A Upper Bound for the Number of Solutions of Ternary Purely Exponential Diophantine Equations

TL;DR

This paper establishes an upper bound on the number of solutions to the exponential Diophantine equation a^x + b^y = c^z for coprime positive integers, showing at most three solutions when the maximum base exceeds 5×10^27.

Contribution

It provides a new upper bound on the number of solutions for the equation using a combination of Gel'fond-Baker method and elementary techniques.

Findings

At most three solutions when max{a,b,c} > 5×10^27

Combines Gel'fond-Baker method with elementary approaches

Establishes a significant bound for large bases in the equation

Abstract

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 $(x,y,z)$.