Under which (nice) hypotheses on two polynomials $f, g \in \mathbb{Z}[X]$ can we say that there exists a prime number $p$ such that $p$ divides $f(n)$ for some positive integer $n$, and $p$ does not divide $g(m)$ for all positive integers $m$ ?
It is easy to find necessary conditions, for example $g$ cannot be a power of $f$. However I am stuck in finding a (non-trivial) sufficient condition.
Thanks.