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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.,
∀ (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
What you must prove
import FormalConjectures.ErdosProblems.«137»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos137.erdos_137.variants.multiple_powerful_factors") := by
sorry
end Bounty
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
Erdős problem 137 - multiple powerful factors · Conjectures.io