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

Chris Busenhart

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 $M \in \mathbb{N}$.

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

Abstract

In the following we consider Diophantine equations of the form for given and discuss the number of its (primitive) solutions as well as the construction of them. To reach this goal we introduce -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 -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 -rings into two main categories, i.e. the regular and irregular elements. We show that the irregular elements are prime in the corresponding -rings and we identify that most of the -rings cannot be unique factorization domains. We determine the number of positive, primitive solutions of the above Diophantine equation if is a product of irregular elements in the corresponding -ring for . We also give an overview how many primitive and non-primitive solutions in a given quadrant we can find for arbitrary , especially, if is a power of any irregular element. Furthermore, we consider the case , determine the regular and irregular elements as well as the number of positive, primitive solutions of the Diophantine equation depending on .

Paper Structure

This paper contains 16 sections, 45 theorems, 229 equations, 13 figures.

Table of Contents

  1. Introduction and motivation
  2. Construction of $z$-rings
  3. Conjugate, norm, real and imaginary parts
  4. $z$-rings and their application
  5. Extension of $z$-rings
  6. The functions $\boldsymbol{\mathrm{I}}_{+},\boldsymbol{\mathrm{I}}_{-}$ and their properties
  7. Partition and local solution theorems
  8. Unit group of $\mathbb{Z}[i_z]$
  9. Primes in $\mathbb{Z}$ with respect to $\mathbb{Z}[i_z]$
  10. The irregular elements in $\mathbb{Z}[i_z]$
  11. Special elements in $\mathbb{Z}[i_z]$
  12. Many $z$-rings are not unique factorization domains
  13. Positive, primitive solutions of the Diophantine equation $x^2 + zxy + y^2 = M$ for $M$ being a product of irregular primes
  14. The general case
  15. The ring $\mathbb{Z}[i_3]$ and solutions to $x^2 + 3xy + y^2 = M$
  16. ...and 1 more sections

Key Result

Proposition 2.2

$\boldsymbol{\mathrm{R}}_{z}$ is a commutative and unitary ring for all $z \in \mathbb{Z}$.

Figures (13)

  • Figure 1: Geometric interpretation of conjugation
  • Figure 2: Some level sets in $\mathbb{R} \times \mathbb{R}i_4$
  • Figure 3: Units of the Eisenstein integers
  • Figure 4: Multiplication with the imaginary unit
  • Figure 5: Units of $\mathbb{Z}[i_{-4}]$ on $S_{1} \subseteq \mathbb{R}[i_{-4}]$
  • ...and 8 more figures

Theorems & Definitions (121)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 111 more