RE: Goldbach's Conjecture (fwd)

["Blake Buzzini" <bab282@psu.edu>]

First, let me add yet another reference. This one is from _Number Theory in
Science and Communication_ by M.R. Schroder, pg. 96:

"One of the most enduring (if not endearing) conjectures is the famous
Goldbach conjecture, asserting that every even number >4 is the sum of two
primes."

I would argue they're all essentially equivalent.  Here's how they all
agree:

Goldbach's conjecture is that every even number is expressible as the sum of
TWO primes.

The variations are: (1)Goldbach's "original" conjecture versus his revised
conjecture after communication with Euler and (2) excluding four while
adding an odd-primes-only condition (four IS the sum of two EVEN primes,
namely 2 and 2). I think these variations are simply different depths/points
of view of the same idea and one who focuses on semantic differences is
missing the point.

Cutting to the chase, Goldbach's conjecture concerns the sum of EXACTLY TWO
primes stands, and your statement "If we allow repetition *and* the number 2
as a prime then all even numbers can be written as a string of 2's summed
appropriately" is thereby contradicted.

Let me conclude with this inspirational quote from a man whose name escapes
me: "Young men should write proofs, old men should write books." ; )

Blake Buzzini, PSU

-----Original Message-----
From:	owner-cypherpunks@Algebra.COM [mailto:owner-cypherpunks@Algebra.COM]
On Behalf Of Jim Choate
Sent:	Friday, November 20, 1998 12:20 AM
To:	Cypherpunks Distributed Remailer
Subject:	RE: Goldbach's Conjecture (fwd)


Forwarded message:

> From: "Blake Buzzini" <bab282@psu.edu>
> Subject: RE: Goldbach's Conjecture (fwd)
> Date: Thu, 19 Nov 1998 23:37:09 -0500

> >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.)"

Ok, so this one says it was Goldbach himself and in particular states two
odd primes completely eliminating 4 from the get go ...

> >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?"

This one is the second version...

> >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."

And finaly a third completely different slant. They at least get Fermats
contribution right.

> Am I misreading somewhere?

Well I'd say that all three of your references tended to contradict each
other. Which one do you want to stand on?

This is my quote:

> 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).

I'll stand by this statement.


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