A note on ternary purely exponential diophantine equations
Volume 171 / 2015
Acta Arithmetica 171 (2015), 173-182
MSC: Primary 11D61.
DOI: 10.4064/aa171-2-4
Abstract
Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$, and let $m=\max \{a,b,c\}$. Using the Gel'fond–Baker method, we prove that all positive integer solutions $(x,y,z)$ of the equation $a^x+b^y=c^z$ satisfy $\max \{x,y,z\}<155000(\log m)^3$. Moreover, using that result, we prove that if $a,b,c$ satisfy certain divisibility conditions and $m$ is large enough, then the equation has at most one solution $(x,y,z)$ with $\min\{x,y,z\}>1$.
