Erdős problem 930 · problems.science

Is it true that, for every $r$ , there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$ , then $\prod_{1\leq i\leq r}\prod_{m\in I_i}m$ is not a perfect power?

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3 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

930.lean

Retained formal statement3 of 3

Let $k$ , $l$ , $n$ be integers such that $k \ge 3$ , $l \ge 2$ and $n + k \ge p^{(k)}$ , where $p^{(k)}$ is the least prime satisfying $p^{(k)} \ge k$ . Then there is a prime $p \ge k$ for which $l$ does not divide the multiplicity of the prime factor $p$ in $(n + 1) \ldots (n + k)$ .

Theorem 2 from [ErSe75].

[ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.

FormalConjectures/ErdosProblems/930.leanErdos930.erdos_930.variants.consecutive_strong5 linesExact file

1∀ (k l n : ℕ),2  3 ≤ k →3    2 ≤ l →4      Erdos930.nextPrime k ≤ n + k →5        ∃ p, k ≤ p ∧ Nat.Prime p ∧ ¬l ∣ (∏ m ∈ Finset.Icc (n + 1) (n + k), m).factorization p

SolvedStatement only, no proof

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