Determination of Gaussian Integer Zeroes of F(x, z) = 2x 4 − z 3

In this paper the zeroes of the polynomial F(x, z) = 2x
4 −z
3
in Gaussian integers Z[i] are determined, a problem equivalent to finding the solutions of the Diophatine equation x
4 + y
4 = z
3
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) = 2x
4 −z
3 under the general
form x = (1 + i)η
3
and c = −2η
4
for η ∈ Z[i].