  Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and
$\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the
Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$.
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 \leq
n \leq p -1 $, in finite fields $\mathbb{Z}_p$.
