2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64875We give a light-hearted look at the topic in the title. Most solutions are in semi-boring parametric families, but some are sporadic and come from Thue equations. One amusing example is m = 2, x = (4/5)^128, y = (4/5)^125.12 pagesNumber Theory11D41Positive rational solutions to $x^y = y^{mx}$: a number-theoretic excursiontext