A Question of Erdős and Graham on Covering Systems

TL;DR

The paper investigates whether the divisors of an integer $n>1$ can form a distinct CD covering system, addressing a question posed by Erdős and Graham. It develops a density-based framework for CD congruence sets and uses Euler-product bounds to show that no such CD covering exists for any $n$, by analyzing structural forms of $n$ and bounding the CD-density below 1 in all cases. Beyond the nonexistence result, it analyzes when divisors of $n$ can serve as moduli of a CD congruence set, establishing a necessary condition for non-intersecting $n$ and proving non-intersecting results for several families (notably $n=p^k$ and certain $n=qp^k$ cases) with explicit density computations. The work thus rules out the original Erdős–Graham CD-covering possibility and provides partial classifications and conjectures guiding future study of CD covering systems and divisor-based moduli.

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, $a\mod d$ and $a'\mod d'$, then $\gcd(d,d')=1$. 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.

Theorems & Definitions (15)

• Lemma 2.1
• proof : Proof of Lemma \ref{['good']}
• Lemma 2.2
• proof
• Proposition 2.3: Folklore
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof