Results for distinct covering systems with a prime minimum modulus $q \ge 11$

Source abstract

We prove that if the smallest modulus of a covering system is a prime $q \ge 11$, then the sum of the reciprocals of the moduli is at least $1+10^{-28}$. This improves on a previous more general bound of the first two authors in this particular case. We also show that it is impossible to construct a covering system using distinct multiples of $q$ as moduli. Finally, we show as a corollary that it is impossible to construct $10$ distinct covering systems with the property that no modulus is used in more than one of the covering systems.