  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
$\max\{a,b,c\}\geq 10^{62}$, then the equation has at most two positive integer
solutions $(x,y,z)$.
