论文标题
关于马尔可夫号码的顺序
On the ordering of the Markov numbers
论文作者
论文摘要
马尔可夫号是出现在公式$ x^2+y^2+z^2 = 3xyz $的解决方案中的正整数。这些数字是数字理论的经典主题，并且在双曲几何形状，代数几何和组合学方面具有重要的分歧。 众所周知，马尔可夫数字可以在第一个象限中的晶格点$（q，p）$标记，其坐标为codrime的对角线。在本文中，我们考虑以下问题。给定两个晶格点，我们可以说哪个相关的马尔可夫数字更大？对这个问题的完整答案将解决弗罗贝尼乌斯在1913年提出的独特性猜想。我们根据连接两个晶格点的线段的斜率给出了部分答案。我们证明，如果坡度至少为$ - \ frac {8} {7} $，那么带有$ x $坐标的马尔可夫号大于另一个，并且如果坡度最多的斜率最多是$ \ frac {5} {4} {4} $，则它比另一个小。 作为特殊情况，即，当斜率等于0或1时，我们从Aigner的书“ Markov的定理和100年的独特性猜想”中获得了两个猜想的证明。
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 $-\frac{8}{7}$ 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".
