[Beowulf] The recently solved Lie Group problem E8

Peter St. John peter.st.john at gmail.com
Thu Mar 22 12:36:53 PDT 2007


Ah, I see where you're coming from. Yes indeed, computers were used to
verify that x^n + y^n = z^n has no nontrivial solutions (with n > 2)  for
certain values of n: n into the millions :-) There were classical proofs for
n = 3 and n = 4 that can be covered in an undergraduate lecture, and results
for classes of numbers, like, n composite. The computers were used to verify
for a huge huge range of n.

 All of this added to the widespread presumption of Fermat, but contributed
nothing, I don't think, to Wiles' work, which links Elliptic Curves to
Modular Forms in a way first noticed by ...um... a late Japanese
mathematician, I'm thinking of the go player Takemiya, I think it was
Shimura? OK Wiki has saved me: Taniyama and Shimura. Shimura is alive and at
Princeton now.

There were two guys who worked together on the conjecture (which led to
Wiles), which started with just noticing that the first few numbers
describing something that no sane person would ever contemplate, coincided
with the first few numbers describing something else that was completely
unrelated and that no sane person would ever contemplate. It was just a
miracle that a single human being would recognize a connection between two
such fields. It would be like a microbiologist studying nucleotide
sequencing, noticing numbers matching up with the spectral decomposition of
a star photographed by Hubble.

Computers were big in the history of the subject but Wiles didn't touch 'em
for the final conclusion. The link you give (very amusing btw) mentions that
just recognizing all the lemmas in Wiles, who uses material from nigh every
branch of mathematics, would be a hard job for a machine, before it could
even start trying to apply automatic theorem prover technology (to verify
Wiles).

Peter


On 3/22/07, Robert G. Brown <rgb at phy.duke.edu> wrote:
>
> On Thu, 22 Mar 2007, Peter St. John wrote:
>
> > First, a picky but pertinent point: Fermat's Last Theorem wasn't done by
> > machine. It was Andrew Wiles, at Princeton. The story is that he broke
> up a
>
> There are computer efforts along these lines, though:
>
>   http://www.cs.rug.nl/~wim/fermat/wilesEnglish.html#a5
>
> and it was my understanding that part of the exploration that led to the
> result involved computation.  Perhaps I am mistaken.
>
>    rgb
>
> --
> Robert G. Brown                        http://www.phy.duke.edu/~rgb/
> Duke University Dept. of Physics, Box 90305
> Durham, N.C. 27708-0305
> Phone: 1-919-660-2567  Fax: 919-660-2525     email:rgb at phy.duke.edu
>
>
>
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