Given the nature of Wile's proof of FLT, do any of u still think Fermat may have had a proof?

Answer 1

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Author: Pubkeybreaker
Subject: Re: Simple proof of FLT: Why is it impossible?


[e-mail] wrote:
> Hi.
>
> I have a question. Why does everyone believe that a "simple" proof like
> Fermat claimed to have for his famous theorem, does not exist?
> So, why is it believed that such a proof is impossible?

Not impossible. Merely highly unlikely.

Many excellent mathematicians have tried elementary proofs. All
have failed. A lot of interesting mathematics has been discovered in
the
process.

It is known that certain elementary approaches can not work.
Methods based upon factoring the equation do not work because
unique factorization fails.

It is known that descent arguments (such as the Fermat argument for
exponent = 4) can't work for n > 4. The reason for this is very
deep.
The elliptic curve that is associated with the Fermat equation has a
group associated with it, known as the Selmer group. It is known that
only when this group is trivial can descent arguments work. It is
known
that the group is NOT trivial for n > 4.

It is known that modular approaches (look at the equation mod some set
of
primes, then try to piece together a solution for the integers based on
solutions
mod the primes) can not work. The reason is similar to why descent
won't work.
In this case, the non-trivial Selmer group prevents the Hasse-Minkowski
theorem
from working. This theorem tells us when one can construct solutions
over Z
using known solutions modulo primes.

Most amateurs know nothing about these results, so continue trying
these approaches. Their efforts will always be futile.

Fermat never claimed publicly that he had a proof. He only had a
private
note to himself. The fact that he *later* did make public a proof for
the
special case n=4 suggests he realized that he was wrong. Why publish a
special case if you already have a proof for the general case?



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Answer 2

Perhaps Fermat was being literal when he said the margin was too small for his proof. :) While I suppose it might have been possible that he had come up with a much more simple proof, it seems difficult to believe nobody has found it after 300+ years. Either way, Fermat's genius is still unquestionable.

Answer 3

Nope, I don't believe he could have worked out a proof with the maths of the time when it took Wiles 7 years using tools like elliptic integrals. There isn't a single instance in the whole history of science of somebody having this much more insight than everybody else. Not even Newton, Einstein or Darwin.

Answer 4

They (mathematical historians) believe that Fermat proved it for a couple of cases, and assumed that he could easily generalize it to all cases. The problem is that even though we don't have his "proof that was too big for the margin", historians have found some of his notes, where he thought he proved it for some of the cases, but it turns out that he made some mistakes. Therefore they (at least the ones who have told me this) believe that he probably thought that he could prove it elementarily, but in actuality, he couldn't, and thus his "proof that was too big for the margin" was probably flawed.

Answer 5

absolutely yes because he knew the fact that any given number having any given power can be expressed as difference of infinite sets of two perfect squares like
3 ^ (4) = 41 ^ (2) - 40 ^ (2).

Answer 6

whoma whatti huh? u must be smart....