On the ordering of the Markov numbers

Abstract

The Markov numbers are the positive integers that appear in the solutions of the equation x 2 + y 2 + z 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 − 8 7 and that it is smaller than the other if the slope is at most − 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".

On the monotonicity of the generalized Markov numbers

“…In this section, we review the Markov distance, generalized Markov numbers defined in [10], and some properties.…”

“…Every Markov number appears as the maximum of some Markov triple, The Markov Uniqueness Conjecture by Frobenius from 1913 asserts that each Markov number appears as the maximum of a unique Markov triple [5,1,10,6,16].…”

“…To study the ordering of the Markov numbers and the Uniqueness Conjecture, Lee-Li-Rabideau-Schiffler [10] introduced the Markov distance on the plane. By comparing the Markov tree and the Farey tree, one sees that the Markov numbers can be indexed by the rational numbers between zero and one, equivalently, the set {(q, p) ∈ Z 2 >0 | p < q, g.c.d.…”

Boundary slopes for the Markov ordering on relatively prime pairs

“…This is the result of two distinct normalizations for the Farey labelling λ F in the literature. In one, the root of the tree is incident to 0/1, 1/0 and 1/1, as in Figure 2 and [ Aig15 , LLRS20 ,McS21 ] (observe that this is natural by virtue of the integral homology Z 2 ), and in the other the root sees 0/1, 1/1, and 1/2, as in [ Foc97 , SV17 , SV19 ] (somewhat more natural when considering the holonomy of the modular torus in PSL(2, Z)). We've chosen a presentation of…”

“…While an initial draft of[ LLRS20 ] does not contain these conjectures, they can be found in a revised manuscript which was shared in private communication. This author thanks those authors for their openness.…”