Determination of Gaussian Integer Zeroes of F(x,z)=2x4−z3

Abstract

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].