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

Abstract

Let $a,b,c$ be fixed coprime positive integers with $\min \{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max \{a,b,c\}\ge 10^{62}$, then the equation has at most two positive integer solutions $(x,y,z)$.