Markoff m-Triples with k-Fibonacci Components

Abstract

We classify all solution triples with k-Fibonacci components to the equation x2%2By2%2Bz2=3xyz%2Bm, 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%2B2)), 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%2B3, where (F3(a),F3(b),F3(a%2Bb)) and (F3(a%2B1),F3(b−1),F3(a%2Bb)) share the same m.We classify all solution triples with k-Fibonacci components to the equation x2%2By2%2Bz2=3xyz%2Bm, 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%2B2)), 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%2B3, where (F3(a),F3(b),F3(a%2Bb)) and (F3(a%2B1),F3(b−1),F3(a%2Bb)) share the same m.