Congruence and Uniqueness of Certain Markoff Numbers

An elementary proof of uniqueness of Markoff numbers which are prime powers

We present a very elementary proof of the uniqueness of Markoff numbers which are prime powers or twice prime powers, in the sense that it uses neither algebraic number theory nor hyperbolic geometry.

Nonabelian level structures, Nielsen equivalence, and Markoff triples

In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show that for all but finitely many primes $p$, the group of Markoff automorphisms acts transitively on the nonzero $\mathbb{F}_p$-points of the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$. This yield…

On the uniqueness conjecture for Markoff triples

This paper has been withdrawn by author due to an error in the proof.

关于马尔可夫图的陈氏定理的一个新证明

A new proof of Chen's theorem for Markoff graphs

本研究针对模$p$马尔可夫图相关的陈氏定理的马尔可夫推论，给出了一种替代证明，该推论与Baragar关于模$p$马尔可夫图对绝大多数素数连通的猜想相关。

A simple proof of the Markoff conjecture for prime powers

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.

关于某些替换相关的拉格朗日常数的序关系与单射性

On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions

本文研究参数化替换生成的连分数的拉格朗日常数，建立序关系并证明在 $a\ge2$ 时映射的单射性，类比马尔可夫唯一性猜想，并提出关于一般 $b>a$ 的序关系猜想。