Is there any integer polynomial with smooth values except constants?

Question

In this problem, an integer $x$ is called smooth for some set $S$ if and only if it equals $0$ or all prime divisors of $|x|$ belong to $S$.

I suspect that for any finite set $S$, there isn't any integer non-constant polynomial $F(x)=\sum_{i=0}^n a_i x^i$ ($n>0,a_n \neq 0, a_i \in Z$) that $F(x)$ is smooth for $S$ for all $x \in Z_+$, but I'm having trouble proving it.

Answer 1

Let $P(F)$ denote the set of all primes $p$ such that $p|F(x)$ for some integer value of $x$. It is an old theorem of Schur that $P(F)$ is infinite in size, for any nonconstant polynomial $F$. This implies the desired property.

You can see a citation and brief proof of this fact (and much more) in a lovely article in the American Mathematical Monthly, vol.78 (3) 1971, by Gerst and Brillhart, titled On the Prime Divisors of Polynomials. This article is free to download, and the theorem above is titled Theorem 1, on p.253.