Fermat's Last Theorem

Who was Fermat and what was his Last Theorem?

Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. His proposition was about an equation which is closely related to Pythagoras' equation. Pythagoras' equation gives you:

x² + y² = z²

You can ask, what are the whole number solutions to this equation, and you can see that:

3² + 4² = 5²and 5² + 12² = 13²

And if you go on looking then you find more and more such solutions. Fermat then considered the cubed version of this equation:

x³ + y³ = z³

He raised the question: can you find solutions to the cubed equation? He claimed that there were none. In fact, he claimed that for the general family of equations:

"xⁿ + yⁿ = zⁿ where n is bigger than 2,
it is impossible to find a solution."

That's Fermat's Last Theorem.

Extended Fermat's Theorem?

"xⁿ + yⁿ + wⁿ = zⁿ where n is bigger than 2, it is impossible to find a solution."

No, Naom Elkies of Harvard University discovered the following counter-example in 1988.

2,682,440⁴ + 15,365,639⁴ + 18,796,760⁴ = 20,615,673⁴

From: Simon Singh, Fermat's Enigma, Anchor Books Inc., 1997, p. 159.

Proof of Fermat's Last Theorem

The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation . He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states

The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

In March 1994 Faltings, writing in Scientific American , said

If it were easy, he would have solved it by now. Strictly speaking, it wasn't a proof when it was announced.

Weil, also in Scientific American , wrote

I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.

In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.

In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.

On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem

From: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

Disclaimer: This page is just a personal note. Some of the above were copied from other web pages.