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?

One solution curve is obviously x=y. For the other, let's suppose a=p and b=p^q. Since a^b=b^a, then p^(p^q)= (p^q)^p= p^qp, so we have p^q=pq, or p^(q-1)=q. For this to be an equality, we need p=q^(1/(q-1)), and we got another solution curve: x=q^(1/(q-1)) and y=q^(q/(q-1)).

For q=1, this produces the first solution curve.

Posted by e.g. on 2005-04-24 16:41:23