On Fermat’s Last Theorem and Gӧdel’s Incompleteness Theorem

A proof of Fermat's last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat's last theorem in the case of í µí±›í µí±› = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat's approach of infinite descent. The infinite descent is linked to induction starting from í µí±›í µí±› = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat's last theorem. As far as Fermat had been proved the theorem for í µí±›í µí±› = 4, one can suggest that the proof for í µí±›í µí±› ≥ 4 was accessible to him.

The Bulletin of Symbolic Logic, 2010

This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.

English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat's Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time -- such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles's proof employs very advanced mathematical tools and methods that were not at all available in the known World during Fermat's days. Given that Fermat claimed to have had the `truly marvellous' proof, this fact that the proof only came after $358$ years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat's time, this has led many to doubt that Fermat actually did possess the `truly marvellous' proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat's Last Theorem actually yields to our efforts to prove it. This proof is so elementary that anyone with a modicum of mathematical prowess in Fermat's days and in the intervening 358 years could have discovered this very proof. This brings us to the tentative conclusion that Fermat might very well have had the `truly marvellous' proof which he claimed to have had and his `truly marvellous' proof may very well have made use of elementary arithmetic methods.

SSRN Electronic Journal, 2021

English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat's Last Theorem} which had for 358 years notoriously resisted all efforts to prove it. Sir Professor Andrew Wiles's proof employs very advanced mathematical tools and methods that were not at all available in the known World during Fermat's days. Given that Fermat claimed to have had the `truly marvellous' proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat's time, this has led many to doubt that Fermat actually did possess the `truly marvellous' proof which he claimed to have had. In this short reading, via elementary arithmetic methods which make use of Pythagoras theorem, we demonstrate conclusively that Fermat's Last Theorem actually yields to our efforts to prove it.

This work contains two papers: the first published in 2022 and entitled "On the nature of some Euler's double equations equivalent to Fermat's last theorem" provides a marvellous proof through the so-called discordant forms of appropriate Euler's double equations, which could have entered in a not very narrow margin and the second instead published in 2024 and entitled "Some Diophantus-Fermat double equations equivalent to Frey's elliptic curve" provides the possible proof, which Fermat has not published in detail, but which uses the characteristic of all right-angled triangles with sides equal to whole numbers, or the famous Pythagorean identity. Some explanations in session(III) are provided: the first makes evident the nature of the "proof a' la Fermat" and the subsequent sessions clarify the direct and interesting connection of the two elementary proofs and it is necessary if you want to understand how two different elementary proofs of Fermat's Last Theorem are possible. Regarding the first paper, a method is used that simplifies Wiles' theory, a theory that has received much honors from the entire mathematical community. More precisely, through the aid of a Diophantine equation of second degree solved at first not directly, but as a consequence of the resolution of the double Euler equations that originated it and finally in a direct way, the author was able to obtain the following result: the intersection of the infinite solutions of Euler's double equations gives rise to an empty set and this only by exploiting a well-known Legendre Theorem, that is a criterion which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. It must be observed that this proof must in no way be interpreted as a sort of absurd revenge of elementary number theory over more modern analytic and algebraic treatments. The author himself has added a section in which he connects his concepts with some of those used by Wiles in his complex demonstration.

Current Developments in Mathematics, 1995