On the Solutions of the Diophantine Equation $x^n+y^n=z^n$ In the Finite Fields ${\mathbb{Z}}_p$

Abstract

Let $p$ be a prime integer, ${\mathbb{Z}}_p$ the finite field of order $p$ and ${\mathbb{Z}}_p^{\ast }$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n+y^n=z^n$ with $1\le n\le \frac{p-1}{2}$. Our main aim in this paper is to give optimal conditions or relationships between the exponent $n$ and the prime $p$ to determine the existence of nontrivial solutions of the diophantine equation $x^n+y^n=z^n$ with $1\le n\le p-1$, in finite fields ${\mathbb{Z}}_p$.