Browsing by Autor "Shahrina Ismail"

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Determination of Gaussian Integer Zeroes of F(x,z)=2x4−z3
(2022) Shahrina Ismail; Kamel Ariffin Mohd Atan; Diego Sejas Viscarra; Z. Eshkuvatov
In this paper the zeroes of the polynomial F(x,z)=2x4−z3 in Gaussian integers Z[i] are determined, a problem equivalent to finding the solutions of the Diophatine equation x4+y4=z3 in Z[i], with a focus on the case x=y. We start by using an analytical method that examines the real and imaginary parts of the equation F(x,z)=0. This analysis sheds light on the general algebraic behavior of the polynomial F(x,z) itself and its zeroes. This in turn allows us a deeper understanding of the different cases and conditions that give rise to trivial and non-trivial solutions to F(x,z)=0, and those that lead to inconsistencies. This paper concludes with a general formulation of the solutions to F(x,z)=0 in Gaussian integers. Results obtained in this work show the existence of infinitely many non-trivial zeroes for F(x,z)=2x4−z3 under the general form x=(1+i)η3 and c=−2η4 for η∈Z[i].
Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y
(College of Science for Women, University of Baghdad, 2023) Shahrina Ismail; Kamel Ariffin Mohd Atan; Diego Sejas Viscarra; Kai Siong Yow
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.
On the integral solutions of the Diophantine equation x4 + y4 = 2kz3 where k > 1
(American Institute of Physics, 2021) Shahrina Ismail; Kamel Ariffin Mohd Atan; Kai Siong Yow; Diego Sejas Viscarra
This paper is concerned with the existence, types, and the cardinality of the integral solutions of the Diophantine equation x4 + y4 = 2kz3, for k > 1. The objective of this paper is to develop methods to be used in finding all integer solutions to this equation. Results of the study show the existence of infinitely many integral solutions to this type of Diophantine equation for both cases, x = y and x ≠ y. For the case when x=y, the form of the solutions is given by (a, b, c) = (2k−1n3, 2k-1n3, 2k−1n4) when 1 ≤ k < 5, and (a, b, c) = (2k−1−3tn3, 2k−1−3tn3, 2k−1−4tn4), for t≤k−14 when k ≥ 5. Meanwhile, for the case when x ≠ y, the form of solutions is given by (a, b, c) = (2kun2, 2kvn2, 2kn3) or (a, b, c)= (2kdu, 2kdv, 2kdn), depending on the value of k. The main result obtained is a formulation of the generalized method to find all the solutions for this type of Diophantine equation.