Diophantine equations via cluster transformations

TL;DR

This paper explores a new approach to solving a variant of the Markov equation using cluster transformations inspired by cluster algebra theory, revealing how solutions can be generated from initial solutions.

Contribution

It introduces a novel variant of the Markov equation and demonstrates that all solutions can be obtained through cluster transformations, linking algebraic solutions to cluster algebra structures.

Findings

Solutions generated from initial solutions via cluster transformations

Connection between Markov equations and cluster algebra theory

All natural solutions arise from initial solutions through transformations

Abstract

Motivated by Fomin and Zelevinsky's theory of cluster algebras we introduce a variant of the Markov equation; we show that all natural solutions of the equation arise from an initial solution by cluster transformations.