20071130, 11:38  #1 
Nov 2007
home
5^{2} Posts 
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/2Y. A Diophantine equation can be generalized from this: (X/2Y)(X/2+Y)=M; (X^2)/4Y^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 semiprime 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 20071130 at 11:46 
20071130, 12:12  #2 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
> The chance that any of those M's is a semiprime 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 
20071130, 12:21  #3 
"Brian"
Jul 2007
The Netherlands
2×3×5×109 Posts 
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 counterexamples.

20071130, 13:32  #4  
Nov 2007
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5^{2} Posts 
Quote:
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. 

20071130, 14:36  #5  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×937 Posts 
Quote:
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. 

20071201, 14:43  #6 
Nov 2007
home
5^{2} Posts 
nevermind
Last fiddled with by ewmayer on 20071203 at 23:17 Reason: Don't worry  we didn't. 
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