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

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.