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 e.g.)

This problem seems to be pretty well wrapped up.  Kudos so far to e.g., Richard, Larry.

One observation:
The solution method that was popular in this thread involves introducing a  value q, and then evaluating x and y when q = 1.  x and y are undefined when q = 1, so a limit was taken.  This is very clever, and completely correct.

Note that this does not imply, however, that there is anything odd about the "interesting part" of the curve.  At the risk of being wrong about my own problem, I think the curve that excludes x = y but includes the intersection point is continuous and differentiable at all points.  I haven't done the math, but it looks regular enough.

Which reminds me.  I wasn't sure how to classify this problem.  I stuck it in calculus, because most of the solution methods I thought of involved limits or derivatives.  Any comments on the appropriate classification?

Thanks, Steve

Edited on April 27, 2005, 1:34 pm

Posted by Steve Herman on 2005-04-27 13:30:15