The sum of Lagrange numbers

TL;DR

This paper links McShane's identity and Schmutz's work to establish an equivalence between the Markov Uniqueness Conjecture and a specific infinite sum involving Lagrange numbers, revealing a deep connection in hyperbolic geometry and number theory.

Contribution

It introduces a novel identity connecting Lagrange numbers with the Markov Uniqueness Conjecture, bridging hyperbolic geometry and number theory in a new way.

Findings

Establishes the equivalence of MUC with a specific infinite sum involving Lagrange numbers.

Derives a new identity linking hyperbolic geometry and number theory.

Provides a mathematical connection between McShane's identity and the MUC.

Abstract

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4 - \varphi - \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $\varphi=\frac{1+\sqrt5}2$ is the golden ratio.