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) = (Fm, Fn) with the property that m ≤ n and the pair (m,n) 6∈ {(1, 2r− 1), (1, 2), (2, 2r+ 1), (2r+ 1, 2r+ 3) : r ≥ 1} such that (x, y, z) is a Markov triple for some integer z. Mathematics Subject Classification (2010). Primary: 11B39; Secondary: 11D61.