Asked by Erdős 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, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

Hough and Nielsen [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, Bollobás, Morris, Sahasrabudhe, and Tiba [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 [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 [FFK00] report that Erdős, '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.'

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This page was last edited 11 January 2026.