While experimenting with random polynomials I've found this conjecture:
A polynomial $f\in\mathbb Z[X]$ of degree $n$ with co-prime coefficients have no fixed prime divisor $p> n$.
A fixed prime divisor is a prime $p$ such that $p|f(m)$ for all $m\in\mathbb Z$.
Is this known? Proved? Or are there counterexamples?