This may be a well known problem:

Let $f$ be a non-constant 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? ( To be clear, I need the answer (if known) for all $f(x)\in \mathbb{Z}[x]$. )

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

Thanks!