Analysis Seminar | |

Topic: | The minimum modulus problem for covering systems |

Speaker: | Robert Hough |

Affiliation: | Member, School of Mathematics |

Date: | Wednesday, May 4 |

Time/Room: | 4:30pm - 5:30pm/S-101 |

Video Link: | https://video.ias.edu/analysis/2016/0504-Hough |

A distinct covering system of congruences is a finite collection of arithmetic progressions to distinct moduli \[ a_i \bmod m_i, 1 < m_1 < m_2 < \cdots < m_k \] whose union is the integers. Answering a question of ErdÅ‘s, I have shown that the least modulus $m_1$ of a distinct covering system of congruences is at most $10^{16}$. I will describe aspects of the proof, which involves the theory of smooth numbers and a relative form of the Lovász local lemma.