Is the notorious $n^2 + n + 41$ prime generator the last of its type? The polynomial $n^2+n+41$ famously takes prime values for all $0\le n\lt 40$. I have read that this is closely related to the fact that 163 is a Heegner number, although I don't understand the argument, except that the discriminant of $n^2+n+41$ is $-163$. The next smaller Heegner number is 67, and indeed $n^2+n+17$ shows the same behavior, taking prime values for $0\le n\lt 16$.  
But 163 is the largest Heegner number, which suggests that this is the last such coincidence, and that there might not be any $k>41$ such that $n^2+n+k$ takes on an unbroken sequence of prime values.
Is this indeed the case?  If not, why not?
Secondarily, is there a $k>41$, perhaps extremely large, such that $n^2+n+k$ takes on prime values for $0<n<j$ for some $j>39$?

Yes, below is the key result.
Theorem $ $ $\rm\ (x\!-\!\alpha)\:(x\!-\!\alpha') = x^2\! + x + k\ $ is prime  for $\rm\, 0 \le x \le k\!-\!2  \iff \mathbb Z[\alpha]\, $ is a PID
Hint $\, (\Rightarrow)\ $ Show all primes $\rm\ p \le \sqrt{n},\; n = 1\!-\!4k\ $ satisfy $\rm\ (n/p) = -1\ $ so none split/ramify
For proofs, see e.g. Cohn, Advanced Number Theory, pp. 155-156, or Ribenboim, My numbers, my friends, 5.7 p.108.  Note that both proofs employ the bound $\rm\ p < \sqrt{n}\ $ without explicitly mentioning that this is a consequence of the Minkowski bound - presumably assuming that is obvious to the reader based upon earlier results. Thus you'll need to read the prior sections on the Minkowski bound. Compare Stewart and Tall, Algebraic number theory and FLT, 3ed, Theorem 10.4 p.176 where the use of the Minkowski bound is mentioned explicitly.
Alternatively, see the self-contained paper [1] which proceeds a bit more simply, employing Dirichlet approximation to obtain a generalization of the Euclidean algorithm  (the Dedekind-Rabinowitsch-Hasse criterion for a PID). If memory serves correct, this is close to the approach originally employed by Rabinowitsch when he first published this theorem in 1913.
[1] Daniel Fendel, Prime-Producing Polynomials and Principal Ideal Domains,
Mathematics Magazine, Vol. 58, 4, 1985, 204-210
