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# Disjunctive numbers

From OeisWiki

**rich number**or

**disjunctive number**(in french:

*nombre univers*; this might suggest the english translation:

**universe number**, although this term is not used in the literature) is a real number whose expansion, in a given base

b |

{0, ..., b − 1} |

b |

b |

b |

b |

b |

n |

n |

n |

A number that is disjunctive to every base is called **absolutely disjunctive** or is said to be a **lexicon**. A set is called “residual” if it contains the intersection of a countable family of open dense sets. The set of absolutely disjunctive reals is a residual.^{[1]} It is conjectured that every real irrational algebraic number is absolutely disjunctive.^{[2]}

## Contents

## Examples

The Champernowne constant (0.123456789101112...) is a disjunctive number, and normal, in base 10. The number defined byn |

n |

n + 1 |

## Is π a disjunctive number?

It is not known whetherπ |

^{[3]}

### π approximations within the decimal expansion of π

approximations within the decimal expansion ofπ π *n*Digits Starting position **1**3 9 **2**31 137 **3**314 2120 **4**3141 3496 **5**31415 88008 **6**314159 176451 **7**3141592 25198140 **8**31415926 50366472

a (n) |

n |

π |

- {9, 137, 2120, 3496, 88008, 176451, 25198140, 50366472, ...}

### *e* approximations within the decimal expansion of π

approximations within the decimal expansion of*e*π *n*Digits Starting position **1**2 6 **2**27 28 **3**271 241 **4**2718 11706 **5**27182 28024 **6**271828 33789 **7**2718281 1526800 **8**27182818 73154827

n |

e |

- {6, 28, 241, 11706, 28024, 33789, 1526800, 73154827, ...}

## Universe numbers for all fixed base *b*

(...)

## Notes

- ↑ Calude & Zamfirescu (1999).
- ↑ Adamczewski & Bugeaud (2010) p. 414.
- ↑ André Boileau, Les nombres univers, Mathématiques ludiques et technologie, AMQ, octobre 2012 (Cégep de Ste-Foy). (Contains a tool to search for a base 10 digit sequence within the decimal expansion of π.)

## References

- Adamczewski, Boris; Bugeaud, Yann (2010). “8. Transcendence and diophantine approximation”. in Berthé, Valérie; Rigo, Michael.
*Combinatorics, automata, and number theory*. Encyclopedia of Mathematics and its Applications.**135**. Cambridge: Cambridge University Press. p. 410–451. Zbl pre05879512. ISBN 978-0-521-51597-9.

- Calude, C.S.; Zamfirescu, T. (1999). “Most numbers obey no probability laws”.
*Publicationes Mathematicae Debrecen***54**(Supplement): pp. 619–623.

## External links

- http://villemin.gerard.free.fr/Wwwgvmm/Type/Univers.htm (in french)
- Jean-Paul Delahaye, «Les nombres univers»,
*Pour la Science*, juillet 1996, pp. 104–107.