Date: Wed, 19 Jun 2002 12:03:30 -0400
From: Steve Smith <sgs at aginc.net>
To: WSFA members <WSFAlist at keithlynch.net>
Subject: [WSFA] Re: Goldbach's Conjecture
Reply-To: WSFA members <WSFAlist at keithlynch.net>

ronkean at juno.com wrote:

> 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?

Because proofs are done by reasoning about the problem, not brute
calculation.  There are specialized "therom proving" computer programs,
but I don't know how effective they are in the general case.  My
experience with similar programs is that it's almost as much work to
code the problem into a form that the computer can solve as it is to
solve it by hand.

Simple algebraic proof:

We want to prove that the sum of the first "n" numbers is given by the
formula
S = n(n+1)/2.

First, we show that it works for one special case.  If n=1, the sum is
obviously 1.  S = 1(1+1)/2 = 1.  Okay, that works.

Now we show that, if it works for "n", it will work for "n+1":
S' = S + (n+1) = n(n+1)/2 + (n+1).  Letting "y" = n+1

S' = (y-1)y/2 + y = [(y-1)/2 + 1 ]y = [(y-1) + 2]y/2 = [y+1]y/2

So if the formula works for "n", it will work for "n+1".  We've shown
that it works for n=1.  Since it works for 1, it will work for 2.  Since
it works for 2, it will work for 3.  Since ...

So we've shown, in just a couple of lines, that our formula will work
for *any* number.  No calculation required; just a bit of algebra.

--
Steve Smith                                           sgs at aginc.net
Agincourt Computing                            http://www.aginc.net
"Truth is stranger than fiction because fiction has to make sense."