2021/01/09 by McShane, Greg · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2101.03316
Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic x2 + y2 + z2 - 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.