Orderings of k-Markov Numbers

TL;DR

This paper extends known conjectures about Markov numbers to the broader class of k-Markov numbers, showing they satisfy similar uniqueness properties.

Contribution

It demonstrates that k-Markov numbers adhere to Aigner's conjectures, generalizing properties known for classical Markov numbers.

Findings

k-Markov numbers satisfy Aigner's conjectures

Extension of Markov number properties to k-Markov numbers

Supports the unicity conjecture in a broader context

Abstract

The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures.