# Knödel number

The Knödel numbers ${K}_{n}$ for a given positive integer $n$ are the set of composite integers $m>n$ such that any $$ coprime^{} to $m$ satisfies ${b}^{m-n}\equiv 1modm$. The Carmichael numbers^{} are ${K}_{1}$. There are infinitely many Knodel number ${K}_{n}$ for a given $n$, something which was first proven only for $n>2$. Erdős speculated that this was also true for $n=1$ but two decades passed before this was conclusively proved by Alford, Granville and Pomerance.

## References

- 1 W. R. Alford, A. Granville, and C. Pomerance. “There are Infinitely Many Carmichael Numbers” Annals of Mathematics 139 (1994): 703 - 722
- 2 P. Ribenboim, The Little Book of Bigger Primes, (2004), New York: Springer-Verlag, p. 102.

Title | Knödel number |
---|---|

Canonical name | KnodelNumber |

Date of creation | 2013-03-22 16:06:54 |

Last modified on | 2013-03-22 16:06:54 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A51 |

Synonym | Knodel number |