Positive rational solutions to $x^y=y^{mx}$: a number-theoretic excursion

Abstract

We 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 pages