This may be a well known problem:

Let $f$ be a polynomial with integer coefficients. Is it possible for the set of prime divisors of values of $f$ on an infinite set of integers, to be finite?

I guess such a set exists for only polynomial $f$ of the form $f(x) = c(ax+b)^n, a,b,c\in \mathbb{Z}, c\neq 0$. But I can't give a proof.

I'm specially interested in the case $f(x) = x^2+1$.

Thanks!