There exists infinitely many $n\in\mathbb{Z}$ such that $f(n)$ is a prime. I found in a number theory book the following lines 
Let $f(x)$ be a non-constant polynomial with integer coefficients such that none of the following hold for it
1) There is an integer $d>1$ such that $d|f(n)$ for all $n\in \mathbb{Z}$
2) $f(x)$ is reducible over $\mathbb{Z}[x]$
Then it is conjectured that there exists infinitely many $n\in\mathbb{Z}$ such that $f(n)$ is a prime.
If deg$(f)=1$ we get Dirichlet's Theorem
The book is not a really new one so I wanted to know if this is still open or has been solved. In any case please share some links for more information on it.
 A: This is known as Bunyakovsky conjecture, and as far as I know, almost no progress has been made on it. In fact, we don't even know if $p(n)=n^2+1$ produces infinitely many primes.
A: To add to Alex R.'s answer, this conjecture is further generalized by the Bateman-Horn conjecture, which not only predicts infinitely many primes for polynomials, but it also predicts the asymptotic density of primes, not only for an individual polynomial, but even if you require multiple polynomials to be simultaneously prime.  Again, almost no progress has been made on this conjecture.  
A: Add a little culture, given integers $a,b,c$ with, say, $a > 0,$ $\gcd(a,b,c) = 1$ and $b^2 - 4 a c $ not zero or one or a square, then
$$ f(x,y) = a x^2 + b x y + c y^2 $$
takes on infinitely many prime values. Just that it takes two variables to accomplish that. If  $b^2 - 4 a c < 0,$ the form is positive definite. If  $b^2 - 4 a c > 0,$ the form is indefinite; it can sometimes be extremely difficult to say which (positive) primes $p$ can be expressed as $ p = a x^2 + b x y + c y^2, $ and for which we have instead $ -p = a x^2 + b x y + c y^2. $
