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. <= Back to Main