From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: possibly what Fermat may have had in mind?
Date: Fri, 29 Oct 1999 13:42:39 -0500 (CDT)
Newsgroups: sci.math
To: lemmerm@mpim-bonn.mpg.de
Keywords: Descent proof of Fermat's Last Theorem for exponent 3

In article <3818641F.2DE9@mpim-bonn.mpg.de> you write:
>As a matter of fact, I even doubt that Fermat had a correct
>proof of FLT for n=3; in one of his letters, he made a list
>of results he claimed he could prove using his descent: it 
>was FLT for n=3, the solution of y^2 = x^3 - 2 and y^2 = x^3 - 4
>in integers, and the claim that Fermat numbers are always prime. 
>I am not aware of proofs for any of these claims that were 
>accessible to Fermat.

Well, FLT for n=3 amounts to a quest for rational points on an elliptic
curve which turns out to have rank 0; that's a reasonably natural sort 
of thing to prove by descent, isn't it? (In fact the gibberish posted by
a certain member of this newsgroup amounts to showing that any rational
points would have to be in the image of a certain isogeny. What JSH is
lacking is a proof that rational points on this other curve would
in turn have to be in the image of an isogeny too, so that descent
would imply the rank is zero). I have often intended to do just this
in the hopes of educating potential FLT-provers, but it started to
look like too much work to describe the isogenies needed; do you
know any good place to find this approach worked out?

THe other two curves you describe have rank 1 over the rationals,
so descent is going to give a very different kind of conclusion,
with which it is difficult to argue about _integer_ points even
with the hindsight of 20th century mathematics. I think these are
of a class of problem which is easily approached with algebraic
number theory, but of course that wasn't available to him either.

dave
==============================================================================

From: Franz Lemmermeyer <lemmerm@mpim-bonn.mpg.de>
Subject: Re: possibly what Fermat may have had in mind?
Date: Mon, 1 Nov 1999 14:51:33 +0100 (MET)
Newsgroups: [missing]
To: Dave Rusin <rusin@math.niu.edu>

> Well, FLT for n=3 amounts to a quest for rational points on an elliptic
> curve which turns out to have rank 0; that's a reasonably natural sort 
> of thing to prove by descent, isn't it? 

 Yes it is. But it's a 3-descent, and to the best of my knowledge
these have not been used before our century (Selmer ?), at least 
not in diophantine analysis.

> I have often intended to do just this
> in the hopes of educating potential FLT-provers, but it started to
> look like too much work to describe the isogenies needed; do you
> know any good place to find this approach worked out?
 
 Do you refer to p = 3 here? BTW, there's a short discussion 
of a proof for p = 7 using 2-descent on an elliptic curve in 
my preprint on "Pepin's counter examples to the Hasse Principle 
for curves of genus 1".

bye,

  franz 
==============================================================================

From: Franz Lemmermeyer <lemmerm@mpim-bonn.mpg.de>
Subject: Re: possibly what Fermat may have had in mind?
Date: Tue, 2 Nov 1999 11:33:55 +0100 (MET)
Newsgroups: [missing]
To: Dave Rusin <rusin@math.niu.edu>

> Yes. There is the algebraic-number-theory proof (e.g. in Niven and Zuckerman,
> but this is the same as the Euler's original proof, cast in Kummer's 
> framework). It's obviously a descent proof, deriving from a solution
> of x^3+y^3=(unit)*z^3 in  Q[zeta_3] a "smaller" solution. 

 That's Gauss's posthumously published proof. And Kummer's proof 
of the second case is also a descent proof.

> I've only
> wondered if one could do the proof using only rational arithmetic, for
> the benefit of neophytes who don't really follow complex arithmetic.
> Probably it's not worth the effort to disguise something simple.

 There certainly is such a proof . Basically you have to show 
that the curve has trivial Selmer group and torsion; explicit 
computation of the torsors was provided by Selmer, Cassels
and Satge, so all you have to do is collect the results 
there. I'll have a look at it. 

> >BTW, there's a short discussion 
> >of a proof for p = 7 using 2-descent on an elliptic curve in 
> >my preprint on "Pepin's counter examples to the Hasse Principle 
> >for curves of genus 1".
> 
> I would like to see this. Is it online or can you send me a copy? 

 oops, I forgot to give the URL:

   http://www.rzuser.uni-heidelberg.de/~hb3/prep.html

 or article 175 in the Algebraic Number Theory Archives

   http://www.math.uiuc.edu/Algebraic-Number-Theory/

franz