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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