Generalization of Markov Diophantine equation via generalized cluster algebra

TL;DR

This paper introduces algorithms to find all positive integer solutions to two classes of Diophantine equations, extending the classical Markov equation and related equations using generalized cluster algebra structures.

Contribution

It provides novel algorithms for enumerating solutions and explores the underlying generalized cluster algebra structures of these equations.

Findings

Algorithms successfully enumerate all positive solutions.

Identification of generalized cluster algebra structures behind the equations.

Extension of classical Markov equation to broader classes.

Abstract

In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+ky^2z^2+2xz^2+2xy^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.