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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 statement4 of 4

Erdős was unable to prove that if the two products have the same factors then there must exist a prime between n1n_1 and n2n_2.

FormalConjectures/ErdosProblems/931.leanErdos931.erdos_931.variants.exists_prime6 linesExact file
∀ (k₁ k₂ n₁ n₂ : ℕ),  k₂ ≤ k₁ →    3 ≤ k₂ →      n₁ + k₁ ≤ n₂ →        (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors →          ∃ p, Nat.Prime p ∧ n₁ ≤ p ∧ p ≤ n₂
OpenStatement only, no proof