Circularity in Finite Fields and Solutions of the Equations ${\mathbf{x}}^{\mathbf{m}}+{\mathbf{y}}^{\mathbf{m}}-{\mathbf{z}}^{\mathbf{m}}=1$

Abstract

An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper.