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

2002/09/06 by Michael A. Bennett, Bennett, Michael A., Bruce Reznick +1
Mathematics · #11D41 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11D41

paper · pdf · doi:10.48550/arxiv.math/0209072

12 pages

arxiv created 2002/09/06 · openalex publication_date 2002/09/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• An Ancient Diophantine Equation with applications to Numerical Curios and Geometric Series
• Rationality and power
• Computing rational solutions to Ax^p+By^p+Cz^p=0
• On the equation x²-2^m=yⁿ
• The number of solutions to y²=px(Ax²+2)