# proof that the set of sum-product numbers in base 10 is finite

This was first proven by David Wilson, a major contributor to Sloane’s Online Encyclopedia of Integer Sequences.

First, Wilson proved that ${10}^{m-1}\le n$ (where $m$ is the number of digits of $n$) and that

$$\sum _{i=1}^{m}{d}_{i}\le 9m$$ |

and

$$\prod _{i=1}^{m}{d}_{i}\le {9}^{m}$$ |

. The only way to fulfill the inequality ${10}^{m-1}\le {9}^{m}9m$ is for $m\le 84$.

Thus, a base 10 sum-product number can’t have more than 84 digits. From the first ${10}^{84}$ integers, we can discard all those integers with 0’s in their decimal representation. We can further eliminate those integers whose product^{} of digits is not of the form ${2}^{i}{3}^{j}{7}^{k}$ or ${3}^{i}{5}^{j}{7}^{k}$.

Having thus reduced the number of integers to consider, a brute force search by computer yields the finite set^{} of sum-product numbers in base 10: 0, 1, 135 and 144.

Title | proof that the set of sum-product numbers in base 10 is finite |
---|---|

Canonical name | ProofThatTheSetOfSumproductNumbersInBase10IsFinite |

Date of creation | 2013-03-22 15:46:58 |

Last modified on | 2013-03-22 15:46:58 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Proof |

Classification | msc 11A63 |

Synonym | Proof that the set of sum-product numbers in decimal is finite |