Author 
Message 
stargene@sbcglobal.net science forum beginner
Joined: 27 Sep 2005
Posts: 8

Posted: Mon Jul 17, 2006 5:02 am Post subject:
Distribution of Goldbach pairs as even integers n increases



A question about Goldbach's Conjecture...
In the Wikipedia article "Goldbach's Conjecture", which relates
the status of the "strong" conjecture that all even integers n above
4 are the sum of two primes p + q, there is displayed a striking
curve showing "Number of ways to write an even number n as
the sum of two primes (4 = n = 1,000,000)". Thus, n = p+q ,
p' + q' , p" + q" , etc.
It is at
http://en.wikipedia.org/wiki/Goldbach%27s_conjecture
It of course reflects the fact that as the even integers n increase
without bound, in general the number of ways each one can be re
solved into one or more distinct pairs of primes will also increase.
What I found most striking about the actual spread of points in
the curve is the increasing resolution into a rich spectrum as
n gets higher and higher.
What accounts for this structure in the distribution of prime pair
sums for n, as n increases indefinitely? 

Back to top 


Jens Kruse Andersen science forum beginner
Joined: 23 Jul 2005
Posts: 40

Posted: Mon Jul 17, 2006 2:41 pm Post subject:
Re: Distribution of Goldbach pairs as even integers n increases



stargene wrote:
Quote:  What accounts for this structure in the distribution of prime pair
sums for n, as n increases indefinitely?

The factorization of n, primarily the tiny factors 3, 5, 7.
If the prime r divides n, and p<n is any other prime, then q = np is not
divisible by r, and thus q has increased chance of being prime.
If r does not divide n, then np is divisible by r for around 1 out of r1
prime values of p.
That's 1/2 for r=3, so this alone halves the expected number of Goldbach
partitions compared to n divisible by 3.
This effect is mentioned at
http://en.wikipedia.org/wiki/Goldbach's_conjecture#Heuristic_justification
Each "band" in the graphs should correspond to numbers with the same
divisibility for tiny primes.
The lower half of the graph is n not divisible by 3.
The four main bands in that half, listed in increasing order of Goldbach
partitions:
Not divisible by 5 and 7.
Not divisible by 5, but by 7.
Divisible by 5, but not 7.
Divisible by 5 and 7.
The four main bands in the upper half is the same four possibilities
in the same order.

Jens Kruse Andersen 

Back to top 


Google


Back to top 



The time now is Sun May 27, 2018 1:34 pm  All times are GMT

