General sharp bounds for the number of solutions to purely exponential equations with three terms

TL;DR

This paper establishes sharp bounds on the number of solutions to the exponential equation a^x + b^y = c^z, showing that for fixed c, only finitely many pairs (a, b) can have multiple solutions, extending previous results in number theory.

Contribution

It proves that for any fixed c, there is at most one solution to a^x + b^y = c^z for all but finitely many pairs (a, b), generalizing prior work on related exponential equations.

Findings

For fixed c, at most one solution exists for all but finitely many (a, b).

The proof uses p-adic methods and deep Diophantine approximation theorems.

Complete description of solutions to certain polynomial-exponential systems included.

Abstract

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. In this paper, we prove that for any fixed $c$ there is at most one solution to the equation, except for only finitely many pairs of $a$ and $b.$ This is regarded as a 3-variable generalization of the result of Miyazaki and Pink [T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two unknowns, III, arXiv:2403.20037 (accepted for publication in Math. Proc. Cambridge Philos. Soc.)] which asserts that for any fixed positive integer $a$ there are only finitely many pairs of coprime positive integers $b$ and $c$ with $b>1$ such that the Pillai's type equation $a^x-b^y=c$ has more than one solution in positive integers $x$ and $y$. The proof of our result is based on a certain $p$-adic idea of Miyazaki and Pink and relies on many deep theorems on the theory of Diophantine approximation, and it also includes the complete description of solutions to some interesting system of simultaneous polynomial-exponential equations. We also discuss how effectively exceptional pairs of $a$ and $b$ on our result for each $c$ can be determined.