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

TL;DR

This paper provides a simple, elementary proof demonstrating the uniqueness of Markoff numbers that are prime powers or twice prime powers, avoiding complex algebraic or geometric methods.

Contribution

It introduces a novel elementary proof for the uniqueness of certain Markoff numbers, bypassing traditional advanced mathematical tools.

Findings

Proves uniqueness of Markoff numbers that are prime powers or twice prime powers

Avoids algebraic number theory and hyperbolic geometry in the proof

Simplifies understanding of Markoff number properties

Abstract

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.