|p Statistics: Difference of Sums|
Compared with the SUM OF DIGITS we go one step further.|
Taking two consecutive substrings of length L we can calculate the difference of these sums to proof if there would be any correlation between these two values that is a dependence of the 2*L digits.
Here is an example for p with L=5.
First sequence X=14159 -> SUM (X)=20
First sequence Y=26535 -> SUM (Y)=21
Difference of sums = SUM(X)-SUM(Y) = 20-21 = -1
We have a simple recursive law for the expected distribution.
Let w(L,d) be the probability that two chains of length L have a difference of sums d.
Then w(L+1,d) = w(L,d) / 10 + sum w(L,d-i) x (10-i)/10 + sum w(L,d+i) x (10-i)/10
where sums taken for i=1 to 9.
Starting condition is: w(1,d) = (10-abs(d))/100 for d= -9,-8,..+8,+9.
For any L the difference of sums of digits is located between -9*L (X=000..0,Y=999..9) and S=9*L (X=999..9, Y=000..0). Again, when going to longer and longer chains the min and max difference became extremely improbable because the likelihood for a single digit long run falls like 10-L.
|Digits analyzed: 4.2 * 10 9|
Analysis started at digit: 1
Ellapsed computer time for one class: 3 min 40 sec
|Length of |
|Number of examined pairs of|
K = N/(2*L)
|Chi2||Number of statistical |