Well, it's nice to know I'm not the only one who starts flame wars.

I have an idea. Why doesn't everyone start responding to the question that was originally posed. Even if it's not worded quite the way it is needed for precise mathematics, to me it's clear what the O.P. intended:

Find prime forms with few primes but be reasonable about it by not allowing such outlandish forms that they become large very quickly. In other words, don't make it some stupid form that gets large so quickly that while it cannot be proven to never have a prime, it has such a miniscule chance that it is uninteresting for mathematical discussion.

RDS, I realize this is still very vague. I'm only stating what it APPEARS that the O.P. intended. I claim no understanding of math higher than high school calculus and freshman level algebra.

This appears to be an interesting topic to expound upon. Let's set up some parameters and rules about what constitue an 'interesting rare prime form', i.e. not something stupid like k*2^(n^n^n^n^n^n)-1. Then go from there. The O.P. gave us a starting point with some interesting forms. We just need to frame the parameters for 'rare primes' of other forms.

Gary