A classification of Markoff-Fibonacci m-triples

Abstract

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.

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.