Yes. It's Wiles' proof of the Taniyama-Shimura conjecture. This is incredibly elegant and also incredibly profound.Does it exist in its elegant and marvelous form?
If you mean "did Fermat have a proof slightly larger than could be written in that famous margin", the answer is "No". What happened, I suspect, is what happens to mathematicians all the time- when he wrote that, he had what he thought was an insight into the problem that only require a little "expanding" to give a proof- and the next morning when he tried to work out the "expansion", he found it didn't work!
And the evidence is that, years later, he published proofs of the special cases, n= 3, 4, and 5. He wouldn't have done that if he already had a proof for all n.
Er...why? Did you know him? Mathematicians are wrong all the time, they just don't publish anything until it's been thoroughly examined. Wiles had to construct entirely new fields of mathematics to prove the theorem; there is no reason to believe that Fermat had such a wonderfully simple proof, especially given that, as HallsofIvy said, he felt the need to publish proofs of special cases years later (which would have been completely pointless if he did, in fact, have a proof of the general case).Thanks for your answers. I just think Fermat wouldnt lie nor to be wrong.
"Possiblility", yes. But you don't seem to see the difference between saying something is possible and saying it is true.Perhaps, he was an idealist and didnt want to publish general proof. He anyway did more than a lot. I believe there are some creative Mathematicians, who have wonderful proofs at home in their drawer, I am not applying to anything, but there is a possibility that someone in Russia has proved something important, but being so unsatisfied with the world he lives in, he is not willing to share his proof. I think there are few such in the world. Can you stand this possibility?
Yes, creative people exist in all fields. But being "creative" is not enough. A creative pizza maker has to have skills in pizza making, a creative artist has to have painting skills, a creative musician has to have music skills. And a creative mathematician has to have mathematics skills.But somebody who has proof of something in Math doesnt have to be a mathematician after all. I see creativity as universal fact and there are some very creative lets say Pizza makers. So the creativity occurs in various fields. So creative people in majority follow their interests. I mean its independent of the skills.
I have no idea what you mean by this. It certainly is possible to find a "general proof" of many things. The fact that we have general proofs of many things shows that. No, I don't believe that Wile's proof is the "only way to prove it". There are manys to prove anything in mathematics. But I doubt that anyone will find a proof of "Fermat's last theore" that is much simpler than Wiles' proof.But to be honest, I doubt it too. I have found a pattern what is wrong with most of the proofs attempts with algebra. It is impossible to find general proof. When you find semi proof and you think its ok, there is a counterexample or some special case it does not work for. When you think you proved that, there is a counterexample in this sub group of proof 2. And so on... At the end you get I believe n "proofs" for Fermats last theorem :) And this has no sense, since its proved already :) maybe this is the right way to prove, that the general proof other than Wiles proof does not exist :) Maybe in Wiles solution its hidden the fact, that this is the only way to prove it. This would have some additional added value.
"Possiblility", yes. But you don't seem to see the difference between saying something is possible and saying it is true.
Yes, creative people exist in all fields. But being "creative" is not enough. A creative pizza maker has to have skills in pizza making, a creative artist has to have painting skills, a creative musician has to have music skills. And a creative mathematician has to have mathematics skills.
I have no idea what you mean by this. It certainly is possible to find a "general proof" of many things. The fact that we have general proofs of many things shows that. No, I don't believe that Wile's proof is the "only way to prove it". There are manys to prove anything in mathematics. But I doubt that anyone will find a proof of "Fermat's last theore" that is much simpler than Wiles' proof.