Several Diophantine Equations in Some Rings of Integers of Quadratic Imaginary Fields

Abstract

In this paper, it is proved that the Diophantine equation x4-y4 =z2 has no non-trivial coprime solutions in the rings of integers of quadratic imaginary fields [Formula: see text] for d=11, 19, 43, 67, 163, which implies that the Fermat equation x4+y4 =z4 has no non-trivial solutions in these fields either. Then all the solutions of the Pocklington equation x4-x2y2+y4 =(-1)σz2 (σ =0 or 1) in the ring of integers of [Formula: see text] are determined, and as an application, the result is applied to K2 of a field.