Erdős problem 931 · problems.science

Let $k_1 \geq k_2 \geq 3$ . Are there only finitely many $n_2\geq n_1 + k_1$ such that $\prod_{1\leq i\leq k_1}(n_1 + i)\ \text{and}\ \prod_{1\leq j\leq k_2} (n_2 + j)$ have the same prime factors?

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

931.lean

Retained formal statement4 of 4

Erdős was unable to prove that if the two products have the same factors then there must exist a prime between $n_1$ and $n_2$ .

FormalConjectures/ErdosProblems/931.leanErdos931.erdos_931.variants.exists_prime6 linesExact file

1∀ (k₁ k₂ n₁ n₂ : ℕ),2  k₂ ≤ k₁ →3    3 ≤ k₂ →4      n₁ + k₁ ≤ n₂ →5        (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors →6          ∃ p, Nat.Prime p ∧ n₁ ≤ p ∧ p ≤ n₂

OpenStatement only, no proof

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