RE: Goldbach's Conjecture (fwd)

Forwarded message:

> From: "Blake Buzzini" <bab282@psu.edu>
> Subject: RE: Goldbach's Conjecture
> Date: Thu, 19 Nov 1998 22:17:54 -0500

> I could be wrong, but I thought Goldbach's conjecture was that every even
> number could be expressed as the sum of *two* primes. This doesn't prohibit

No, that was Fermat, Goldbach just says every even number greater than two
can be represented as a sum of primes. Basicaly Fermat says that if we have
n primes we can reduce them to 2 primes only, in all cases. Which happens to
exclude using equilateral triangles as a test bed since you can't tile a
equilateral with only two other equilaterals, you could use rectangles though.
So basicaly from a geometric perspective Fermat says that given a rectangle of
even area it is possible to divide it with a bisector into two rectangles of
prime area.

It's interesting that Fermat doesn't mention that the only prime that can
use two as a factor is 4. And you can't factor 2 at all since we eliminate
1 as a potential candidate (another issue of symmetry breaking simply so we
don't have to write '....works for every prime but 1' on all our theorems).

> repetition. Therefore, under Goldbach's conjecture:
> 
> 4 -> 2 + 2
> 6 -> 3 + 3 but NOT 2 + 2 + 2
> 8 -> 5 + 3 but NOT 2 + 2 + 2

The real issue for me is the interaction of primes (ie n * 1 = n) and the
identity theorem (ie n * 1 = n). They're opposite sides of the same coin.

It doesn't really matter now since it doesn't look like I'm going to get a
copy of EURISKO in this lifetime to play with.


    ____________________________________________________________________
 
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>From _Elementary Theory of Numbers_ by William J. LeVeque, pg. 6:

"It was conjectured by Charles Goldbach in 1742 that every even integer
larger than 4 is the sum of two odd primes.  (All primes except 2 are odd,
of course, since evenness means divisibility by two.)"


>From _Excursions in Number Theory_ by C. Stanley Ogilvy and John T.
Anderson, pg. 82:

"Goldbach's conjecture. Is every even number expressible as the sum of two
primes?"


>From _Goldbach's Conjecture_ by Eric W. Weisstein
(http://www.astro.virginia.edu/~eww6n/math/GoldbachConjecture.html):

"Goldbach's original conjecture, written in a 1742 letter to Euler, states
that every Integer >5 is the Sum of three Primes. As re-expressed by Euler,
an equivalent of this Conjecture (called the ``strong'' Goldbach conjecture)
asserts that all Positive Even Integers >= 4 can be expressed as the Sum of
two Primes."


Am I misreading somewhere?

Blake Buzzini, PSU

-----Original Message-----
From:	owner-cypherpunks@Algebra.COM [mailto:owner-cypherpunks@Algebra.COM]
On Behalf Of Jim Choate
Sent:	Thursday, November 19, 1998 10:39 PM
To:	Cypherpunks Distributed Remailer
Subject:	RE: Goldbach's Conjecture (fwd)


Forwarded message:

> From: "Blake Buzzini" <bab282@psu.edu>
> Subject: RE: Goldbach's Conjecture
> Date: Thu, 19 Nov 1998 22:17:54 -0500

> I could be wrong, but I thought Goldbach's conjecture was that every even
> number could be expressed as the sum of *two* primes. This doesn't
prohibit

No, that was Fermat, Goldbach just says every even number greater than two
can be represented as a sum of primes. Basicaly Fermat says that if we have
n primes we can reduce them to 2 primes only, in all cases. Which happens to
exclude using equilateral triangles as a test bed since you can't tile a
equilateral with only two other equilaterals, you could use rectangles
though.
So basicaly from a geometric perspective Fermat says that given a rectangle
of
even area it is possible to divide it with a bisector into two rectangles of
prime area.

It's interesting that Fermat doesn't mention that the only prime that can
use two as a factor is 4. And you can't factor 2 at all since we eliminate
1 as a potential candidate (another issue of symmetry breaking simply so we
don't have to write '....works for every prime but 1' on all our theorems).

> repetition. Therefore, under Goldbach's conjecture:
>
> 4 -> 2 + 2
> 6 -> 3 + 3 but NOT 2 + 2 + 2
> 8 -> 5 + 3 but NOT 2 + 2 + 2

The real issue for me is the interaction of primes (ie n * 1 = n) and the
identity theorem (ie n * 1 = n). They're opposite sides of the same coin.

It doesn't really matter now since it doesn't look like I'm going to get a
copy of EURISKO in this lifetime to play with.


    ____________________________________________________________________

            Lawyers ask the wrong questions when they don't want
            the right answers.

                                        Scully (X-Files)

       The Armadillo Group       ,::////;::-.          James Choate
       Austin, Tx               /:'///// ``::>/|/      ravage@ssz.com
       www.ssz.com            .',  ||||    `/( e\      512-451-7087
                           -====~~mm-'`-```-mm --'-
    --------------------------------------------------------------------

At 09:39 PM 11/19/98 -0600, Jim Choate instructed:
>It's interesting that Fermat doesn't mention that the only prime that can
>use two as a factor is 4. And you can't factor 2 at all since we eliminate
>1 as a potential candidate (another issue of symmetry breaking simply so we
>don't have to write '....works for every prime but 1' on all our theorems).

I thought I was following along until I got here, and got very lost. First
question: I think the first sentence implies 4 is prime, so I must have
the emphasis wrong.

Unless you are saying that you cannot factor 4 as 2*2 because < of
something I missed >. So the only factorization of 4 is 4*1, hence 
four is prime. 

The other explanation is "Whoosh" the whole conversation when over
my head and I'm lost.

-MpH


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