Searching for Disjoint Covering Systems with Precisely One Repeated Modulus

Abstract

A set of arithmetical sequences $$a_1(\mathrm{mod}{m}_1),a_2(\mathrm{mod}{m}_2),\dots ,a_k(\mathrm{mod}{m}_k),$$ with $$m_1\le m_2\le \cdots \le m_k,$$ is called a {\it disjoint covering system} (alias {\it exact covering system}) if every positive integer belongs to {\bf exactly} one of the sequences. Mirski, Newman, Davenport and Rado famously proved that the moduli can't all be distinct. In fact the two largest moduli must be equal, i.e. $m_{k-1}=m_k$ This raises the natural question:"How close can you get to getting distinct moduli?", in other words, can you find all such systems where all the moduli are distinct except the largest, that is repeated $r$ times, for any, specific given $r$? It turns out (conjecturally, but almost certainly) that excluding the trivial case where the smallest modulus is 2, for any number of repeats $r$, there are only finitely many such systems. Marc Berger, Alexander Felzenbaum and Aviezri Fraenkel found them all for $r$ up to $9$, and Mekmamu Zeleke and Jamie Simpson extended the list for systems up to $12$ repeats. In the present article we continue the list up to $r=32$. All our systems are correct, but we did not bother to formally prove completeness, but we know for sure that the lists are complete if the largest modulus is $\le 600$, and we are pretty sure that they are complete.

Accompanied by a Maple package available from http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ecs.html