Minimal triples for a generalized Markoff equation

TL;DR

This paper studies solutions to a generalized Markoff equation, introducing minimal triples as a finite generating set, and provides formulas and algorithms to count and find these triples using quadratic forms and fundamental solutions.

Contribution

It establishes a correspondence between minimal triples and fundamental solutions of quadratic forms, enabling efficient computation and characterization of solutions.

Findings

Infinite solutions are generated by finitely many minimal triples.

A formula relates minimal triples to fundamental solutions of quadratic forms.

Criteria and formulas for minimal triples of the form (1, b, c) are provided.

Abstract

For a positive integer $m>1$, if the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ has a solution triple, then it has infinitely many solutions. We show that all positive solution triples are generated by a finite set of triples that we call minimal triples. We exhibit a correspondence between the set of minimal triples with first or second element equal to $a$, and the set of fundamental solutions of $m-a^2$ by the form $x^2-3axy+y^2$. This gives us a formula for the number of minimal triples in terms of fundamental solutions, and thus a way to calculate minimal triples using composition and reduction of binary quadratic forms, for which there are efficient algorithms. Additionally, using the above correspondence we also give a criterion for the existence of minimal triples of the form $(1, b, c)$, and present a formula for the number of such minimal triples.