Let $P(x)$ be a polynomial and $P(x)=a_n x^n+...+a_1 x +a_0$, with $a_0,a_1,...,a_n \in \{ 1,0,-1\}, a_n \neq 0$. Consider a sequence $S$ : $$P(0), P(1), ...,P(n)$$ Is it true that there are infinite prime numbers in the sequence $S$? If not, are there any conditions of $P(x)$ so that there are infinite prime numbers in the sequence $S$?
(Sorry, English is my second language)