|Aug6-12, 08:09 PM||#1|
Direction of Goldbach Partitions
I have been investigating goldbach partitions for some time.
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:
I ran a program to determine a histogram of the number of occurences of the subsequences of the following form
No. of Occurences
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?
|Aug8-12, 01:15 AM||#2|
What exactly does 'x,1,x' mean? Are the two x's the same or independent? Either way, the count seems much too low. How can it be less than x,1,1,x?
|Aug9-12, 05:27 AM||#3|
I am counting the number of consecutive runs of similar digits in the sequence. So the run x1x refers to either the subsequence 0,1,0, or 0,1,-1 or -1,1,0 or -1,1,-1.
As another example the consecutive run of two positive ones viz x,1,1,x refers to the subsequences -1,1,1,0 or -1,1,1,-1 or 0,1,1,0 or 0,1,1,-1.
All the counts are mutually exclusive, and add up to the correct number of digits [give or take one or two, as I am having problems with the start and end of the sequence]
But the question remains why this particular pattern.
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