p Statistics: Long Runs
by JVSchmidt


General considerations
A LONG RUN (LR) is an uninterrupted line of a certain number in a digit sequence. If the digit repeats itself M times it is also called an M-repeat. Testing LR is the opposite view to the gap test which proofs nonoccurance.
An M-repeat is a rare event because the probabilty to find exactly the same digit on the next place is 1/10 and so the chance for a LR falls down very quickly like 10-M.
When testing about 4 billion digits we will expect max long runs of 9 or 10 digits length.

A famous early LR in p is the FEYNMAN POINT at position 762 where a six digits repeat of "9" can be observed.


Results
Digits analyzed: 4.2 * 10 9
Analysis started at digit: 1
Ellapsed computer time for each digit: 3 min 45 sec

The estimated average LR-length is 1,111...
Proofing the distribution for the entire number of LR's we found a Z-value for the normal distribution
Z = 1,8395.
Table shows the Chi2-value of the LR distribution for each digit.
The position of the 10 digits long runs are given additionally.

Digit Chi2 (f=6) Number of LR>=8 Ten digits long run at position
0 7,26464 46 -
1 2,01710 42 3.961.184.001
2 7,25172 37 -
3 5,68390 28 -
4 3,38456 44 -
5 2,12706 41 -
6 7,00881 31 386.980.412
7 3,59519 43 -
8 3,08914 43 3.040.319.543
9 13,24100 37 -


Download LongRuns-Details as an Excel file here.

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