Branches of Markoff $m$-triples with two $k$-Fibonacci components

TL;DR

This paper classifies certain infinite solutions to a generalized Markoff equation involving $k$-Fibonacci numbers, revealing their structure and distribution among specific branches and trees.

Contribution

It provides a complete classification of Markoff $m$-triples with two $k$-Fibonacci components not originating from Markoff trees and describes their branch structure.

Findings

Classified Markoff $m$-triples with two $k$-Fibonacci components.

Proved that infinite paths are contained in specific branches.

Identified the distribution of branches among $2r$ trees.

Abstract

We study infinite paths of Markoff $m$-triples, that is, solutions to the generalised Markoff equation \[ x^2+y^2+z^2=3xyz+m, \] with $m>0$, with at least two $k$-Fibonacci components. First, we obtain a complete classification of Markoff $m$-triples whose last two entries are $k$-Fibonacci numbers and that are not roots of any Markoff trees. We then prove that every such infinite path is contained in a branch, starting at a triple of the form \[ \left(\frac{F_k(4r)}{3F_k(2r)},\,F_k(\ell+2r),\,F_k(\ell+4r)\right), \] where $r$ is an odd integer, $\ell\in\{1,2,\ldots, 2r\}$ and $3\nmid k$. These branches are distributed among exactly $2r$ distinct trees.