What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$? Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value is not prime and count the number of distinct primes your polynomial has generated. 

Question: what is the maximum number of distinct primes a polynomial of degree $a$ can generate by the process described above? Furthermore, what is the general form of such a polynomial $p(n)$?

This question was inspired by this article.
Thanks, 
Max
[Please note that your polynomial does not need to generate consecutive primes, only primes at consecutive positive integer arguments.]
 A: Here is result by Rabinowitsch for quadratic polynomials.

$n^2+n+A$ is prime for $n=0,1,2,...,A-2$ if and only if $d=1-4A$ is squarefree and the class number of $\mathbb{Q}[\sqrt{d}]$ is $1$.

See this article for details.
http://matwbn.icm.edu.pl/ksiazki/aa/aa89/aa8911.pdf
Also here is a list of imaginary quadratic fields with class number $1$
http://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one#Imaginary_quadratic_fields
There are many other articles about prime generating (quadratic) polynomials that you can google.
A: The Green-Tao Theorem states that there are arbitrarily long arithmetic progressions of primes; that is, sequences of primes of the form
$$ b , b+a, b+2a, b+3a,... ,b+na $$
Since such a progression will be the first $n$ values of the polynomial $ax+b$, this implies that even for degree 1, there is no upper bound to how many primes in a row a polynomial can generate.
A: Here is a related fact which might also be of interest.  There exists a polynomial in 26 variables with the property that, if you plug in integers for all 26 variables, and the output is a positive number (it will always be an integer), then the output is prime.  The polynomial can be found here:
http://en.wikipedia.org/wiki/Formula_for_primes
Furthermore, every prime occurs as an output of this polynomial.  This is a very indirect way to generate primes, because there is no easy way to know if a given input will give a positive output.
There is apparently another polynomial in 10 variables that does this, but Wikipedia doesn't explicitly write it.
