The Diophantine equation $b(b+1)(b+2)=ta(a+1)(a+2)$ 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 $b\gg \frac{a\log a{)}^{1\mathrm{/}6}}{\log \log a{)}^{1\mathrm{/}3}}$. 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 effective Liouville-Baker-Feldman theorem.