I have been struggling to solve this exercise but with no result:
$$x^y-y^x=xy^2-19,$$ $x,y\in{\mathbb Z}$
I have started to think it has no solutions at all. I have no idea how to solve it so I was wondering if anyone could be so kind to help?
Edit, William C. Jagy
Graphed on actual paper. In the first quadrant, the zero set is two curves, one with vertical and horizontal asymptotes, one with vertical and slanted asymptotes. It follows, easily enough, that there are finitely many integer solutions in the first quadrant.
