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Gaussian Integer Solutions Of The Diophantine Equation 𝒙^𝟒+𝒚^𝟒=𝒛^𝟑 For X Not Equal To Y
Journal
Baghdad Science Journal
Date Issued
2022
Author(s)
Shahrina Ismail
Kamel Ariffin Mohd Atan
Diego Sejas-Viscarra
Kai Siong Yow
DOI
10.21123/bsj.2022.ID
Abstract
The investigation of determining solutions for the Diophantine equation 𝑥𝑥4 + 𝑦𝑦4 = 𝑧𝑧3 over the Gaussian integer ring for the specific case of 𝑥𝑥 ≠ 𝑦𝑦 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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Gaussian Integer Solutions Of The Diophantine Equation 𝒙^𝟒+𝒚^𝟒=𝒛^𝟑 For X Not Equal To Y.pdf
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