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?
Best Answer
Since $p$ does not divide each of the coefficients of $f(x)$, $f(x)$ reduces to a non-trivial polynomial $\overline f(x)$ of degree $≤n$ $\pmod p$. But if $p>n$ then $\overline f(x)$ can have at most $n$ roots $\pmod p$, hence it is non-zero on at least one residue $a \pmod p$. But then $f(a)\not \equiv 0 \pmod p$ and we are done.