Odd incongruent covering systems

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.

Can a finite covering system have pairwise distinct odd moduli $q_1,\ldots,q_k>1$? The conjecture says that no integers $a_1,\ldots,a_k$ and such moduli satisfy \[ \mathbb Z=\bigcup_{i=1}^k(a_i+q_i\mathbb Z). \] [S2]

Short English statement

Must every distinct covering system with moduli greater than one contain an even modulus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3381 (OPG-491). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Odd incongruent covering systems, OPG-491. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/odd_incongruent_covering_systems.