was fermet telling truth when he told he has a small proof to his theorem,but margin could not hold?

Question

I am refering to Fermat s last equation.....
x^n+y^n!=z^n
x,y,z are integers
n>2

Answer 1

Fermat did tell the truth.

http://en.wikipedia.org/wiki/Fermat%27s_last_theorem

Answer 2

My understanding is that Fermat believed he had a correct proof, but that it did not actually work. It is not too hard to see that to prove this, you need to prove it for all n such that n is 4 or an odd prime.

If you have had a fair it of number theory recently (at one point I could, but it has been too long ago now), it is not too hard to show this for 4, as well as 3 and some other small odd primes. However, the same technique does not work for all odd primes. I would guess that Fermat simply guessed that it would, jotted down a note, and went on. This is not the first time Fermat did that sort of thing. He also conjectured that all numbers of the form 2^(2^n) + 1 are primes (the ones that are are called Fermat primes). He checks this for n = 0, 1, 2, 3, 4 and stopped. For n=5, this is not prime, but for n=5 it is over 4 billion and the smallest prime factor is something like 641, so I do not blame him =)

Fermat was a very good mathematician for his day, but from what I understand of Wiles proof (and that is very little), there is no way Fermat had anything close to it.

Wikipedia probably has some cool info about Fermat and this as well...

Answer 3

I'd go with Vincent's answer. Although I might add in: he could have been that creative and had an actual proof. But this is unlikely, because if there was a method to solve it, I think some of the brilliant people who have tried, over the many years, would have found an "easy proof." It is highly unlikely that he devloped all the concepts that went into Wiles proof. So he probably didn't lie, but just made a mistake and didn't recognise it (like unique factoriation).

Answer 4

It's hard to say.

He was a great mathematician, so he could have known the proof to his theorem but just didn't take the time to write it out. His proof would involved concepts prior to Andrew Wiles' usage of modern mathematical techniques.

On the other hand, he could have been lying as well, putting what he put in the margin as a "pathetic give-up excuse" for not writing down the proof but conjecturing it to be true.

The only way we can really know is if we can speak to deceased people, or if he tucked away the real proof somewhere where it can stand the test of time. Oh, and also if we can time travel and ask him ourselves.

Answer 5

He never claimed it was small, what he wrote was that he found a "marvelous proof".
As to why he would have lied in a margin of his own book, one can speculate endlessly, but what would have been the point?
He probably did not lie, but he could certainly have been mistaken.
Clearly, to have a proof that would not fit in margin, and that he would not make any reference as to where he would have published or kept it, and the fact his personal papers never mentioned that proof, he could have believed that he knew how to develop it, without having really worked the details, and could have been mistaken about the applicability of the method, which he could have seen was flawed once he would have started writing down the details.

Answer 6

I think that he was telling the truth. If he was mistaken, then maybe there still is another way to solve it. I also think that the four color theorem could be proved on paper.

Answer 7

I think it got solved eventually.