Erdős problem 137 · problems.science

Let $k\geq 3$ . Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$ ?

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FormalConjectures/ErdosProblems/

137.lean

Retained formal statement2 of 3

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.

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

FormalConjectures/ErdosProblems/137.leanErdos137.erdos_137.variants.multiple_powerful_factors6 linesExact file

1∀ (k : ℕ),2  ∀ᶠ (n : ℕ) in Filter.atTop,3    ∀ (m : ℕ),4      0 < m →5        ∃ P,6          P.card = k ∧ ∀ p ∈ P, Nat.Prime p ∧ p ∣ ∏ x ∈ Finset.Ioc m (m + n), x ∧ ¬p ^ 2 ∣ ∏ x ∈ Finset.Ioc m (m + n), x

OpenStatement only, no proof

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