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.