Powers in $\prod\limits_{k=1}^n (ak^{2^l\cdot3^m}+b)$

Abstract

Let $f(x)=ax^{2^l\cdot3^m}+b\in \mathbb{Z}[x]$ be a polynomial with $l\geq 1, l+m\geq 2, ab\neq 0$ and such that $f(k)\neq 0$ for any $k\geq 1$. We prove, under $ABC$ conjecture, that the product $\prod_{k=1}^n f(k)$ is not a $2^l\cdot3^m$-th power for $n$ large enough.