Geometric Aspects to Diophantine Equations of the Form $x^2 + zxy + y^2 = M$ and $z$-Rings

TL;DR

This paper explores solutions to specific quadratic Diophantine equations using novel algebraic structures called z-rings, analyzing their properties, solutions, and prime classifications to understand solution existence and count.

Contribution

Introduces z-rings and their extensions to study solutions of quadratic Diophantine equations, including prime classifications and solution counts, especially for the case z=3.

Findings

z-rings are useful tools for analyzing Diophantine solutions

Most z-rings are not unique factorization domains

Number of solutions depends on the factorization properties of M

Abstract

In the following we consider Diophantine equations of the form $x^2+ zxy + y^2 = M$ for given $M,z \in \mathbb{Z}$ and discuss the number of its (primitive) solutions as well as the construction of them. To reach this goal we introduce $z$-rings which turn out to be a useful tool to investigate these Diophantine equations. Moreover, we will extend these rings and study the algebraic curves defined by them on a plane by methods inspired by the complex plane. Then we define the so called subbranches which are bounded and connected parts of the algebraic curves containing a representative of each solution of the Diophantine equations with respect to association in $z$-rings. With the help of them we can easily prove the existence or non-existence of solutions to the above Diophantine equations. Then we divide the integer primes with respect to the different $z$-rings into two main categories, i.e. the regular and irregular elements. We show that the irregular elements are prime in the corresponding $z$-rings and we identify that most of the $z$-rings cannot be unique factorization domains. We determine the number of positive, primitive solutions of the above Diophantine equation if $M \in \mathbb{N}$ is a product of irregular elements in the corresponding $z$-ring for $z \in \mathbb{N}$. We also give an overview how many primitive and non-primitive solutions in a given quadrant we can find for arbitrary $M,z \in \mathbb{Z}$, especially, if $M$ is a power of any irregular element. Furthermore, we consider the case $z = 3$, determine the regular and irregular elements as well as the number of positive, primitive solutions of the Diophantine equation $x^2 + 3xy + y^2 = M$ depending on $M \in \mathbb{N}$.