Covering systems of congruences

13 February 2014
Bob Hough
A distinct covering system of congruences is a collection \[ (a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k \] whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.
  • Number Theory Seminar