Finiteness of Disjoint Covering Systems with Precisely One Repeated Modulus

TL;DR

This paper proves that for each fixed number of repetitions, only finitely many disjoint covering systems with a minimum modulus of at least 3 exist where only the largest modulus is repeated exactly m times.

Contribution

It establishes the finiteness of such covering systems for each fixed repetition count, advancing understanding of their structural limitations.

Findings

Finitely many such systems exist for each fixed m ≥ 2.

The largest modulus is the only repeated one in these systems.

Minimum modulus in these systems is at least 3.

Abstract

We prove that for each fixed $m \ge 2$, there are only finitely many disjoint covering systems with minimum modulus at least $3$ in which precisely one modulus is repeated, namely the largest modulus, and it occurs exactly $m$ times.