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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

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?
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 = n-p is not
divisible by r, and thus q has increased chance of being prime.
If r does not divide n, then n-p is divisible by r for around 1 out of r-1
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

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