Markov triples with two Fibonacci components

Abstract

In this paper, we prove that there are at most finitely many pairs of Fibonacci numbers $(x,y)=(F_m,F_n)$ with the property that $m\le n$ and the pair $(m,n)\notin \{(1,2r-1),(1,2),(2,2r+1),(2r+1,2r+3):r\ge 1\}$ such that $(x,y,z)$ is a Markov triple for some integer $z$.