This is a preview and has not been published.

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.


Download data is not yet available.


Szabó S. Some fourth degree diophantine equations in gaussian integers. Integers Electron J Comb Number Theory. 2004; 4(A16): A16.

Najman F. The Diophantine equation x4±y4= iz2 in Gaussian integers. Am Math Mon. 2010; 117(7): 637–41.

Emory M. The diophantine equations X4 + Y4 = D2Z4 in quadratic fields. Integers Electron J Comb Number Theory. 2012; 12: A65.

Ismail S, Mohd Atan KAM. On the Integral Solutions of the Diophantine Equation x4 + y4 = z3. Pertanika J Sci Technol. 2013; 21(1): 119–26. .

Izadi F, Naghdali RF, Brown PG. Some quartic diophantine equations in the gaussian integers. Bull Aust Math Soc. 2015; 92(2): 187–94.

Izadi F, Rasool NF, Amaneh AV. Fourth power Diophantine equations in Gaussian integers. Proce Math Sci. 2018; 128(2): 1–6.

Söderlund GA. Note on the Fermat Quartic 34x4 + y4 =z4. Notes Number Theory Discrete Math. 2020; 26(4): 103–5.

Jakimczuk R. Generation of Infinite Sequences of Pairwise Relatively Prime Integers. Transnat J Math Anal Appl. 2021; 9(1): 9–21.

Ismail S, Mohd Atan KA, Sejas Viscarra D, Eshkuvatov Z. Determination of Gaussian Integer Zeroes of F(x,z) = 2x^4- z^3. Malaysian J Math Sci. 2022; 16(2): 317–328.

Li A. The diophantine equations x4+2ny4=1 in quadratic number fields. Bull Aust Math Soc. 2021; 104(1): 21–28.

Somanath M, Raja K, Kannan J, Sangeetha V. on the Gaussian Integer solutions for an elliptic diophantine equations . Adv Appl Math Sci. 2021; 20(5): 815-822.

Ahmadi A, Janfada AS. On Quartic Diophantine Equations With Trivial Solutions In The Gaussian Integers. Int Electron. J Algebra. 2022; 31:134-142.

Tho NX. The equation x4+2ny4=z4 in algebraic number fields. Acta Math Hungar. 2022; 167(1): 309-331.

Tho NX. Solutions to x^4+py^4=z^4 in cubic number fields. Arch. Math. 2022; 119: 269–277.