Counting Minimal Triples for a Generalized Markoff Equation

A. Srinivasan, L.A. Calvo

Experimental Mathematics Vol. 34, nº. 2, pp. 200 - 211

Original summary:

If the generalized Markoff equation a²+b²+c²=3abc+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−a² by the form x²−3axy+y². 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.

English summary:

If the generalized Markoff equation a2+b2+c2=3abc+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−a2 by the form x2−3axy+y2. 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.

Spanish layman's summary:

En este trabajo se estudia una generalización de la ecuación de Markoff y se propone un procedimiento para contar sus triples minimales. Estos triples se relacionan con el número de árboles distintos en los que se distribuyen las soluciones. El artículo aporta nuevos resultados y métodos combinatorios y de teoría de números que permiten comprender mejor la estructura y el crecimiento de estas familias de soluciones diofánticas.

English layman's summary:

We study a generalization of the Markoff equation and introduce a procedure to count its minimal triples. These triples are linked to the number of distinct trees that organize the solutions. The paper provides new results and combinatorial and number-theoretic methods that improve the understanding of the structure and growth of these families of Diophantine solutions.

Keywords: Markoff triples; fundamental solutions; generalized Markoff equation; binary quadratic forms

JCR-JIF Impact Factor and WoS quartile: 0,500 - Q3 (2025)

DOI reference: https://doi.org/10.1080/10586458.2024.2338279

Published on paper: April 2025.

Published on-line: April 2024.

Citation:
A. Srinivasan, L.A. Calvo, "Counting Minimal Triples for a Generalized Markoff Equation", Experimental Mathematics, Vol. 34, nº. 2, pp. 200 - 211, April 2025. [Online: April 2024] doi: 10.1080/10586458.2024.2338279