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Source. Item 8 on p. 7 of the author survey, an unnumbered summary. The survey attributes both statements to Zhi-Wei Sun, On covering systems with distinct moduli, J. Yangzhou Teachers College (Nat. Sci. Ed.) 11 (1991), no. 3, 21--27.

Conventions

A cover is a finite system $\{a_s(n_s)\}_{s=1}^k$ of residue classes $a_s(n_s)=a_s+n_s\mathbb Z$ whose union is $\mathbb Z$. It is minimal (irredundant) when no proper subsystem is a cover.

Statement

1. The survey reports that the following two questions are equivalent. The question of V. Billik and H. M. Edgar (1973) asks whether, for every $d\in\mathbb Z^+$, there is a minimal cover whose moduli are all distinct and have greatest common divisor $d$. The question of Erdős asks whether, for every $c>0$, there is a cover whose moduli are all distinct and all greater than $c$.

2. Let $\{a_s(n_s)\}_{s=1}^k$ be a cover with $n_1<\cdots<n_k$, and suppose that no cover has all its moduli distinct and greater than $n_1$. Then $3n_1$ or $4n_1$ divides some $n_s$.

Proof scope. The survey states both results without proof; it credits them to the 1991 paper, which is their proof source.

Bears on

• Problem 2: the survey states Erdős's question in the distinct-moduli form, reports its equivalence with the Billik–Edgar question, and reports a divisibility property of the moduli of a cover with least modulus $n_1$ under the hypothesis that no cover has distinct moduli all greater than $n_1$. Neither statement answers the question.