Convexity and Aigner's Conjectures

Abstract

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic $$x^2+y^2+z^2-3xyz=0.$$ 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.

9 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2003.05967

References in corpus (1)

• Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space