Green-Tao style theorem for quadratic regressions (Ulam Spiral)

This is a naive question about number theory.

Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a remarkable structure that on some lines there seems to be a higher density of prime values. This is addressed in Hardy-Littlewood's conjecture F.

My question is: Can we find sequences of consecutive prime number

$an^2+bn+c, a(n+1)^2+b(n+1)+c,\ldots, a(n+k)^2+b(n+k)+c$

for arbitrary large $k$ if we allow $a,b,c$ to vary? Or, maybe more restrictively, varying $c$ for fixed $a$ and $b$ (for example $a=4$, $b=2$ as in the Ural spiral)? Does this follow from any other conjectures?

-
I am slightly confused. You mention Green--Tao and then talk about consecutive primes. Do you mean the primes are consecutive primes (this is not even known for arithemetic progressions) or do you just mean the consectuive values of the quadratic polynomial are all prime. If the latter (at least the less restrictive version) this seems like a special case of a result of Tao--Ziegler on polynomial progressions in the primes (maybe even the more restrictive but I think what can be said more restrictive there is slightly different). –  quid Sep 1 '13 at 12:22
Thanks, this is helpful. I was asking for the case with $P_1, \ldots, P_k$ s.t. $P_k(m)=a(km)^2+b(km)$ and $c=x$. The statement I tried to formulate above would be the case where $m=1$ which is stronger than their statement. In pictures like here it seems that consecutive progressions of primes occur frequently and I was wondering if these are expected to get arbitrarily long. –  Zahlendreher Sep 1 '13 at 13:14