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Statement

Setting (pp. 1--4). A cover of $\mathbb Z$ is a finite system of residue classes $a_i\pmod{n_i}$, $1\le i\le k$, whose union is $\mathbb Z$; it is minimal if no proper subsystem covers. A positive integer $n$ is a covering number (Definition 1.1, p. 2) if some cover of $\mathbb Z$ 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. For a predicate $P$, $[\![P]\!]$ is $1$ if $P$ holds and $0$ otherwise (p. 3).

Theorem 1.1 (p. 3). Let $p_1,\ldots,p_r$ be distinct primes and $\alpha_1,\ldots,\alpha_r$ positive integers. If

$$\prod_{0<t<s}(\alpha_t+1)\ \ge\ p_s-[\![r\ne s]\!]\qquad\text{for all }s=1,\ldots,r,\qquad(1.3)$$

then $p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ is a covering number.

Remark 1.1 (p. 3). The empty product is $1$, so (1.3) at $s=1$ forces $p_1=2$ and $r\ge2$.

Corollary 1.1 (p. 3). For any distinct primes $p_1=2<p_2<\cdots<p_r$ with $r>1$ there are positive exponents $\alpha_1,\ldots,\alpha_r$ making $p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ a covering number. The paper takes $\alpha_t=\lceil(p_{t+1}-[\![t\ne r-1]\!])/(p_t-1)\rceil-1$ for $t<r$ and checks (1.3) by a telescoping product; the paper calls the Erdős--Selfridge conjecture the converse of this corollary.

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, Remark 1.1 and Corollary 1.1 were read clause by clause on the page images of the print, and the proof was followed. Nothing here is independently reviewed.

Proof pointer

Pp. 5--7. Write $m_s=\prod_{t<s}p_t^{\alpha_t}$. Condition (1.3) says $m_s$ has at least $p_s-[\![r\ne s]\!]$ divisors, which supply distinct cofactors $d^{(s)}_j$. For each $s$ and each $\alpha\le\alpha_s$ the classes with moduli $d^{(s)}_jp_s^{\alpha}$, $j<p_s$, cover the integers divisible by $m_sp_s^{\alpha-1}$ but not by $m_sp_s^{\alpha}$; stacking these over $\alpha$ and $s$ covers every integer not divisible by $p_1^{\alpha_1}\cdots p_r^{\alpha_r}$, and one more class $0\pmod{d^{(r)}_{p_r}p_r^{\alpha_r}}$, available because $s=r$ gives one spare divisor, covers the rest. All moduli are distinct. The paper credits the basic ideas to Znám and to its author's 1990 paper (Remark 2.1, p. 6).

Dependencies

None beyond the divisor-count formula $d(n)=\prod(\alpha_t+1)$.

Bears on

No problem directly. It is the construction behind Theorem 1.2 and Theorem 1.3, and Conjecture 1.1 proposes its converse for primitive covering numbers.