Covering systems with big moduli

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For integers $a$ and $q\ge1$, let $a+q\mathbb Z=\{a+qt:t\in\mathbb Z\}$. A covering system is a finite family of these residue classes whose union is $\mathbb Z$. It is distinct if its moduli are pairwise unequal.

Statement recorded in the supplied source.

For every integer $N\ge1$, does there exist a finite covering system \[ \mathbb Z=\bigcup_{i=1}^k(a_i+q_i\mathbb Z) \] with pairwise distinct moduli $q_i\ge N$? [S2]

Short English statement

Can distinct covering systems have arbitrarily large minimum modulus? The historical question has a negative answer: Hough proves a uniform upper bound of $10^{16}$ for the least modulus. [S3]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3382 (OPG-493). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Covering systems with big moduli, OPG-493. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/covering_systems_with_big_moduli.
• S3 Bob Hough, Solution of the minimum modulus problem for covering systems, Annals of Mathematics 181 (2015), 361--382; arXiv version 3 (2016). Introduction, Theorem 1, PDF page 1. https://arxiv.org/abs/1307.0874v3.