The Diophantine equation $b (b+1) (b+2) = t a (a + 1) (a + 2)$ and gap principle

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 gaps between them, namely b a log a)1/6log log a)1/3. We also consider other polynomial sequences such as a² (a² + l) dividing b² (b² + l) for some fixed integer l. Our method is based on effective Liouville-Baker-Feldman theorem.