In this article, we are interested in whether a product of three consecutive integers \(a (a+1) (a+2)\) divides another such product \(b (b+1) (b+2)\) . If this happens, we prove that there is some gap between them: \(b \gg \frac{a (\log a)^{1/6}}{(\log \log a)^{1/3}}\) . We also consider other polynomial sequences such as \(a^2 (a^2 + l)\) dividing \(b^2 (b^2 + l)\) for some fixed integer l. Our method is based on the effective Liouville–Baker–Feldman theorem.
