Convexity and Aigner's Conjectures

TL;DR

This paper provides a new unified proof of conjectures related to Markov numbers, connecting number theory, geometry, and algebra through geodesic lengths on a punctured torus.

Contribution

It introduces a novel proof approach linking Markov numbers to geodesic lengths, resolving longstanding conjectures in number theory.

Findings

Proof of certain Markov conjectures established

Connection between Markov numbers and geodesic lengths demonstrated

Unified approach simplifies understanding of Markov number properties

Abstract

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic $$x^2 + y^2 + z^2 - 3x y z = 0.$$ 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. One can associate to each a positive rational number a Markov number in a natural way. We give a new unified proof of certain conjectures from Martin Aigner's book, Markov's Theorem and 100 Years of the Uniqueness Conjecture. Our proof relies on a relationship between Markov numbers and the lengths of closed simple geodesics on the punctured torus discovered by H. Cohn.