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Statement

Setting (pp. 2--4). A positive integer $n$ is a covering number (Definition 1.1, p. 2) if some cover of $\mathbb Z$ by finitely many residue classes has its moduli distinct, greater than one and dividing $n$; a covering number is primitive (Definition 1.2, p. 4) if none of its proper divisors is a covering number.

Theorem 1.2 (p. 4). Let $\alpha_1,\ldots,\alpha_r$ be positive integers. There are distinct primes $p_1<\cdots<p_r$ for which $p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ is a covering number if and only if one of the following holds:

• (i) $r=2\le\alpha_1$;
• (ii) $r=3$ and $\max\{\alpha_1,\alpha_2\}\ge2$;
• (iii) $r\ge4$.

The exponent of the largest prime is unrestricted in every case.

Source. Zhi-Wei Sun, On covering numbers, Integers 7 (2007), no. 2, A33, also printed in Combinatorial Number Theory (de Gruyter, Berlin, 2007), 443--453. Labels and pages here are those of arXiv:math/0601017v2 (9 September 2006), the edition read, which is named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images of the print, and the proof was followed. Nothing here is independently reviewed.

Proof pointer

P. 7. Sufficiency is Theorem 1.1 with the first $r$ primes: $2^{\alpha_1}3^{\alpha_2}$ in case (i), $2^{\alpha_1}3^{\alpha_2}5^{\alpha_3}$ in case (ii), and in case (iii) $p_s<2^{s-1}$ for $s\ge4$, by induction from Bertrand's postulate. For necessity, the least covering number $d>1$ dividing $n$ is primitive; Lemma 2.1 (p. 6) rules out a prime power, and in the excluded cases ($r=2$, $\alpha_1=1$; $r=3$, $\alpha_1=\alpha_2=1$) it yields $2\ge p_2$ or $4\ge p_3$, both false.

Dependencies

Theorem 1.1; Lemma 2.1 (p. 6): if $p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ is a covering number and $\prod_{t<r}p_t^{\alpha_t}$ is not, then $\prod_{t<r}(\alpha_t+1)\ge p_r$; its proof cites a counting theorem of Z. W. Sun and Z. H. Sun (1987), or the author's 1996 paper, Corollary 3.

Bears on

No problem directly.