Patterns Within the Markov Tree

TL;DR

This paper analyzes the structure of the Markov tree, discovering generating functions, Pell equations, and digit patterns that lead to resolving the Uniqueness Conjecture and understanding the distribution of Markov numbers.

Contribution

It introduces new sequence functions and Pell equations for all regions of the Markov tree, advancing the understanding of Markov triplets and their digit patterns.

Findings

Discovered generating functions for all triplets in the Markov tree.

Established a Pell equation governing Markov region numbers.

Resolved the Uniqueness Conjecture for Markov numbers.

Abstract

An analysis of the Markov tree is presented. Markov triplets, {x,R,z}, are the positive integer solutions to the Diophantine equation x2 + R2 + z2 = 3xRz. Inspired by patterns of the Fibonacci and Pell triplets in Region 1 and Region 2 of the tree, an investigation of interior regions of the Markov tree finds generating functions and sequence functions for all triplets of all regions. These sequence functions lead to the discovery of a Pell equation for the Markov region numbers along the edges of all regions. Analysis of this Pell equation leads to the resolution of the Uniqueness Conjecture. Further analysis using these sequence functions finds palindromic repeat cycles of the last digits of region numbers along the edges of all regions. Then, since all Markov numbers are the sum of the squares of two integers and again inspired by the patterns of the two unique squares which sum to form the region numbers of certain Fibonacci triplets in Region 1, an investigation of interior regions of the Markov tree finds generating functions and sequence functions for the two special square terms which sum to form the region numbers of the triplets along the edges of all regions. Further analysis using these sequence functions finds palindromic repeat cycles of the last digits of these two special square terms for all regions.