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Source labels openChecked July 26, 2026

Erdős ProblemsNumber theory

Erdős Problem 18

Conjecture 1.* Are there infinitely many practical numbers mm such that h(m)<(log⁡log⁡m)O(1)h(m) < (\log \log m)^{O(1)}?

Number theoryAMS 11

Mathematical statement

Conjecture 1.* Are there infinitely many practical numbers mm such that h(m)<(log⁡log⁡m)O(1)h(m) < (\log \log m)^{O(1)}?

More precisely: does there exist a constant C>0C > 0 such that for infinitely many practical numbers mm, we have h(m)<(log⁡log⁡m)Ch(m) < (\log \log m)^C?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_18a

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_18a : answer(sorry) ↔    ∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧      (practicalH m : ℝ) < (log (log m)) ^ C := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalizationerdosproblems.com/18background