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.

• It is open whether there exists a polynomial in one variable of degree $>1$ that represents infinitely many primes. This is mentioned here. – Jeremy Rouse Jun 6 '15 at 20:33
• @MicahMilinovich, how does one deduce that from Baier & Zhao's result? To the best of my knowledge, these results typically require that the number of polynomials in the family increase with other parameters in the average and thus do not imply asymptotic statements about any single (even a typical) polynomial. This is analogous to the fact that while Bombieri–Vinogradov is an averaged form of the RH, B-V does not imply that the RH holds even a single L-function. – Mark Lewko Jun 6 '15 at 22:14
• @MarkLewko: Correct, I have retracted my comments. – Micah Milinovich Jun 6 '15 at 23:03
• @Gerhard Paseman: $x^2+y^2$ even assumes all the infinitely many primes in $4\mathbb N+1$ ... – Peter Mueller Jun 8 '15 at 17:11
• Along the lines of Friedlander-Iwaniec, Heath-Brown proved that $x^3+2y^3$ represents infinitely many primes, with $x$ and $y$ positive. With Moroz, he extended this to general irreducible binary cubic forms. – Gerry Myerson Jun 9 '15 at 6:27

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.
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.