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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.

Corollary 1.2 (p. 4, quoted). "For any $r=2,3,\ldots$ there are infinitely many primitive covering numbers having exactly $r$ distinct prime divisors."

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 on the page image of the print, and the two-line proof was followed. Nothing here is independently reviewed.

Proof pointer

P. 4. By Dirichlet's theorem, for every positive $m$ there are infinitely many primes $p$ with $m\mid p-1$, so chains $2=p_1<\cdots<p_r$ meeting the hypotheses of Theorem 1.3 can be chosen with $p_r$ arbitrarily large, and the theorem gives a distinct primitive covering number for each.

Dependencies

Theorem 1.3; Dirichlet's theorem on primes in arithmetic progressions.

Bears on

No problem directly.