Source labels openChecked July 26, 2026

Erdős Problem 137: Multiple Powerful Factors

Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$ , there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.

Number theory AMS 11

Mathematical statement

[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,

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_137.variants.multiple_powerful_factors

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_137.variants.multiple_powerful_factors (k : ℕ) : ∀ᶠ n in Filter.atTop,    ∀ (m : ℕ) (hm : 0 < m),    letI N := ∏ x ∈ Finset.Ioc m (m + n), x    ∃ P : Finset ℕ, P.card = k ∧ ∀ p ∈ P, p.Prime ∧    p ∣ N ∧ ¬ p ^ 2 ∣ N := 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 sourceformalization erdosproblems.com/137background