From: "Eric Jablow" <erjablow at netacc.net>
To: "WSFA members" <WSFAlist at keithlynch.net>
Subject: [WSFA] Re: Goldbach's Conjecture
Date: Wed, 19 Jun 2002 07:26:31 -0500
Reply-To: WSFA members <WSFAlist at keithlynch.net>

In number theory, 1 is not considered prime.  In an algebraic number
domain, (of which the integers Z is the best example), we put numbers
into 3 classes:

units - numbers that have multiplicative inverses in the domain
primes - numbers p such that p | ab implies p | a or p | b
composites - the rest.

[The schoolboy definition of prime says that p is prime if p = ab
implies p = a or p = b.  This is inadequate, as in domains without
unique factorization the two concepts differ, and for various
reasons you really want the first property.  Consider Z[\sqrt{-5}].
Then 2 * 3 = (1 + \sqrt{-5}) (1 - \sqrt{-5}).  These 4 numbers
are irreducible but not prime.]

> Continuing further by hand would be increasingly tedious, but presumably
> a modern computer could extend the computations to some very large
> number.  But, since a computer could never count to infinity (in a finite
> time), it seems that it would be impossible for a computer to prove the
> conjecture for all even numbers.  But, a human has allegedly done so.
> Assuming that has been done, what is the difference between a human
> mathematician and a computer which explains why the human can do what the
> computer cannot?
>
> Ron Kean

Because that's not how people prove results in mathematics.  Usually.
The computer-aided proof of the Four-Color Conjecture could be
considered an exception.  Mathematics isn't arithmetic.

I guess you need to look at various mathematical proofs.  Start with the
proof of the exponent 4 case of the Fermat Conjecture.  You show that
any "solution" of x^4 + y^4 = z^4 in positive integers leads to a smaller
solution.  This is absurd, and leads to a contradiction.  You don't find a
way to plug all possible triples (x, y, z) into the equation.  I'll find a
good presentation of this if you want.

On the other hand, people have announced such proofs in the past.
Sometimes they're wrong.  Sometimes the authors are cranks.  Someone
announced a proof of the Poincare Conjecture earlier this year [one of the
two great unproved conjectures in mathematics]. He was wrong.

In fact, I just looked at the paper--it has the whiff of crank-hood.  I'm
not
going to get my hopes up.  When I was a grad student, I had to deal with
at least one crank Fermat-solver.  It was unpleasant.

Respectfully,
Eric Jablow