On Fermat curves modulo a finite number

TL;DR

This paper establishes a connection between solutions to Fermat equations modulo a prime and solutions to a specific system of recursion relations in modular arithmetic, providing a new perspective on Fermat's Last Theorem in finite fields.

Contribution

It introduces a novel equivalence between Fermat solutions modulo a prime and solutions to a system of recursion relations, offering a new approach to understanding Fermat curves over finite fields.

Findings

Equivalence between Fermat solutions and recursion systems

Characterization of Fermat curves via over-determined equations

Potential new methods for studying Fermat's Last Theorem in finite settings

Abstract

We show that the existence of a non-trivial solution of $x^n+y^n=p^n$, with $p$ a prime number, is equivalent to the existence of a solution of a certain (over-determined) system of $(n-1)$-recursion relations ("zipper" equations) in $\mathbb{Z}_{p-1}$.