  Markov numbers are integers that appear in the solution triples of the
Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. 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.
  There is a natural map from the rational numbers between zero and one to the
Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's
book, Markov's theorem and 100 years of the uniqueness conjecture, that
determine an ordering on subsets of the Markov numbers based on their
corresponding rational.
  The proof relies on a relationship between Markov numbers and continuant
polynomials which originates in Frobenius' 1913 paper.
