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On a conjecture concerning the number of solutions to $a^x+b^y=c^z$ (2206.14032v2)

Published 28 Jun 2022 in math.NT

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 $ax + by = cz$. 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.

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