On the special solutions of an equation in a finite field

摘要

The main purpose of this paper is to prove the followingconclusion: letpbe a prime large enough and letkbe a fixed positive integer with2k|p−1. Then for any finite fieldFpand any element0≠c∈Fp, there exist three generatorsx,y, andz∈Fpsuch thatxkyk+ykzk+xkzk=c.