  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.
