Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Primitive Covering Numbers   In 2007, Zhi-Wei Sun defined a \emph{covering number} to be a positive
integer $L$ such that there exists a covering system of the integers where the
moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is
called \emph{primitive} if no proper divisor of $L$ is a covering number. Sun
constructed an infinite set $\mathcal L$ of primitive covering numbers, and he
conjectured that every primitive covering number must satisfy a certain
condition. In this paper, for a given $L\in \mathcal L$, we derive a formula
that gives the exact number of coverings that have $L$ as the least common
multiple of the set $M$ of moduli, under certain restrictions on $M$.
Additionally, we disprove Sun's conjecture by constructing an infinite set of
primitive covering numbers that do not satisfy his primitive covering number
condition.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol