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 re: Solution & Spoiler by Federico Kereki)

e. g.'s solution amounts to introducing a parameter q that ranges over the open interval (0,infinity) and relates a to b by b=qa and then figuring out in terms of q what a^b=b^a means. The result is

a=q^(1/(q-1)) and b=qa.

This gives a curve parameterized by q. When a=b, q=1, so it is q=1, not q=0, that is the parameter value of interest.  However, q=1 presents a small problem in the formulas.  It turns out that a limit must be taken if we are to get a result. Taaking the limit will give a=b=e=2.718... .

Posted by Richard on 2005-04-24 21:48:56