Prove that for $P(X) \in \mathbb{Z}[X]$ the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for } n \in \mathbb{Z}^+\right\}$ is infinite

Question

Prove that for $P(X) \in \mathbb{Z}[X]$, $P(x)$ non-constant, the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for some } n \in \mathbb{Z}^+\right\}$ is infinite.

could someone please give me some advice on how to start this proof.

Answer 1

Outline: The result is obvious if the constant term of $P$ is $0$. So assume the constant term is non-zero.

Let $p_1,\dots,p_k$ be any primes that do not divide the constant term of $P$, and let $M_k$ be their product. Show that for large enough $w$, $P(wM_k)$ is divisible by a prime different from the $p_i$.