Markoff Equation and Nilpotent Matrices

TL;DR

This paper demonstrates that in the Markoff equation, the largest element of a Markoff triple uniquely determines the other two when represented as integral upper triangular matrices, resolving a century-old open question.

Contribution

It recasts the Markoff triples within the framework of upper triangular matrices to prove the uniqueness of the other two elements based on the largest one.

Findings

Largest member of a Markoff triple determines the other two uniquely.

Recasting Markoff triples as upper triangular matrices provides new insights.

Answers an open question from nearly 100 years ago.

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 intergral 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 almost 100 years.