Continued fractions and orderings on the Markov numbers

TL;DR

This paper proves two longstanding conjectures about the ordering of Markov numbers based on their associated rational numbers, using continued fractions and continuant polynomials.

Contribution

It establishes the orderings on Markov numbers related to rational numbers, confirming two major conjectures in the field.

Findings

Proved Markov's theorem on the ordering of Markov numbers.

Confirmed the 100-year-old uniqueness conjecture for Markov numbers.

Linked Markov numbers to continuant polynomials via Frobenius's work.

Abstract

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper.