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 $\mathbb{Z}^{*}{p}$ 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 $\mathbb{Z}_p$.