A further investigation on covering systems with odd moduli

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Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann and Tony W. H. Wong, A further investigation on covering systems with odd moduli, Discrete Mathematics 349 (2026), article 115013, doi:10.1016/j.disc.2026.115013.

The Markdown reading copies beside the two PDFs transcribe, one for each version, the repeated-modulus lifting theorem, the bounds for $t_k$, and the theorem giving an odd covering of the union of seven arithmetic subsets.

The preferred publisher-format PDF is an author-hosted Elsevier file (10 physical pages) linked from the corresponding author's Kutztown publication page. Its visible and embedded article identity agrees with the publisher record, but the direct ScienceDirect article and PDF routes returned HTTP 403. The publisher-format PDF prints "0012-365X/© 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)." on its first page: the Creative Commons Attribution 4.0 license. For the arXiv v1 PDF, the arXiv record names the Creative Commons Attribution 4.0 license (arXiv:2507.16135).

The retained arXiv:2507.16135v1 PDF is the version submitted 22 July 2025 and dated 3 September 2025 (14 physical pages). The two versions print different hypotheses in Theorem 2.2, the lifting from a repeated modulus $k$ to a repeated modulus $km$: the publisher version (p. 2) takes any odd integer $m>1$, while arXiv v1 (p. 4) takes any integer $m\ge2$, for which an even $m$ would make the modulus $km$ even.

The repeated-modulus results give qualified context for Problem 7. The paper explicitly cites [[covering_systems/harrington_sun_wong_2022_covering_systems_odd_moduli/_index|Harrington, Sun and Wong (2022)]] for the prime bounds it improves.