The “Difficulties” in Fermat's Original Discourse on the Indecomposability of Powers Greater Than a Square: A Retrospect

Grigory Dedenko

doi:10.7176/mtm/9-7-04

Abstract

A possible version has been identified of the original proof of the decomposability of whole degrees above the square which Pierre Fermat spoke of. This reconstructed evidence is discussed with some extra conclusions drawn from it. Keywords: number theory, Fermat’s Big Theorem/Last Theorem DOI : 10.7176/MTM/9-7-04 Publication date : July 31 st 2019

Highlights

The present work is the result of an attempted reconstruction of Fermat’s original discourse along with an explanation of why he might have not written it down
A purely mathematical challenge was www.iiste.org that he had to operate the entirely new notions of binomials and logarithms, both having just appeared for use and to be learnt “on the fly”
Fermat was obviously “playing” with the new notions, decomposing powers of differences into sums of powers and suddenly found out that as one confines oneself with positive integers in the power, the logarithmic equation yields immediately that xn + yn = zn is correct for whole x, y, z only and if only n = 1 or 2

Summary

Introduction

The present work is the result of an attempted reconstruction of Fermat’s original discourse along with an explanation of why he might have not written it down. The author had performed it within a one-year period of time – between 1990 and 1993 – trying proving the theorem When completed, it did look like a proof of Fermat’s epoch, as it only involved the knowledge and techniques available and utilised by Fermat’s contemporary and pre-Fermat mathematical world.[1]. Tries at demonstrations of that conjecture were either based on triangular deliberations (earlier, inclusive of Fermat’s own proof of the conjecture for n = 4) or modular theory techniques (later, inclusive of Andrew Wiles’ eventual proof of 1995 (Faltings, Abramov)) These all are methods that deal with transformation properties of special curves over particular types of space (e.g. rational numbers), underscoring the ‘stability’ of elliptic curves with respect to modular transformations. In order for the y to be a positive integer, n 2 must leave, since for n > 1 n 2 is an irrational number

It is thus necessary that the expression

Conclusion