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


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



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


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.


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