马氏数的连分数与序
Continued fractions and orderings on the Markov numbers
摘要：
马尔可夫数是出现在丢番图方程（$x^2+y^2+z^2=3xyz$，称为马尔可夫方程）的解三元组中的整数。一个经典的主题在数论中，这些数字涉及到许多领域的数学，如组合数学，双曲几何，逼近论和集群代数。从0到1的有理数到马尔可夫数之间存在一个自然映射。在本文中，我们证明了马丁·艾格纳的书，马尔可夫定理和100年的唯一性猜想，这决定了基于其相应的理性的马尔可夫数的子集的顺序中看到的两个命题。证明依赖于马尔可夫数和连续多项式之间的关系，这起源于弗罗贝纽斯1913年的论文。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
