Fermat's Last Theorem for Special Case

TL;DR

This paper provides an elementary proof for a special case of Fermat's Last Theorem, focusing on prime exponents where 4 divides n+1, showing no solutions exist under certain conditions.

Contribution

It introduces a new elementary proof for a specific case of Fermat's Last Theorem involving prime exponents with particular divisibility properties.

Findings

No solutions for the equation when n is prime, 4 divides n+1, and abc are not divisible by n.

The proof applies to a specific subset of Fermat's Last Theorem cases.

Provides an elementary approach to a special case of a famous theorem.

Abstract

In this paper we present an elementary proof for a special case of Fermat's last theorem for specific category of a, b and c. In fact, we assume that $n$ is prime and $4\rvert (n+1),$ then for $a,b$ and $c$ that $ n\nmid abc$ the equation $a^{2n} = b^{2n} + c^{2n}$ does not have any solution in natural numbers.