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

Michael A. Bennett·Bruce Reznick
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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.

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Michael A. Bennett, Bruce Reznick. 2002-09-06. Positive rational solutions to $x^y = y^{mx}$: a number-theoretic excursion. https://arxiv.org/abs/math/0209072

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