Markoff $m$-triples with $k$-Fibonacci components

TL;DR

This paper classifies solutions to a specific Diophantine equation involving $k$-Fibonacci components, revealing unique or special solution families depending on the parameter $m$ and the Fibonacci index $k$.

Contribution

It provides a complete classification of solutions with $k$-Fibonacci components for the equation, identifying unique solutions and special cases for different $m$ and $k$ values.

Findings

For $m=8$, Markoff triples with Pell components are characterized.

Most $m$ values admit at most one such solution, with exceptions.

Special solution families occur when $k=3$, with specific parity and size conditions.

Abstract

We classify all solution triples with $k$-Fibonacci components to the equation $x^2+y^2+z^2=3xyz+m,$ where $m$ is a positive integer and $k\geq 2$. As a result, for $m=8$, we have the Markoff triples with Pell components $(F_2(2), F_2(2n), F_2(2n+2))$, for $n\geq 1$. For all other $m$ there exists at most one such ordered triple, except when $k=3,$ $a$ is odd, $b$ is even and $b\geq a+3$, where $(F_3(a),F_3(b),F_3(a+b))$ and $(F_3(a+1),F_3(b-1),F_3(a+b))$ share the same $m$.