This is a collection of questions and replies on the subject of Fermat's Last Theorem.

Hello, my name is Bodin.
I am a 12 years old boy that is curious about the FLT theorem. I know that it was made by Fermat when he was reading his book. He wrote on the margin of the book that there are no solutions of x,y,z in the form of xⁿ + yⁿ = zⁿ when n is more than 2. Fermat himself prove in the case that n = 4 by using the method of infinite descent that leads to the impossibility of reducing the z solution. Because z is a positive whole number and there are the smallest prositive whole number that is 1, so it is impossible to reduce the z solution to infinite times. That is the FLT is true when n = 4.
I would like to know that, did other famous mathematicians try to prove it? And if they did, why can't they prove the theorem, did they made mistakes?
And if you can answer me that what method Andrew Wiles use in his proof of FLT. Because now, the proof is confirmed right after fixing in 1993 and the complete proof in 1994. I would like to know what different methods he used to proof FLT, the methods that weren't there in the past 350 years.

There is an article about the history of mathematicians trying to prove Fermat's Last Theorem on the St Andrew's History of Maths site .

I wanted to find out if there's a way to prove that:



$x^n+y^n=r^n$

has no integral solutions if $n\ge 3$.

Saurabh Sinha

Hi Saurabh,

Aside from the fact that you haven't said 0 can't be a solution, the short answer to this question is ``yes, there is''.
Unfortunately, it's a bit complicated.

The statement you give is Fermat's Last Theorem, which was finally proved very recently (1995, I believe) by Wiles. The actual proof involves many branches of pure mathematics at university level and higher (it's close to beyond my understanding, anyway!). It basically revolves around the fact that two theorems about a particularly strange curve, both of which we know are true, must contradict one another if a solution to the equation exists, and hence no solution can possibly exist.

If you want more details, you might look up the web page at:

http://www.best.com/~cgd/home/flt/flt01.htm


which has a very good introduction, provided that your maths is advanced enough. But after all, this took some of the greatest mathematical minds in the world a few hundred years to prove, so it would be disappointing if it was easy, wouldn't it?

Best wishes,

Chris

Maybe it is not a right place to ask this question, but I really want to know where I can find the whole solution for Fermat Last Theorem.

Two papers, one by Wiles and one by Wiles and Taylor, in a journal called "The Annals of Mathematics". More precisely, you want the May 1995 issue, which contains both. (And nothing else, I think.)

Wiles's paper is called "Modular elliptic curves and Fermat's Last Theorem"; the joint paper is called "Ring-theoretic properties of certain Hecke algebras". Between them they take up about 130 pages.

A warning: unless you are a professional mathematician you will almost certainly find these papers extremely incomprehensible. Most professional mathematicians would too, in fact. (I certainly would.)

You can find a summary (intended for professional mathematicians who aren't specialists in this particular area) of the proof in the July 1995 "Notices of the American Mathematical Society". That's just a couple of pages long, and quite clearly written. (But still, as I said, for professionals.)

Is there an accessible interpretation of the Andrew Wiles' proof of Fermat's Last Theorem for Sixth Formers?

Many thanks

Jeff. Davies

It depends on what you mean by 'interpretation', clearly the proof is too difficult to explain in simple terms.
There is an excellent book, very readable and accessible to sixth formers:

Fermat's Last Theorem
The Story of a Riddle that Confounded the World's Greatest Minds for 358 years.
by Simon Singh
Published in 1997
by The Fourth Estate Ltd. 9 Salem Rd, London W2 4BU
ISBN 1-85702-521-0

This book was written by the maker of the BBC Horizon programme on Fermat's Last Theorem and has a forward by John Lynch (Editor of BBC Horizon).

It is a good starting point and gives a long list of suggestions for further reading.