Generalization of Fermat's Last Theorem

2015

Current Developments in Mathematics, 1995

This paper develops a framework of algebra whereby every Diophan-tine equation is made quickly accessible by a study of the corresponding row entries in an array of numbers which we call the Binomial triangle. We then apply the framework to the discussion of some notable results in the theory of numbers. Among other results, we prove a new and complete generation of all Pythagorean triples (without necessarily resorting to their production by examples), convert the collection of Bi-nomial triangles to a Noetherian ring (whose identity element is found to be the well-known Pascal triangle) and develop an easy understanding of the original Fermat's Last Theorem (F LT). The application includes the computation of the Galois groups of those polynomials coming from our outlook on F LT and an approach to the explicit realization of arithmetic groups of curves by a treatment of some Diophantine curves.

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.

As is well known FERMAT reading the Commentaria in Diophantum by C. G. BACHET DE MEZIRIAC, had made a habit of annotating them in the margin. Concerning the eighth Diophantum problem, which requis1es and gives the resolution in rational numbers of the equation x^2 + y^2 = a^2, FERMAT postulate : <<On the contrary, it is impossible to divide a cube into the sum of two cubes, a fourth power into two fourth powers, and, in general, any power of degree greater than two, into two powers of the same degree.

Древности Северного Причерноморья, Кавказа и Средней Азии: от открытий Н. И. Веселовского к современной науке: Материалы международной научной конференции, посвященной 175-летию Николая Ивановича Веселовского. СПб., 2024

Alternative per il socialismo, No. 72, 2024

On the centenary of his birth, who remembers Lenin, even in that part of the world that had transformed him in an icon? In this sense Lenin is a tragic icon. From 1902 when with his What to do? he gave identity to his 'Bolshevik' party and up to the 1923 essay Better less but better, which laid down the conditions for running power: his actions and thinking have a tragic significance, for how they were disregarded-first by those who took over his role after him, in his country-and then in those parts of the world attracted by the Russian '17. The USSR-and not Lenin-served as a reference for men and countries where social conflicts were attempted and where the political system allows the activity of socialist and communist parties.