On the special solutions of an equation in a finite field

Abstract

The main purpose of this paper is to prove the following conclusion: let p be a prime large enough and let k be a fixed positive integer with 2k|p−1. Then for any finite field Fp and any element 0≠c∈Fp, there exist three generators x, y, and z∈Fp such that xkyk+ykzk+xkzk=c.