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Old 2007-11-30, 11:38   #1
vector
 
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Default Proof of Goldbach Conjecture

I think I proved the Goldbach conjecture, here is proof:

This conjecture states that every even number greater than 2 can be expressed as the sum of two primes.

This conjecture can be restated as:
If X is even and Y is an integer then there exist prime numbers of the form X/2+Y and X/2-Y.

A Diophantine equation can be generalized from this:
(X/2-Y)(X/2+Y)=M;
(X^2)/4-Y^2=M where M is a positive integer, and ((X^2)/4) is a fixed constant.
The total number of positive M's is sqrt((X^2)/4).

The chance that any of those M's is a semi-prime has a determined lower bound of ~1/4 for moderately sized M and increases as M gets larger.

By finding M which is a semiprime the two primes that add to X can be found.

Therefore the total number of 2 prime groups that sum to an even integer X has a lower bound of Floor[(sqrt((X^2)/4)/4], which is always greater than one for sufficiently large even integer.

Last fiddled with by vector on 2007-11-30 at 11:46
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Old 2007-11-30, 12:12   #2
akruppa
 
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> The chance that any of those M's is a semi-prime has a determined lower bound of ~1/4 for moderately sized M and increases as M gets larger.

What exactly do you mean by that?

Alex
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Old 2007-11-30, 12:21   #3
Brian-E
 
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You seem to be using a probability distribution argument. This would be sufficient to show that "almost all" even numbers are the sum of two primes, but it does not show the complete absence of counter-examples.
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Old 2007-11-30, 13:32   #4
vector
 
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Quote:
Originally Posted by akruppa View Post
> The chance that any of those M's is a semi-prime has a determined lower bound of ~1/4 for moderately sized M and increases as M gets larger.

What exactly do you mean by that?

Alex
see: http://www.research.att.com/~njas/se...e?a=1358&fmt=5

The proof can be made deterministic by using theorem 3 from http://arxiv.org/PS_cache/math/pdf/0506/0506067v1.pdf
Using it the maximum distance between semi primes becomes 26.
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Old 2007-11-30, 14:36   #5
R.D. Silverman
 
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Quote:
Originally Posted by vector View Post
see: http://www.research.att.com/~njas/se...e?a=1358&fmt=5

The proof can be made deterministic by using theorem 3 from http://arxiv.org/PS_cache/math/pdf/0506/0506067v1.pdf
Using it the maximum distance between semi primes becomes 26.
It (the claimed proof) is codswallop.

There already exist probabilistic results regarding Goldbach. Look
up 'Goldbach exceptions'. For example, it is known that exceptions,
*if they exist* have asymptotic density 0. Indeed, the number of
possible primes less than P for which exceptions might exist is known
to be at most O(P^1/4+epsilon) for any epsion > 0. This does not
say whether any exceptions DO exist; merely that there can't be too many
if they do. The exponent 1/4 may have been improved since I last looked
at this problem.

The proofs of this and related results are sieve based and run into the
sieve parity problem & the fundamental lemma of the sieve. See
Halberstam & Richert's book.
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Old 2007-12-01, 14:43   #6
vector
 
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nevermind

Last fiddled with by ewmayer on 2007-12-03 at 23:17 Reason: Don't worry - we didn't.
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