No 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".

For cluster algebras from surfaces, there is a known formula for cluster variables and F -polynomials in terms of the perfect matchings of snake graphs. If the cluster algebra has trivial coefficients, there is also a known formula for cluster variables in terms of continued fractions. In this paper, we extend this result to cluster algebras with principal coefficients by producing a formula for the F -polynomials in terms of continued fractions.2000 Mathematics Subject Classification. Primary: 13F60, Secondary: 11A55 and 30B70.

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 uses the cluster algebra of the torus with one puncture and a resulting reformulation of the conjectures in terms of continued fractions. The key step is to analyze the difference in the numerator of a continued fraction when an operation is applied to its entries.

No abstract