Erdős Problem 686

Reference: erdosproblems.com/686

theorem Erdos686.erdos_686 : (∀ N ≥ 2, ∃ k ≥ 2, ∃ (n : ℕ), ∃ m ≥ n + k, ↑N = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))) ↔ sorry

Can every integer $N≥2$ be written as $$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?