Fermat's Equation Has No Solution with Some Prime Components

TL;DR

This paper proves that Fermat's equation has no solutions when certain prime-related conditions are met among the integers, advancing understanding within elementary number theory and providing methods to generate specific coprime integer sets.

Contribution

It establishes a new class of non-solvable cases of Fermat's equation based on prime components, covering a special case of Abel's conjecture.

Findings

No solutions for Fermat's equation under specified prime conditions

Provides a method to construct infinitely many coprime integers not satisfying Fermat's equation

Advances the understanding of Fermat's Last Theorem in elementary number theory

Abstract

Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result covers a special case of a conjecture of Abel, and furnishes a definite way to construct infinitely many setwise coprime integers that do not satisfy the Fermat's equation uniformly in $n$.