A Upper Bound for the Number of Solutions of Ternary Purely Exponential Diophantine Equations

Yongzhong Hu; Maohua Le

paper

A Upper Bound for the Number of Solutions of Ternary Purely Exponential Diophantine Equations

arXiv:1702.03424

Abstract

Let $a, b, c$ be fixed coprime positive integers with $min {a, b, c} > 1$ . In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $max {a, b, c} > 5 \times 1 0^{27}$ , then the equation $a^{x} + b^{y} = c^{z}$ has at most three positive integer solutions $(x, y, z)$ .