Markoff m-Triples with k-Fibonacci Components

D. Alfaya, L.A. Calvo, A. Martínez de Guinea García, J. Rodrigo, A. Srinivasan

Mediterranean Journal of Mathematics Vol. 22, nº. 3, pp. 76-1 - 76-19

Summary:

We classify all solution triples with k-Fibonacci components to the equation x²+y²+z²=3xyz+m, where m is a positive integer and k≥2. As a result, for m=8, we have the Markoff triples with Pell components (F₂(2),F₂(2n),F₂(2n+2)), for n≥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≥a+3, where (F₃(a),F₃(b),F₃(a+b)) and (F₃(a+1),F₃(b−1),F₃(a+b)) share the same m.

Spanish layman's summary:

Clasificamos todas las soluciones con componentes k-Fibonacci de la ecuación x²+y²+z²=3xyz+m para m≥0 y k≥2. Para m=8 tenemos las ternas de Markoff-Pell (F₂(2), F₂(2n), F₂(2n+2)). Para otros m hay a lo sumo una terna ordenada de este tipo, excepto algunos casos para k=3 también clasificados.

English layman's summary:

We classify all solutions with k-Fibonacci components to the equation x²+y²+z²=3xyz+m, with m≥0 and k≥2. For m = 8 we have the Markoff-Pell triples (F₂(2), F₂(2n), F₂(2n+2)). For all other m there is at most one such ordered triple, except in some cases when k = 3, which are also classified.

Keywords: Markoff triples, generalized Markoff equation, k-Fibonacci numbers, Markoff tree.

JCR Impact Factor and WoS quartile: 1,100 - Q1 (2023)

DOI reference: https://doi.org/10.1007/s00009-025-02845-y

Published on paper: May 2025.

Published on-line: April 2025.

Citation:
D. Alfaya, L.A. Calvo, A. Martínez de Guinea García, J. Rodrigo, A. Srinivasan, Markoff m-Triples with k-Fibonacci Components. Mediterranean Journal of Mathematics. Vol. 22, nº. 3, pp. 76-1 - 76-19, May 2025. [Online: April 2025]