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Number theory
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∣m(m+1)⋯(m+n)
and yet
p2
does not divide the right hand side.
[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,
$2,690 bounty·1 piece from 1 person·last one 17 days ago
week of 27 Sept 2025 · nothing publishedweek of 4 Oct 2025 · nothing publishedweek of 11 Oct 2025 · nothing publishedweek of 18 Oct 2025 · nothing publishedweek of 25 Oct 2025 · nothing publishedweek of 1 Nov 2025 · nothing published
Solve it
A complete proof or refutation in Lean. The first one the validator accepts takes the whole bounty.
∀ (k : ℕ),
∀ᶠ (n : ℕ) in Filter.atTop,
∀ (m : ℕ),
0 < m →
∃ P,
P.card = k ∧ ∀ p ∈ P, Nat.Prime p ∧ p ∣ ∏ x ∈ Finset.Icc m (m + n), x ∧ ¬p ^ 2 ∣ ∏ x ∈ Finset.Icc m (m + n), x
27 Sept 2025a year of work on this problem19 Sept 2026
erdosproblems.com/137
Something wrong with this formalization?
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