Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes
infinitely many prime numbers as values. Is it known so far whether there is at all any
polynomial $P \in \mathbb{Z}[x]$ of degree $\geq 2$ which takes infinitely many prime values?
-- Note that obtaining this result would not necessarily require to prove that any particular polynomial has this property.
 A: As far as I am aware, it is unknown whether any irreducible polynomial of degree greater than one assumes infinitely many prime values. Certainly this is the case if one insists that the polynomial be given explicitly. I merely add that what is conjectured is that if an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ satisfies $1=\mathrm{gcd}\{f(1), f(2), f(3), f(4), \dots\}$ then $f(n)$ is prime for infinitely many $n$. This is known as Bunyakovsky's conjecture. It has not been proven for any polynomial of degree greater than $1$. Generalizations include Schinzel's hypothesis H and the Bateman-Horn Conjecture.
A: The comments and the answer are related to the probably hopeless Bunyakovsky conjecture. It seems to me that Stefan Kohl had a different idea in mind, maybe something like the following: Let $A_p$ be the set of polynomials $f\in\mathbb Z[X]$ of degree $\ge2$ with $p\in f(\mathbb Z)$. The question amounts to asking if there is an infinite set $P$ of primes such that $\cap_{p\in P}A_p$ is not empty.
So in this form the question is if some density argument (with respect to some measure on $\mathbb Z[X]$) could be strong enough. 
