# Analyzing the DISTANCE of sequences of Length L:

What is the distribution like?

## GENERAL CONSIDERATIONS / EXPECTED VALUES

When testing the difference of sums we knew that this test is not very sensitive due to the combinatorical proberty of the sum, generally: n + m = (n-x) + (m+x).

But we can treat any digit position as an coordinate value and ask about the distance of to chains.

Let us define the DISTANCE OF TWO CHAINS (DOC) as:

DOC = SUM _{i=1,L} ABS(X_{i} - Y_{i})

where

L = length of chains X and Y

X_{i},Y_{i} = i-th digit of chain X respectively Y

Here is an example for Pi with L=5.

First sequence X=14159

First sequence Y=26535

DOC = abs(1-2) + abs(4-6) + abs(1-5) + abs(5-3) + abs(9-5) =

= 1 + 2 + 4 + 2 + 4 = 13

The recursive law for the expected distribution can easily be found.

Let w(L,d) be the probability that two chains of length L have a DOC = d.

Then w(L+1,d) = w(L,d) / 10 + sum w(L,d-i) x 2*(10-i)/100

where sum is taken for i=1 to 9.

Starting condition is:

w(1,d) = 2*(10-d)/100 for d=1-9

w(1,0) = 1/10

We understand that the DOC-value can be an indicator for possible correlations between digits of neighbouring sequences.

RESULTS

Digits analyzed: N = 100 * 10 ^{11}

DOC-Analysis started at digit: 1

L=Length of Chains; P=Number of exam.Calculations=N / (2xL)

Classes=Number of statist. relevant classes