Asked by Erd\H{o}s and Selfridge (sometimes also with Schinzel). They also asked whether there can be a covering system such that all the moduli are odd and squarefree. The answer to this stronger question is no, proved by Balister, Bollob\'{a}s, Morris, Sahasrabudhe, and Tiba \cite{BBMST22}.

Hough and Nielsen \cite{HoNi19} proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollob\'{a}s, Morris, Sahasrabudhe, and Tiba \cite{BBMST22}, who also prove that if an odd covering system exists then the least common multiple of its moduli must be divisible by $9$ or $15$.

Selfridge has shown (as reported in \cite{Sc67}) that such a covering system exists if a covering system exists with moduli $n_1,\ldots,n_k$ such that no $n_i$ divides any other $n_j$ (but the latter has been shown not to exist, see [586]).

Filaseta, Ford, and Konyagin \cite{FFK00} report that Erd\H{o}s, 'convinced that an odd covering does exist, offered \$25 for a proof that no odd covering exists; Selfridge, convinced (at that point) that no odd covering exists, offered \$300 for the first explicit example...no award was promised to someone who gave a non-constructive proof that an odd covering of the integers exists...Selfridge (private communication) has informed us that he is now increasing his award to \$2000.'


References


[BBMST22] Balister, Paul and Bollob\'{a}s, B\'{e}la and Morris, Robert
and Sahasrabudhe, Julian and Tiba, Marius, On the Erd\H{o}s covering problem: the density of the uncovered set. Invent. Math. (2022), 377-414.

[FFK00] Filaseta, M. and Ford, K. and Konyagin, S., On an irreducibility theorem of {A}. {S}chinzel associated
with coverings of the integers. Illinois J. Math. (2000), 633--643.

[HoNi19] Hough, Robert D. and Nielsen, Pace P., Covering systems with restricted divisibility. Duke Math. J. (2019), 3261-3295.

[Sc67] Schinzel, A., Reducibility of polynomials and covering systems of congruences. Acta Arith. (1967/68), 91-101.