Positive rational solutions to $x^y = y^{mx}$: a number-theoretic   excursion
  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. 
Michael A. Bennett (1,2), Bruce Reznick (1) ((1) University of   Illinois at Urbana-Champaign, (2) University of British Col...