I have been investigating goldbach partitions for some time.(adsbygoogle = window.adsbygoogle || []).push({});

One interesting observation I have been able to determine is concerning the "direction" of the goldbach partitions whether they are increasing or decreasing as 2N increases.

To get an idea of this I constructed a function f(2N) where

f(2N) = 1 if G[2N] - G[2N-2] > 0

f(2N) = 0 if G[2N] - G[2N-2] = 0

f(2N) = -1 if G[2N] - G[2N-2] < 0

where G[2N] is the number of partitions for the even number 2N.

I then generated a sequence of numbers f(2N) begining at 2N = 10 and finishing at 2N=75398

[arbitrarily selected]. The sequence is then:

{1,1,-1,1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,-1,1,-1,1,1,-1,1,1,-1,.....}

I ran a program to determine a histogram of the number of occurences of the subsequences of the following form

histogram

subsequence type

No. of Occurences

x,1,x

5563

x,-1,x

7003

x,1,1,x

7003

x,-1,-1,x

5562

x,1,1,1,x

0

x,-1,-1,-1,x

1

x,0,x

0

I have been trying to work out a descriptive name for this sequence. Any suggestions?

The question then arises will a sequence generated from Hardy-Littlewoods equation

for Goldbach partions be similar?

Also , any comments on the histogram? Why the assymetric nature ?

And why only these subsequences? I would have thought there would have been longer runs. Any ideas?

Paul

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# Direction of Goldbach Partitions

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