p Statistics: Nonrepeating Digits
by JVSchmidt


General
Dividing data into substrings of length L we can test whether all digits in a substring are different or not. It is clear that max length of substring with nonrepeating digits in decimal presentation is 10. The chance to find a 10 digits long nonrepeating sequence is very poor - 0,0363%. But still the first max sequence in p is already found at position 60: ...4592307816...
We gonna test the frequencies of nonrepeating digits for L=2 to 10.
They expected frequencies are for
L=2: 9/10
L=3: 9/10 * 8/10
L=4: 9/10 * 8/10 * 7/10
...
L=10: 9/10 * 8/10 * 7/10 * ... * 2/10 * 1/10

and in general w(L)=9!/(10-L)!/10L-1

Graph shows the relation between chains with and without repeats depending on the length L.




Result's Overview
Digits analyzed: (4 or 4.2) * 10 9
Analysis started at digit: 1
Ellapsed computer time for one class: ca. 5 min


Chi2-value for the repeating/nonrepeating-distribution of chains with various length L

Length of
chains
L
Number of examined
chains
K = N/L
Expected number of
chains with
digit repeat
Measured number of
chains with
digit repeat
Chi2
(f=1)
2 2.100.000.000 210.000.000 209.999.269 0,00283
3 1.400.000.000 392.000.000 391.990.137 0,34467
4 1.050.000.000 520.800.000 520.792.690 0,20358
5 840.000.000 585.984.000 585.964.433 2,16063
6 700.000.000 594.160.000 594.156.084 0,17070
7 600.000.000 563.712.000 563.698.467 5,37179
8 525.000.000 515.474.400 515.469.604 2,45934
9 466.666.600 464.973.160 464.971.560 1,51768
10 420.000.000 419.847.590 419.847.00 2,25701

I have also examined the dependence of Chi2 for L=10 from starting position (1-10).

Detailed results for this test you will find here: Nonrepeating Digits (EXCEL file)

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