Browsing by Autor "Z. Eshkuvatov"

Filter results by typing the first few letters
Now showing 1 - 1 of 1
  • Thumbnail Image
    Item type: Item ,
    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].