An upper bound for the number of solutions of ternary purely exponential diophantine equations

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}>5×1027, then the equation ax+by=cz has at most three positive integer solutions (x,y,z).