On the ordering of the Markov numbers

TL;DR

This paper investigates the ordering of Markov numbers associated with lattice points, providing partial criteria based on slopes and proving two conjectures related to their ordering in special cases.

Contribution

It offers a partial solution to the long-standing Frobenius conjecture on Markov number ordering, using geometric slope conditions and confirming specific conjectures for particular cases.

Findings

Markov number ordering depends on the slope between lattice points.

If slope ≥ -8/7, the Markov number with larger x is greater.

If slope ≤ -5/4, the Markov number with larger x is smaller.

Abstract

The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points $(q,p)$ in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. We give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater $x$-coordinate is larger than the other if the slope is at least $-\frac{8}{7}$ and that it is smaller than the other if the slope is at most $-\frac{5}{4}$. As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book "Markov's theorem and 100 years of the uniqueness conjecture".