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?

consecutiveprimes. Do you mean the primes areconsecutiveprimes (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). – user9072 Sep 1 '13 at 12:22