25 yrs Ago Andrew Wiles achieved the Challenge to Prove Fermat Last Theorem.

The Peace of God to all that belong to the Light,
Dear Readers,

Fermat's Last theorem was for decades in the Guinness Book of World Records as the "most difficult mathematical problem", one of the reasons being that it has the largest number of unsuccessful proofs.

A quarter of century ago one of the really old pieces of mathematics that remained still unproven was finally showed fully, the last Theorem of Fermat, formulated on 1637 in a margin note at the Arithmetics book written by the famous French mathematician.

There are no three numbers that actually obey the equation X^n+ Y^m= Z^p when n, m, p are integer numbers distinct than 2. Pythagoras theorem is the only particular case on which this relation works.

Please check:
Fermat's Last Theorem

This has been tested exhaustively but for finitely many numbers using supercomputers, the impossibility to show it for all numbers required a formal mathematical proof.

Only on the Summer of 1993 the Mathematician Andrew Wiles, from Cambridge University was able to prove that, the great challenge the defied for centuries many of the most brilliants minds on mathematics including Gauss, Euler, Legendre, Lebesgue, Hilbert, Riemann, Dirichlet, etc.

Wiles proof is based on three important preliminary results: the Taniyama-Shimura well conjecture, Ribet Theorem and Frey's curve, that he applied using Galois representations of elliptic curves.

Taniyama-Shimura well Conjecture: Every elliptic curve is a modular form in disguise.

Elliptic curves correspond to problems whose solutions can be plotted on the surface of a Torus.

Modular forms are functions on the complex plane that have incredible large numver of intricate symmetries.

It was in proving this conjecture that Andrew Wiles established the proof of Fermat's Last theorem.

The reason they are connected is as follows. Gerhard Frey showed that IF there was a solution in integers to x^n + y^n = z^n, say A^n + B^n = C^n then we could get an elliptic curve of the form y^2 = x^3 + (A^n-B^n)x^2 - (A^n.B^n)x

Another mathematician, Ken Ribet, showed that this equation could not be modular.

So now we have the following chain of reasoning:

(1) If the Taniyama-Shimura conjecture can be proved, then every elliptic curve is modular.

(2) If every elliptic curve must be modular, then the Frey elliptic curve is forbidden to exist.

(3) If the the Frey elliptic curve does not exist, then there can be no solutions to the Fermat equation.

(4) Therefore Fermat's Last Theorem is true.

please check:
mathforum.org...

Different Attempts to prove the last theorem of Fermat were fundamental to push the development of Number theory along 2 centuries.

Prof. Wiles is currently faculty of Princeton University in New Jersey, recipient of the Abel, Fermat, Wolf, Ostrowski, Cole, Clay Prizes, among other honors.

Please Check:

Andrew Wiles Definitive Mathematical Proof of Legendary Last Theorem of Fermat

Here is a formal paper published of Sir Andrew Wiles where he explains his Historic Mathematical proof of Fermat's last Theorem.

Please check:
scienzamedia.uniroma2.it...

The thread is open to mathematicians, statisticians and other scientists and people in general motivated by Science advances to discuss the power of Mathematical proofs to attack problems that can not be fully verified or completely solved using computational or other technological means.

Thanks for your attention,

The Angel of Lightness

I has teh book - a very good read