Numerical Approach for Fermat's last theorem

TL;DR

This paper introduces a geometric and numerical approach to Fermat's Last Theorem by mapping the problem onto a torus and analyzing the movement of points, revealing an acceleration phase transition near specific parameters.

Contribution

It proposes a novel geometric transformation and numerical analysis method to study Fermat's Last Theorem, offering new insights into the behavior of solutions.

Findings

Identifies an acceleration phase transition near (x,n)=(0,2).

Reframes Fermat's Last Theorem as finding rational intersections on a torus.

Suggests further investigation into the relationship between acceleration transition and solutions for n>2.

Abstract

This research focuses on the Numerical approach for Fermat's Last theorem. We can induce an Alternative form of Fermat's last theorem by using particular geometric mapping $\mathcal{M}$ on a Cartesian plane to a Torus. It transforms the Fermat's Last Theorem to finding a rational cross point between two parametric curves on the torus. In the end, this research shows the movement of the point, on the line $x^n+y^n=1$, has an acceleration phase transition near ($x,n$)=(0,2). Moreover, the studies about the relationship between this acceleration transition and the solution for the Fermat's Diophantine equation in the case of $n>$2, need further investigation.