A classification of Markoff-Fibonacci m-triples

TL;DR

This work classifies all Markoff $m$-triples whose components are Fibonacci numbers. By analyzing the sign of $m(a,b,c)=F(a)^2+F(b)^2+F(c)^2-3F(a)F(b)F(c)$ and distinguishing non-minimal from minimal cases, the authors show that non-minimal Fibonacci $m$-triples occur only for $m=2$ and have the form $(1,F(b),F(b+2))$ with even $b$, while minimal Fibonacci $m$-triples exist for infinitely many $m$, but are unique (up to order) except for $m=2$ and $m=21$, where there are two such triples. The proof combines Fibonacci inequalities, Vajda’s identity, and analytic bounds, supplemented by computational checks for large maximal elements, yielding a complete classification and revealing an infinite set of $m$ with exactly one Fibonacci $m$-triple. This sharpens understanding of Markoff-type equations in relation to Fibonacci sequences and their tree-structural roots.

Abstract

We classify all solution triples with Fibonacci components to the equation $a^2+b^2+c^2=3abc+m,$ 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.

Figures (1)

• Figure 1: Beginning of the Markoff $2$-tree with minimal $2$-triple $(1,1,3)$. The sequence of non-minimal Markoff-Fibonacci $2$-triples is represented in bold.

Theorems & Definitions (46)

• Theorem 1.1
• Lemma 2.1
• proof
• Remark 1
• Remark 2
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2