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

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 $k\ge 2$. 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 $n\ge 1$. For all other $m$ there exists at most one such ordered triple, except when $k=3,$ $a$ is odd, $b$ is even and $b\ge a+3$, 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 $m$.

15 pages, 1 figure