Number of solutions to $a^x+b^y=c^z$, A Shorter Version

Abstract

For relatively prime integers $a$ and $b$ both greater than one and odd integer $c$, there are at most two solutions in positive integers $(x,y,z)$ to the equation $a^x+b^y=c^z$. There are an infinite number of $(a,b,c)$ giving exactly two solutions.

arXiv commentadded explanatory comments, fixed minor errors