# Sloane’s conjecture on multiplicative digital root

It is believed that there is no integer has a multiplicative persistence greater than itself, a conjecture put forth in 1973 by Neil Sloane, and that Sloane meant to limit this conjecture to fixed radix bases.

In 1998, Diamond and Reidpath published a factorial base counterexample, by proving that “it is possible to find a number in factorial base of arbitrarily large persistence,” specifically a number of the form

$$n!n+\sum _{i=1}^{n-1}i!$$ |

Obviously, this number will have a factorial base multiplicative digital root of $n$ and a persistence of also $n$, suggesting an upper bound for the desired counterexample.

## References

- 1 M. R. Diamond, D. D. Reidpath, “A Counterexample to Conjectures by Sloane and Erdos Concerning the Persistence of Numbers”, J. Rec. Math. 29 (1998), 89 - 92.
- 2 N. J. A. Sloane, “The persistence of a number” J. Rec. Math. 6 (1973), 97 - 98.

Title | Sloane’s conjecture on multiplicative digital root |
---|---|

Canonical name | SloanesConjectureOnMultiplicativeDigitalRoot |

Date of creation | 2013-03-22 16:00:45 |

Last modified on | 2013-03-22 16:00:45 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 7 |

Author | CompositeFan (12809) |

Entry type | Conjecture |

Classification | msc 11A63 |

Synonym | Sloane-Erdős conjecture on multiplicative digital root |