On a conjecture concerning the number of solutions to $a^x+b^y=c^z$

TL;DR

This paper investigates the solutions to the exponential Diophantine equation a^x + b^y = c^z for coprime positive integers, providing restrictions on solutions when multiple solutions exist, thus supporting a related conjecture.

Contribution

The authors identify specific conditions and restrictions on triples of primes where multiple solutions occur, advancing understanding of the conjecture about the equation's solutions.

Findings

If multiple solutions exist, the triple must satisfy specific modular conditions.

Such triples are rare and must meet strong size restrictions, including c > 10^14.

The results support the conjecture that solutions are highly constrained.

Abstract

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{ a,b,c \} >1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples then, taking $a<b$, we must have $a=2$, $(b,c) \equiv (1,17)$, $(13,5)$, $(13, 17)$, or $(23, 17) \bmod 24$, and $(a,b,c)$ must satisfy further strong restrictions, including $c>10^{14}$. These results support a conjecture of the last two authors.