# Does the elegant proof to the Fermats last theorem exists?

#### robert80

Just wondering, what do you think? Does it exist in its elegant and marvelous form?

kind regards,

Robert KM

#### D H

Staff Emeritus
Does it exist in its elegant and marvelous form?
Yes. It's Wiles' proof of the Taniyama-Shimura conjecture. This is incredibly elegant and also incredibly profound.

Fermat's Last Theorem was pretty much a useless little conjecture in mathematics. It wasn't pivotal. (Pivotal: a concept from which many others depend, or a concept bridges multiple areas of science/mathematics.) There weren't any theorems of the sort "If Fermat's Last Theorem is true then ..." Compare to the Riemann hypothesis, which is a pivotal conjecture. The Taniyama-Shimura conjecture is of huge importance because it bridges two apparently disconnected fields of mathematics.

The proof of Fermat's Last Theorem in a sense is trivial. It's Ribet's theorem, which quickly showed that if the Taniyama-Shimura conjecture is true then so is Fermat's Last Theorem. The trick was to prove the Taniyama-Shimura conjecture. Wiles' well-deserved fame arises from his proof of this very pivotal conjecture. His proof is elegant. That Wiles indirectly proved Fermat's Last Theorem is secondary.

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#### HallsofIvy

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.

#### DonAntonio

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.

It seems to me Fermat only proved the case n = 4. Cases n = 3,5,7 we re proved way later by Euler, Legendre, Dirichlet, Gauss and others.

DonAntonio

#### robert80

Thanks for your answers. I just think Fermat wouldnt lie nor to be wrong. So I was thinking that there is a proof in its fully divine form, simply waiting somewhere. But after so many years with no success with elegance, or lets say shortness instead (since Wiles proof seem to be very elegant), there is not much hope... But the proof if supposing it exists, could be based more on logic that on mathematics...All in all I dont intend to spend any more time on that issue. I was just curious what other think and searching for someone on that Forum, who believes that the Fermats proof exists :)

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#### Number Nine

Thanks for your answers. I just think Fermat wouldnt lie nor to be wrong.
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).

#### robert80

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?

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.

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#### robert80

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.

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#### HallsofIvy

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?
"Possiblility", yes. But you don't seem to see the difference between saying something is possible and saying it is true.

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.
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 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.
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.

#### robert80

"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.

It seems logical that, when you have 1 proof of some theory or theorem, there are infinite proofs present, the only question which exist is, which is the most simple. So yeah, Perhaps its Wiles, but there will be doubt for a long time still.

Just the explanation of n proofs with algebra. When you try to do it either way. Sooner or later you get product of 2 factors which are not coprime, so making out of them 2 coprime numbers, proving for that, that raises another 2 factors which are not coprime to each other. And so on...This is the main problem with fake Fermats proofs.

But when I first heard for his proof I tried to prove it with geometry, forming the Fermats triangle and putting Pitagora in. But knowing very little of number theory, I couldnt find the contradiction between rational and irrational numbers when I wrote the equations. So yeah, you could try something different, but the lack of skills lead you to absolutely no result.

And there is another problem with Fermats proof with coprime and semi prime solution. If you suppose solution exists, than this solution when exist could be based on billions of primes combined in a equation... So thats another argument, there is no simple solution, not for Fermats and certainly not for Beals conjecture. I hope Wiles will find the proof for it, as far as I understand he is among few who knows his proof in details. So there are not much options as I see it.

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