On Tue, Mar 19, 1996, Steve Anonsen wrote
<<On 3/15/96 Glenn Morton wrote:
> "The question of incompleteness or undecidability in arithmetic has been
>considered by Chaitin (1987a, 1988) by means of diophantine equations. A
>simple example is equation (2.64). The famous conjecture known as 'Fermat's
>last theorem' is that there are no positive integer solutions
><pre>
> n+3 n+3 n+3
>(x+1) +(y+1) =(z+1) (2.64)
>
>Only addition, multiplication and exponentiation (which is an extension of
>multiplication) are allowed. The extensive search for non-negative solutions
>of equation (2.64) has found none, although no proof is known that there are
>none. [quote snipped]" p. 80-81 Yockey, _Information Theory and Molecular
Biology_,
>(Cambridge, 1992).
This has nothing whatsoever to do with Godel's theorem or SETI, but I believe
Fermat's last theorem was proven in the last two years. I believe it was the
first such theorem proven by the extensive use of computation (if I recall the
article on the subject correctly -- anyone have a reference?).>>
I have closely followed the latest endeavor to solve Fermat's last theorem
(FLT). Andrew Wiles announced three summers ago in Cambridge, England, in a
series of lectures, a proof of this theorem. What was amazing about the proof
was that FLT was not his main objective. Instead he presented a proof of
something called the Taniyama-Weil conjecture which asserts that "all elliptic
curves are modular". Here elliptic curve is any curve of the form y^2 = x^3 +
ax^2 + bx + c (topologically, if you add a point at infinity, you get a doughnut
in 3 dimensional space). Unfortunately, modularity is a term I'm only vaguely
familiar with and so I hope you forgive me if I pass on an explanation. What
was noticed in the late eighties by people like K. Ribet and ? Freyd was that if
you had integers a,b,c and a prime number p>2 s.t. a^p + b^p = c^p then the
elliptic curve y^2 = x(x-a^p)(x-b^p) is _not_ modular. This, of course, is a
contradiction to Taniyama-Weil.
In fall of '93, a gap was found in Wiles' proof (having to do with bounding
something called a Selberg group). Over the next year, this gap was determined
to be too wide and so this part of the proof was abandoned. Fortunately, there
was a different approach, which Wiles had discarded early on, which he and a
colleague (whose name escapes me) took up again and managed to fill in the
details (specifically they showed that a certain relevant Hecke algebra was a
complete intersection). The final proof of all this appears in two papers in
the Annals of Mathematics (Princeton University Press) May 1995.
Unfortunately, I can't answer Steve's question as to where a more
straightforward article concerning this stuff is located. I know the Scientific
American has an article on this somewhere. Kenneth Ribet wrote a more technical
summary of all this in the Bulletin of the American Math Society within the past
year. Sorry I can't be more specific.
I hope this has been of some information.
In Christ,
Jim Turner
103531.1532@compuserve.com
jt2n@virginia.edu