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)!/10 ^{L-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 |

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 | Chi^{2}(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 Chi

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

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