A Question of Erdős and Graham on Covering Systems

Abstract

Erdős and Graham (Erdős and Graham, 1980) asked if there exists an $n$ such that the divisors of $n$ greater than 1 are the moduli of a distinct covering system with the following property: If there exists an integer which satisfies two congruences in the system, $a\mathrm{mod}d$ and $a^{\prime }\mathrm{mod}{d}^{\prime }$, then $\gcd (d,d^{\prime })=1$. We show that such an $n$ does not exist. This problem is part of Problem # 204 on the website www.erdosproblems.com, compiled and maintained by Thomas Bloom. We also study when the divisors of $n$ greater than $1$ can form a congruence system satisfying the above condition.

Clarification added to Prop 4.2 10 pages