Reply to: Fermat's Last Theorem.

[doodab]

For simple proofs that aren't actually wrong you can start here:

Proof of Fermat's Last Theorem for specific exponents - Wikipedia, the free encyclopedia

I got to the point of only needing to prove it for prime exponents when I was at uni. I personally think there is some mileage in looking at logarithms.

[NotAllThere]

But if that is the case why hasn't anyone at least come up with a plausible "flawed" proof to show what Fermat might have done. His proof must have appeared to be good at least superficially.

There have been various explanations. He did prove it for n = 4. He probably had in mind generalising that for higher powers.

[BlasterBates]

There we go

Simple proof of Fermat's last theorem

the wonders of the Internet

[doodab]

But if that is the case why hasn't anyone at least come up with a plausible "flawed" proof to show what Fermat might have done. His proof must have appeared to be good at least superficially.

There are thousands of flawed proofs. Some partial ones as well.

[BlasterBates]

In this case it seems that all the simple lines of argument have been explored, but it might happen that some future maths will come along and simplify things.

But if that is the case why hasn't anyone at least come up with a plausible "flawed" proof to show what Fermat might have done. His proof must have appeared to be good at least superficially.

[doodab]

Ian Stewart wrote an accessible book on maths. He talked about a theorem that was proved in many pages of complex maths. Then some guy came along with a completely different argument, and proved it with simple, easy to understand maths, taking about half a page. ( Sorry, can't find the book, so can't tell you which theorem it was).

In this case it seems that all the simple lines of argument have been explored, but it might happen that some future maths will come along and simplify things.

[NotAllThere]

There isn't a prize AFAIK but you would probably make a name for yourself. There are some prizes on offer from the clay maths institute for some other unsolved problems.

It seems likely that if a simple proof existed Fermat would have written it down somewhere else and/or it would have been found already. A lot of very smart people had a crack at it in the intervening 300 odd years. So my guess is that he realised his idea was flawed.

Ian Stewart wrote an accessible book on maths. He talked about a theorem that was proved in many pages of complex maths. Then some guy came along with a completely different argument, and proved it with simple, easy to understand maths, taking about half a page. ( Sorry, can't find the book, so can't tell you which theorem it was).

[DimPrawn]

How about if all the amounts were in pounds and percentages ?

Now yer talking!

[Freamon]

Simon Singh's book is pretty good.

[OwlHoot]

It all sounds quite tedious to me.

How about if all the amounts were in pounds and percentages ?

[DimPrawn]

It all sounds quite tedious to me.

[xoggoth]

Didn't threaded explain the proof of Fermat's last theorem to us in layman's terms a couple of years ago?

Umpteen years ago somebody on TV did explain relativity really well by walking two different ways round a triangle. Fermats theorem could probably be done with a whippet and a bowl of custard or maybe prunes would be better.

[OwlHoot]

The prize for Beal's conjecture might be worth a crack though. Something to do in the evenings when you're working away from home anyway.

Those Clay problems are obviously mind-bogglingly hard. In many cases just reading the problem statement is enough to bring one out in hives.

But there are many unsolved problems (like Fermat's Last Theorem in a way I suppose) which are ridiculously easy to state, but have resisted all attempts to solve them. Here are a couple to work on in that B&B (and one could easily quote a dozen like them) :

* The Erdos-Straus Conjecture Prove that for every integer n > 1 there exist positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z

* The (or a) Ramanujan conjecture: If 2^t and 3^t are both integers for some real t > 0, prove that t is an integer.

The first is a complete mick-taker, because it looks so easy, and it is easy to find solutions with at least one of x, y, z negative, and very easy to prove it for _most_ n; but some values always slip through the net!

But the second is like an impenetrable glass cliff-face. It's no exaggeration to say that hardly any progress has been made on it, which is all the more incredible when one thinks of all the amazing things that _have_ been proved.

[d000hg]

It's also quite possible he thought he had a proof but was wrong. Prrofs get published and dis-proved sometimes.

[OwlHoot]

The modern proof is pretty mind boggling, and is pages long.

Fermat's comment though, written just before he died, was good, sort of "the proof is quite simple, but it doesn't quite fit in the margin"

The modern proof isn't what Fermat was thinking of so it would be interesting to work out what it actually was. I reckon there's a big prize waiting for anyone who can come up with it.

The comment wasn't written just before he died, quite the opposite in fact.

He first became interested in Diophantine Analysis (finding or ruling out integer and rational fraction solutions of underdetermined polynomial equations) when he read a copy of Bachet's commentaries on Diophantus as a student, or not long after, and that was when he scribbled the comment.

It was one of many other comments he scribbled in the margins, and for most of the other equations he claimed to have proved he did indeed later provide proofs or sketches. So given that he never mentioned FLT again, it is very unlikely he had a valid proof and equally likely he soon realized this.

Probably he assumed one could factor x^n + y^n (for prime n) into linear factors x + w^i y, where w is a complex n-th root of unity. If this factorization can be proved to be unique, then the result follows. Unfortunately (or fortunately, given how many advances have followed from the study of this and related equations), the factorization is not unique for all values of n.

However, at least two distinguished mathematicians, Sir Peter Swinnerton-Dyer and Doron Zeilberger are on record as not completely ruling out an elementary solution; but they both say this would almost certainly be extremely subtle and intricate in the unlikely event it exists.

Kummer's original theory of ideals, which he developed in the 1840s, is fairly tough to get one's head round, as is Kronecker's related theory of primary decomposition. But in the 1880s a guy called Dedekind greatly simplified the theory, and it's his definition of ideal that is used today. It's actually quite easy to explain and understand how it works, in everyday terms and examples, and if anyone is interested I will; but I won't waffle on any more now.