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 statement2 of 4

Erdős thought perhaps if the two products have the same factors then $n_2 > 2(n_1 + k_1)$ . It is an open question whether this is true when allowing a finite number of counterexamples.

FormalConjectures/ErdosProblems/931.leanErdos931.erdos_931.variants.additional_condition8 linesExact file

1sorry ↔2  ∀ (k₁ k₂ : ℕ),3    k₂ ≥ 3 →4      k₂ ≤ k₁ →5        {(n₁, n₂) |6            n₁ + k₁ ≤ n₂ ∧7              n₂ ≤ 2 * (n₁ + k₁) ∧8                (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Finite

OpenStatement only, no proof

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