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

TL;DR

This paper provides an explicit formula for counting solutions to the equation x^m + y^m - z^m = 1 over finite fields, improving previous results under specific conditions related to the field's characteristic.

Contribution

The authors derive a new explicit formula for the number of solutions, enhancing earlier formulas by considering particular conditions on the exponent and field characteristic.

Findings

New explicit solution count formula for finite fields

Improved accuracy over previous formulas under certain conditions

Conditions depend solely on exponent and field characteristic

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.