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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 theDiophantine 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 relationshipsbetween the exponent $n$ and the prime $p$ to determine the existence ofnontrivial solutions of the diophantine equation $x^n + y^n = z^n$ with $1 \leqn \leq p -1 $, in finite fields $\mathbb{Z}_p$.

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