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

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.