From: pwh@bradley.bradley.edu (Pete Hartman)
Subject: Fermat's Last Theorem
Organization: Bradley University
Date: Wed, 6 Jan 93 09:22:12 GMT


I was introduced to this problem less than 6 months ago in some
book (perhaps _The Emperor's New Mind_) which I was reading for
completely different reasons.

Old hands surely know this one, but for the benefit of any who
may not:  the theorem basically boils down to the assertion
that there are no numbers x, y, z that satisfy x^N + y^N = z^N
where N>2.  Fermat's statement of the theorem was a little more
convoluted, and took some examination to realize this.

He claimed in some marginal notes that he had a "truly wondrous"
proof of this, but he did not record it, and no one has been able
to reconstruct it.

Until now. :-)

It seems to me to be a pretty obvious statement upon observation,
and I have some feel for how I would demonstrate the proof, but
unfortunately, I'm not a mathematician, and it's been a long time
since Calculus (not that Calc is needed, but it's been even longer
since most of that other stuff).

So I'm wondering if I could ask a few questions and see what response
I get.  Perhaps I'm just completely off base, and should go back to
programming, but please be polite if you must tell me that.

My idea is related to the Pythagorean Theorem, which is the only
non-trivial value of N for which the equation is true (assuming
Fermat was right) (in ordinary geometry anyway; I'll leave the
weird stuff to others).  One proof of the Pythagorean Theorem I
recall involves squares drawn with their sides along the right
triangle in question.  So the two-dimensional "volume" (usually
"area") of the two smaller squares sum to the 2-D "volume" of the third.

I don't recall *why* this is a proof, or maybe I'm confused and it's
just a demonstration, but I'd like an explanation of it if someone
could (maybe just email to me, considering it's probably pretty
simple and I've just forgotten).

Hinging on that, it seems apparent that if we go to N = 3, we are
dealing with normal 3-D volumes, and it's also apparent that the
two lesser volumes would sum to less than the third.  For N = 4,
we have hypercubes with edges adjacent to the right triangle, and
it seems sensible that the two lesser 4-D volumes would be even
smaller than the third.  Etc.  Is there some way to formalize
this proof?  Is it really a proof?

Assuming this works, there's another corollary thing that seems
like it would be possible, but I'm less sure of it's existance
than I am of this approach to Fermat's Last Theorem.  That is,
there may be "right" objects of higher dimension than 2 where
the sum of the volumes adjacent to the lesser "sides" of  the
object is equal to the volume of the greatest "side".  To make
this clearer, in the case of 2 dimensions, we deal with right
triangles (which has the fewest sides possible for a polygon),
with squares adjacent to each edge.  If this can be generalized,
there should exist some kind of 3 dimensional "right tetrahedron"
(tetrahedron having the fewest faces possible for a polyhedron)
that has a relationship between the faces similar to the relationship
between the sides of a right triangle.  Say,
area1^3 + area2^3 + area3^3 = area4^3.  Does this really sound
feasable to someone who isn't an utter layperson like me?  If
so, is it "interesting"?  Has it been demonstrated already?
Does it have any use?  Does anyone care?

Any comments, criticisms, etc, are welcome.  Just my own little
shot in the dark....
-- 
Pete Hartman		       Bradley University	pwh@bradley.bradley.edu
   and it seems your eternal reward is to hang out in heaven eternally bored