On the ordering of the Markov numbers

Abstract: The Markov numbers are the positive integers that appear in the solutions of the equation $x<sup>2+y<sup>2+z<sup>2=3xyz$. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points $(q,p)$ in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. We give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater $x$-coordinate is larger than the other if the slope is at least $-\frac{8}{7}$ and that it is smaller than the other if the slope is at most $-\frac{5}{4}$. As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book "Markov's theorem and 100 years of the uniqueness conjecture".