2024/11/01 by Chris Busenhart, Busenhart, Chris
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2411.00649
openalex publication_date 2024/11/01 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28
In the following we consider Diophantine equations of the form x2+ zxy + y2 = M for given M,z ∈ 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 ∈ ℕ is a product of irregular elements in the corresponding z-ring for z ∈ ℕ. We also give an overview how many primitive and non-primitive solutions in a given quadrant we can find for arbitrary M,z ∈ 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 x2 + 3xy + y2 = M depending on M ∈ ℕ.