An upper bound for the number of solutions of the exponential diophantine equation $a^x + b^y = c^z$

Abstract

Let $a$, $b$, $c$ be coprime positive integers which are power free. In this paper we prove that if $2\nmid c$, then the equation $a^x + b^y = c^z$ has at most $2^{\omega(c)+1}$ positive integer solutions $(x,y,z)$, where $\omega(c)$ is the number of distinct prime factors of $c$. Moreover, all solutions $(x,y,z)$ satisfy $z < 2ab \log(2eab) / \pi$.