Branches of markoff m-triples with two k-ibonacci components

D. Alfaya, L.A. Calvo, P.J. Cazorla, J. Rodrigo, A. Srinivasan

Summary:

We study infinite paths of Markoff m-triples, that is, solutions to the generalised Markoff equation
x2 + y2 + z2 = 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. [...]

Spanish layman's summary:

El artículo estudia una generalización de la ecuación de Markoff, donde al menos dos variables son números k-Fibonacci. Se demuestra que todas las soluciones infinitas siguen una estructura específica llamada “rama principal” y que no existen otros comportamientos posibles fuera de este patrón.

English layman's summary:

The article studies a generalization of the Markoff equation where at least two variables are k-Fibonacci numbers. It proves that all infinite solutions follow a specific structure called the “main branch” and that no other behaviors are possible outside this pattern.

Keywords: Markoff triples, generalised Markoff equation, generalised Fibonacci solutions.

Registration date: 07-abr-2026