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.

> 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

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.

[quote=akruppa;119590]> 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[/quote]

see: [url]http://www.research.att.com/~njas/sequences/table?a=1358&fmt=5[/url]

The proof can be made deterministic by using theorem 3 from [url]http://arxiv.org/PS_cache/math/pdf/0506/0506067v1.pdf[/url]
Using it the maximum distance between semi primes becomes 26.

[QUOTE=vector;119600]see: [url]http://www.research.att.com/~njas/sequences/table?a=1358&fmt=5[/url]

The proof can be made deterministic by using theorem 3 from [url]http://arxiv.org/PS_cache/math/pdf/0506/0506067v1.pdf[/url]
Using it the maximum distance between semi primes becomes 26.[/QUOTE]

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.

nevermind