As a pure mathematician Newton reached his climax in the invention of the calculus, an invention also made independently by Leibniz. But do you know that, Fermat conceived and applied the leading idea of the differential calculus thirteen years before Newton was born and seventeen before Leibniz was born.
Today we are going delve into Fermat’s last work, whose proof took three and a half centuries to hail again after Fermat. Its nothing but Fermat’s Last Theorem.
We will start to take our steps towards the brilliance of this theorem by learning a bit about the Pythagorean Triples.Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. The equation a2 + b2 = c2 is a Diophantine equation.
|(3, 4, 5)||(5, 12, 13)||(8, 15, 17)||(7, 24, 25)|
|(20, 21, 29)||(12, 35, 37)||(9, 40, 41)||(28, 45, 53)|
|(11, 60, 61)||(16, 63, 65)||(33, 56, 65)||(48, 55, 73)|
|(13, 84, 85)||(36, 77, 85)||(39, 80, 89)||(65, 72, 97)|
Fermat’s Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.
After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a “stunning advance” in the citation for Wiles’s Abel Prize award in 2016. Wiles’s proof also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.
The Big Picture
Wiles’ three groundbreaking lectures, in June 1993, at the Isaac Newton Institute in Cambridge, UK were inspirational and shaped mathematics in its own way.
Theorem. Every semistable elliptic curve over Q is modular.
The strategy that ultimately led to a successful proof of Fermat’s Last Theorem was the connection with elliptic curves which arose from the “astounding” Taniyama–Shimura–Weil conjecture, proposed around 1955 which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat’s equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat’s Last Theorem, as well as leading the way to a full proof by others of what is now the modularity theorem.
The Taniyama-Shimura conjecture
The Taniyama-Shimura conjecture, since its proof now sometimes known as the modularity theorem, is very general and important conjecture (and now theorem) connecting topology and number theory which arose from several problems proposed by Taniyama in a 1955 international mathematics symposium.
The Shimura-Taniyama-Weil conjecture relates elliptic curves (cubic equations in two variables of the form y2 = x3 + ax + b, where a and b are rational numbers) and modular forms, objects (to be defined below) arising as part of an ostensibly different circle of ideas. An elliptic curve E can be made into an abelian group in a natural way after adjoining to it an extra “solution at infinity” that plays the role of the identity element. This is what makes elliptic curves worthy of special study, for they alone, among all projective curves (equations in two variables, compactified by the adjunction of suitable points at infinity) are endowed with such a natural group law. If one views solutions geometrically as points in the (x, y)-plane, the group operation consists in connecting two points on the curve by a straight line, finding the third point of intersection of the line with the curve and reflecting the resulting point about the x-axis.
The entire proof is not possible to provide through this post but a link will suffice it.
Courtesies- https://www.wikipedia.org/, THE PROOF OF FERMAT’S LAST
THEOREM by Nigel Boston University of Wisconsin – Madison.