---

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. For a predicate $P$, $[\![P]\!]$ is $1$ if $P$ holds and $0$ otherwise (p. 3).

Conjecture 1.1 (p. 5). Every primitive covering number can be written as $p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with $p_1,\ldots,p_r$ distinct primes and $\alpha_1,\ldots,\alpha_r$ positive integers so that

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

the condition of Theorem 1.1.

Remark 1.4 (p. 5). The author dates the conjecture to 16 July 1988. Since (1.3) forces $p_1=2$, the paper calls Conjecture 1.1 stronger than the Erdős--Selfridge conjecture, which it states on p. 2: a cover of $\mathbb Z$ whose moduli are distinct and greater than one cannot have all moduli odd.

The abstract (p. 1) restates the conjecture as $\prod_{0<t<s}(\alpha_t+1)\ge p_s-1$ for each $s$, "with strict inequality when $s=r$", for the primes in a suitable order; this is the same condition, since for integers a strict inequality over $p_r-1$ means at least $p_r$, the right side of (1.3) at $s=r$.

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 conjecture, Remark 1.4 and the abstract's restatement were read on the page images of the print. The paper gives no proof. Nothing here is independently reviewed.

Bears on

Problem 7: by the paper's Remark 1.4 the conjecture would imply the Erdős--Selfridge conjecture, that no cover of $\mathbb Z$ with distinct moduli greater than one has all moduli odd, and so a negative answer to the problem. The paper proves neither.