The problem "Unequally Yoked" looks for all integral solutions for a^b = b^a. But of course, there are a lot of non-integral solutions. If I graph the solution set where a and b are both greater than 0, I get two intersecting curves. Where do the two curves cross?
(In reply to Solution & Spoiler
Letting q=0 (and thus p=1) produces (1,1) as the common point.