A classification of Markoff-Fibonacci m-triples

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

Fibonacci Quarterly Vol. 63, nº. 3, pp. 517 - 541

Summary:

We classify all solution triples with Fibonacci components to the equation a2 +b2 + c2 = 3abc + m, for positive m. We show that for m = 2 they are precisely (1, F(b), F(b + 2)), with even b; for m = 21, there exist exactly two Fibonacci solutions (1, 2, 8) and (2, 2, 13) and for any other m there exists at most one Fibonacci solution, which, in case it exists, is always minimal (i.e. it is a root of a Markoff tree). Moreover, we show that there is an infinite number of values of m admitting exactly one such solution.

Keywords: Markoff triples; generalized Markoff equation; Fibonacci solutions

JCR-JIF Impact Factor and WoS quartile: 0,400 - Q4 (2024)

DOI reference: https://doi.org/10.1080/00150517.2025.2481607

Published on paper: 2025.

Published on-line: July 2025.

Citation:
D. Alfaya, L.A. Calvo, A. Martínez de Guinea García, J. Rodrigo, A. Srinivasan, "A classification of Markoff-Fibonacci m-triples", Fibonacci Quarterly, Vol. 63, nº. 3, pp. 517 - 541, 2025. [Online: July 2025] doi: 10.1080/00150517.2025.2481607