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 --'- --------------------------------------------------------------------
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 -------- Mark P. Hahn Work: 212-278-5861 mhahn@tcbtech.com Home: 609-275-1834 TCB Technologies, Inc Consultant to: The SoGen Funds 1221 Avenue of the Americas, NY NY
participants (3)
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Blake Buzzini -
Jim Choate -
Mark Hahn