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Erdős problem 931

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Let k1≥k2≥3k_1 \geq k_2 \geq 3. Are there only finitely many n2≥n1+k1n_2\geq n_1 + k_1 such that ∏1≤i≤k1(n1+i) and ∏1≤j≤k2(n2+j) \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 statements36c6d0fef71a

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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 n2>2(n1+k1)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
sorry ↔  ∀ (k₁ k₂ : ℕ),    k₂ ≥ 3 →      k₂ ≤ k₁ →        {(n₁, n₂) |            n₁ + k₁ ≤ n₂ ∧              n₂ ≤ 2 * (n₁ + k₁) ∧                (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Finite
OpenStatement only, no proof