Searching for Disjoint Covering Systems with Precisely One Repeated Modulus

arXiv1511.04293MaRDI QIDQ6267393

Doron Zeilberger, Shalosh B. XIV Ekhad, Aviezri S. Fraenkel

Publication date: 13 November 2015

Abstract: A set of arithmetical sequences ${\displaystyle }$ a_1, (�mod{ ,, m_1}) quad, quad a_2 , (�mod{,, m_2}) quad, quad dots quad , quad a_k , (�mod{,,m_k}) quad quad , ${\displaystyle }$ with ${\displaystyle }$ m_1 leq m_2 leq dots leq m_k quad quad , ${\displaystyle }$ is called a {it disjoint covering system} (alias {it exact covering system}) if every positive integer belongs to {�f 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 $leq600$, and we are pretty sure that they are complete.

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