  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 $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$
has at most three positive integer solutions $(x,y,z)$.
