Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y
DOI:
https://doi.org/10.21123/bsj.2023.7344Keywords:
Algebraic properties, Diophantine equation, Gaussian integer, quartic equation, nontrivial solutions, symmetrical solutions.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.
Received 20/4/2022, Revised 17/9/2022, Accepted 19/9/2022, Published Online First 20/2/2023
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