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arXiv · 2101.03316

Convexity and Aigner's Conjectures

Greg McShane
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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.

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Greg McShane. 2021-01-09. Convexity and Aigner's Conjectures. https://arxiv.org/abs/2101.03316

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