Fermat's Last Theorem admits an infinity of proving ways and two corollaries

TL;DR

This paper demonstrates that Fermat's Last Theorem can be proved through infinitely many methods using the concept of reversors, and it explores related corollaries and classifications of solutions.

Contribution

It introduces the concept of reversors and shows that Fermat's Last Theorem admits infinitely many proofs, along with deriving new relations and classifications of solutions.

Findings

Fermat's Last Theorem has a noncountable infinity of proofs.

Corollaries establish new relations and classifications of solutions.

Numerical examples confirm the theoretical results.

Abstract

Fermat's statement is equivalent to say that if $x$, $y$, $z$, $n$ are integers and $n>2$, then $z^{n}\gtrless x^{n}+y^{n}$. This is proved with the aid of numbers $\lambda $'s, of the form $\lambda =z/\rho $, with $1<\rho<z$, named \emph{reversors} in the text, because their property of multiplying $z^{n-1}$ in $z^{n-1}<x^{n-1}+y^{n-1}$, not only reverses the signal but also gives $z^{n}>x^{n}+y^{n}$ as a solution of the reversed inequality. As the $\lambda ^{\prime }s$ satisfy a compatible opposed sense system of inequalities, the $\lambda$-set is equivalent to the points of an $\mathbb{R}^{+}$ interval. Therefore the theorem admits a noncountable infinity of proving ways, each one given by a particular value of $\lambda$. In Corollary 1 a general relation between $y$, $x$, $z$ and $n$ is derived. Corollary 2 shows that the Diophantine equation in Fermat's statement admits no solutions other than algebraic irrationals and the inherent complexes. Integer triplets can be classified in seven sets, within each one their relation with the respective $n$ is the same as shown in Table 1. Numerical verification with examples taken from all the mentioned seven sets gives a total agreement with the theory.