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.]

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.
