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On Odd Covering Systems with Distinct Moduli
theorem_1: Forces at least twenty-two prime divisors in the least common multiple of any distinct odd covering whose common multiple is square-free.
Song Guo and Zhi-Wei Sun, On odd covering systems with distinct moduli, Advances in Applied Mathematics 35 (2005), no. 2, 182--187. DOI 10.1016/j.aam.2005.01.004.
The copy read for this card is the seven-page arXiv:math/0412217v2, dated 14 September 2005. Its first page carries the journal citation and records receipt on 10 December 2004 and acceptance on 22 January 2005. The statement below is cited to this v2 pagination; no separate publisher PDF was compared. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0412217), every other right reserved.
Theorem 1 improves earlier necessary conditions for a hypothetical distinct odd covering in the square-free case. It says that the least common multiple of the moduli must contain at least twenty-two distinct primes. This is a historical restriction on a special case of Problem 7, not a resolution of the unrestricted question.
Within the square-free case, this older necessary condition is superseded by Balister--Bollobás--Morris--Sahasrabudhe--Tiba's Theorem 1.1, which proves that every finite distinct cover by square-free moduli greater than one contains an even modulus. That later theorem excludes the square-free case of Problem 7 entirely. Its complete proof and certificate scope are recorded on its canonical pages and are not re-reviewed in this filing.
Compiled scope
The title, abstract, introduction, and Theorem 1 on PDF pp. 1--3 were read. The theorem statement is restated below. The proof in Section 2 was not reconstructed or independently checked.
Read status. Claims checked: Theorem 1 (p. 2) was read clause by clause against the print. The proof (pp. 3--6) is unchecked.
Result
Bears on. Problem 7: Theorem 1 is a necessary condition in the square-free case only. A distinct odd covering whose least common multiple is square-free would need at least prime divisors of that multiple. The theorem does not decide that case or the unrestricted question.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.