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The existence of odd covering system with distinct moduli is a famous open question proposed by Erdős and Selfridge.

I wonder whether a restricted condition for the problem that odd covering system without modulus 3 is also an open problem. That is to say, whether an incongruent covering system with the minimum modulus greater than 3 and whose moduli are odd exist or not.

Thanks.

p.s. The same question was posted on MSE. If I get an answer from one site I would delete the other.

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  • $\begingroup$ This is weaker than what you're asking, but Hough and Neilsen have proved that in an odd covering system at least one modulus must be divisible by 3. (arxiv.org/pdf/1703.02133.pdf) $\endgroup$
    – Thomas Bloom
    Commented Mar 3, 2022 at 11:05
  • $\begingroup$ I suspect that your question is still an open problem. $\endgroup$
    – Thomas Bloom
    Commented Mar 3, 2022 at 11:07
  • $\begingroup$ @ThomasBloom : Thanks. It helped me a lot. $\endgroup$
    – user1851281
    Commented Mar 3, 2022 at 12:49
  • $\begingroup$ I note that OP has deleted the math.se post. $\endgroup$
    – Gerry Myerson
    Commented Mar 5, 2022 at 2:42

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