Is $k^2+k+1$ prime for infinitely many values of $k$? Let's define an infinite sequence of positive integers as :

$a_n=k^2+(2n-1)k+2n-1 $ , where $  k,n \in \mathbf{Z^{+}}$

Suppose that one can prove that this sequence contains infinitely many prime numbers for any particular $k$. The first term of the sequence for any $k$ is of the form :
$a_1=k^2+k+1$ 
My question is : what is sufficient condition to prove that this polynomial produces primes for infinitely many values of $k$ ?
Note that there is strong experimental evidence that $k^2+k+1$ is prime for infinitely many values of $k$ and that this polynomial satisfies conditions of Bunyakovsky's conjecture.
 A: I don't think there is any comprehensive conjecture on the distribution of prime values of polynomials in two or more variables.  Other than simply being more general, when there are multiple variables there are algebraic curves along which the density of primes is higher or lower.  This is analogous to algebraic surfaces in which it is possible for all the rational points to lie on a lower-dimensional subset such as a line, or for the surface to have many rational points but some curves on the surface will have no rational points at all.     
Iwaniec and Friedlander proved that there are infinitely primes of the form $x^2 + y^4$ but there are infinitely many values of $x$ for which the polynomial is always composite.  Any asymptotic prediction of the distribution of prime values in the sequence must have some non-uniformity when expressed in terms of $x$ and $y$, and there is no known way to "concentrate" the theorem down to statements about infinitely many primes in subsequences with $x$ (or $y$) constant.
