Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y

Authors

Shahrina Ismail, Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800, Bandar Baru Nilai, Negeri Sembilan, Malaysia.Follow
Kamel Ariffin Mohd Atan, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 UPM, Serdang, SelangorFollow
Diego Sejas Viscarra, Departamento de Ciencias Exactas, Facultad de Ingenierías y Arquitectura, Universidad Privada Boliviana, Cochabamba, Bolivia.Follow
Kai Siong Yow, School of Computer Science and Engineering, College of Engineering, Nanyang Technological University, 50 Nanyang Ave, Singapore 639798.Follow

Abstract

The investigation of determining solutions for the Diophantine equation over the Gaussian integer ring for the specific case of is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

Keywords

Algebraic properties, Diophantine equation, Gaussian integer, quartic equation, nontrivial solutions, symmetrical solutions.

Article Type

Article

How to Cite this Article

Ismail, Shahrina; Atan, Kamel Ariffin Mohd; Viscarra, Diego Sejas; and Yow, Kai Siong (2023) "Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y," Baghdad Science Journal: Vol. 20: Iss. 5, Article 13.
DOI: https://doi.org/10.21123/bsj.2023.7344