On the Markoff Equation

TL;DR

This paper proves that in Markoff triples, the largest element uniquely determines the other two, solving a century-old open problem by analyzing the equation through 3x3 matrices.

Contribution

It introduces a matrix framework to analyze Markoff triples, establishing the unique determination of the other elements by the largest one.

Findings

Largest element of Markoff triple uniquely determines the other two

Matrix representation clarifies structure of Markoff triples

Answers a 100-year-old open question

Abstract

A triple (a, b, c) of positive integers is called a Markoff triple iff it satisfies the Diophantine equation a2+b2+c2=abc . Recasting the Markoff tree, whose vertices are Markoff triples, in the framework of integral upper triangular 3x3 matrices, it will be shown that the largest member of such a triple determines the other two uniquely. This answers a question which has been open for 100 years. The solution of this problem will be obtained in the course of a broader investigation of the Markoff equation by means of 3x3 matrices.