The minimum modulus problem for covering systems

 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 Erdős, 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.