MARKOFF m-TRIPLES WITH k-FIBONACCI COMPONENTS

Abstract

https://doi.org/10.48550/arXiv.2409.13885

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 ≥ 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.