erdos_686

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 $m≥n+k$?

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

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