Abstract
In 2007, Zhi-Wei Sun defined a covering number to be a positive integer [Formula: see text] such that there exists a covering system of the integers where the moduli are distinct divisors of [Formula: see text] greater than 1. A covering number [Formula: see text] is called primitive if no proper divisor of [Formula: see text] is a covering number. Sun constructed an infinite set [Formula: see text] of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given [Formula: see text], we derive a formula that gives the exact number of coverings that have [Formula: see text] as the least common multiple of the set [Formula: see text] of moduli, under certain restrictions on [Formula: see text]. 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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