Counting Minimal Triples for a Generalized Markoff Equation

Abstract

If the generalized Markoff equation $a^2+b^2+c^2=3\mathit{\text{abc}}+m$ has a solution triple, then it has infinitely many solutions. For a positive integer m > 1, 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 the first or second element equal to a, and the set of fundamental solutions of $m-a^2$ by the form $x^2-3\mathit{\text{axy}}+y^2$. 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.

KEYWORDS:

• Markoff triples
• fundamental solutions
• generalized Markoff equation
• binary quadratic forms

Acknowledgments

We would like to thank the referees for an extremely careful review, providing us with several relevant and useful comments.