Markoff m-Triples with k-Fibonacci Components

Abstract

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.

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.