Fermat's Theorem -- a Geometrical View

Abstract

Fermat's Last Theorem questions not only what is a triple, but more importantly, what is an integer in the context of equations of the type $x^n+y^n=z^n$. This paper explores these questions in one, two and three dimensions. It was found that two conditions are required for an integer element to exist in the context of the Pythagoras' theorem in 1D, 2D and 3D. An integer must satisfy the Pythagoras' theorem of the respective dimension -- condition 1. And, it must be completely successfully split into multiple unit scalars -- condition 2. In 1D, the fundamental unit scalar is the line length 1. All integers in 1D satisfy $x+y=z$, and can be decomposed into multiples of the unit line, hence integers exist and can form 1D triples $(x,y,z)$. In 2D, the fundamental unit scalar is the square side 1. Only some groups of integers (called triples) satisfy $x^2+y^2=z^2$, and can be decomposed into multiples of the unit square, forming 2D triples. In 3D, the fundamental unit scalar is the octahedron side 1. The geometry of the 3D Pythagoras' theorem dictates that $x^3+y^3=z^3$ is governed by octahedrons, validating condition 1. However, octahedrons with side length integer cannot be completely divided into unit octahedrons (as tetrahedrons appear), invalidating condition 2. Hence, if integers do not exist in the context of the 3D Pythagoras' theorem, then neither do triples. This confirms Fermat's Last Theorem for three dimensions ($n=3$). The geometrical interdependency between integers in 1D and 2D suggests that all integers of higher dimensions are built, and hence are dependent, on the integers of lower dimensions. This interdependency coupled with the absence of integers in 3D suggests that there are no integers above $n>2$, and therefore there are also no triples that satisfy $x^n+y^n=z^n$ for $n>2$.