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

TL;DR

This paper establishes an upper bound of two solutions for the exponential equation involving coprime integers and odd integers, and shows that infinitely many such triples achieve this maximum.

Contribution

It proves a new upper bound of two solutions for the equation and demonstrates the existence of infinitely many triples attaining this bound.

Findings

Maximum of two solutions for the equation under given conditions

Existence of infinitely many triples with exactly two solutions

Extension of previous bounds in exponential Diophantine equations

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.