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.

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马尔可夫数的连分数和排序

马尔可夫数是出现在丢番图方程的解三元组中的整数，$x^2+y^2+z^2=3xyz$，称为马尔可夫方程。作为数论中的一个经典主题，这些数字与数学的许多领域有关，例如组合学、双曲几何、近似理论和聚类代数。从零到一之间的有理数到马尔可夫数有一个自然映射。在本文中，我们证明了在 Martin Aigner 的书中看到的两个猜想，马尔可夫定理和 100 年唯一性猜想，它们根据马尔可夫数的相应有理数确定对马尔可夫数子集的排序。该证明使用了环面的簇代数，一次穿刺和由此产生的关于连分数的猜想的重新表述。