Covering systems with odd moduli

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corollary_5_2: Harrington, Sun and Wong's corollary that for all primes p at least 7 some covering system uses p exactly p - 1 times as a modulus while its other moduli are odd, distinct, square-free and greater than 1.

lemma_5_1: Harrington, Sun and Wong's lemma that a covering system with a p-node root, distinct moduli except that p is used exactly p - t times, and no modulus divisible by p^2, yields for every prime q > p a covering system using q exactly q - t times, keeping oddness and square-freeness.

question_5_3: Harrington, Sun and Wong's question whether some constant epsilon with 0 <= epsilon < 1 has t_p at most epsilon p for all sufficiently large primes p.

theorem_3_2: Harrington, Sun and Wong's theorem that if, for a prime p at least 3, some covering system has odd, square-free moduli, distinct except that p is used exactly twice, then an odd covering of the integers exists.

theorem_3_4: Harrington, Sun and Wong's construction of a covering system whose moduli are odd, square-free and distinct except that 7 is used exactly six times, so tau_7 is at most 6.

theorem_4_1: Harrington, Sun and Wong's construction of a covering system whose moduli are odd and distinct except that 7 is used exactly four times, so t_7 is at most 4.

theorem_4_2: Harrington, Sun and Wong's construction of a covering system whose moduli are odd and distinct except that 11 is used exactly seven times, so t_11 is at most 7.

theorem_4_3: Harrington, Sun and Wong's construction, for every prime p at least 23, of a covering system whose moduli are odd and distinct except that p is used exactly p - 5 times, so t_p is at most p - 5.

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Joshua Harrington, Yewen Sun and Tony W. H. Wong, Covering systems with odd moduli, Discrete Mathematics 345 (2022), article 112936, doi:10.1016/j.disc.2022.112936.

For an odd prime $p$, the paper writes $t_p$ for the least nonnegative integer $t$ such that some covering system uses $p$ as a modulus exactly $t$ times while its other moduli are odd, distinct and greater than $1$, and $\tau_p$ for the same quantity when the other moduli must also be square-free (Questions 1.4 and 1.5, p. 2). Theorem 3.2 (p. 5) shows that if, for some prime $p\geq3$, a covering system has odd, square-free moduli that are distinct except that $p$ occurs exactly twice, then an odd covering exists. Theorem 3.4 (p. 7) gives $\tau_7\leq6$, and Corollary 5.2 (p. 11) gives $\tau_p\leq p-1$ for every prime $p\geq7$. Theorems 4.1--4.3 (pp. 7--8) give $t_7\leq4$, $t_{11}\leq7$, and $t_p\leq p-5$ for every prime $p\geq23$. Lemma 5.1 (pp. 9--10) moves a cover whose root is a $p$-node with $p-t$ leaves of modulus $p$ ($1\leq t\leq p$), whose moduli are distinct except that $p$ is used exactly $p-t$ times, and none of whose moduli is divisible by $p^2$, to every prime $q>p$, with $q$ used exactly $q-t$ times, no modulus divisible by $q^2$, and oddness and square-freeness kept; with Theorem 4.2 it gives $t_p\leq p-4$ for $11\leq p\leq19$ (Table 1, p. 11). Question 5.3 (p. 11) asks whether $t_p\leq\epsilon p$ for some constant $0\leq\epsilon<1$ and all sufficiently large primes $p$.

The copy read for this card is the published article, twelve physical pages. The arXiv:2104.00602v1 preprint has eighteen physical pages. The arXiv:2104.00602v1 preprint is stamped 1 April 2021 and carries the manuscript date 2 April 2021. The published article is used for the primary page locators; the preprint serves for version comparison. The published article PDF prints "0012-365X/© 2022 Elsevier B.V. All rights reserved." on its first page. For the arXiv v1 preprint PDF, the arXiv record names the Creative Commons Attribution 4.0 license (arXiv:2104.00602).

Read status. Claims checked: Theorems 3.2, 3.4 and 4.1--4.3, Lemma 5.1, Corollary 5.2 and Question 5.3 were read clause by clause on the printed pages. The proofs were read but not checked step by step, and the tree-diagram constructions were not checked to cover the integers.

Bears on. Problem 7: every cover the paper constructs repeats one prime modulus, so none is a distinct odd covering and none answers Problem 7. By Theorem 3.2, the bound $\tau_p\leq2$ for one odd prime $p$ would give a distinct odd covering; the converse is not claimed. The paper's best bound is $\tau_p\leq p-1$ for $p\geq7$. Theorem 1.1 of Balister et al. (2021), that a cover by distinct square-free moduli greater than $1$ has an even modulus, gives $\tau_p\geq2$ for every odd prime $p$, since a cover using $p$ at most once would have distinct, odd, square-free moduli greater than $1$; this paper does not cite it.

Results. Theorem 3.2 (p. 5); Theorem 3.4 (p. 7); Theorem 4.1 (p. 7); Theorem 4.2 (p. 8); Theorem 4.3 (p. 8); Lemma 5.1 (pp. 9--10); Corollary 5.2 (p. 11); Question 5.3 (p. 11). Lemma 3.1 (p. 5) is a proof step of Theorem 3.2, and Lemma 5.4 (pp. 11--12) is a variant of Lemma 5.1 that the paper does not apply; Lemma 3.1 is described on the Theorem 3.2 page and Lemma 5.4 on the Lemma 5.1 page.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.