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Claim. The answer to Problem 2 is no: the smallest modulus of a finite covering system with pairwise distinct moduli greater than one is bounded by an absolute constant. Theorem 1 of B. Hough, Solution of the minimum modulus problem for covering systems, states that every such system has least modulus at most $10^{16}$; more precisely, for every finite set of distinct integers greater than $10^{16}$ and every choice of one residue class to each of them, the uncovered integers form a periodic set of positive density. The published paper and the arXiv v3 state $10^{16}$; the arXiv v2 stated $10^{18}$, and the arXiv v1 of 2013-07-02, which names this page, gave an unspecified absolute bound. Distinctness of the moduli is essential, since repeated moduli cover trivially. The theorem is compiled on the library's Theorem 1 page, with a qualitative version that already settles the question from elementary prime-counting bounds without the numerical certificate; the proof filters the moduli by primes, applies a relative form of the Lovász local lemma on surviving residue fibers, and reweights to control the bias statistics.

Depends on. Nothing in this wiki; the theorem is the paper's own. The precursor of Filaseta, Ford, Konyagin, Pomerance and Yu (J. Amer. Math. Soc. 20 (2007)), which Hough builds on, bounded the uncovered density under extra restrictions on the large moduli and is background here; it is an accepted partial claim on its own claim page.

Acceptance. Refereed: Annals of Mathematics (2) 181 (2015), no. 1, 361--382, doi:10.4007/annals.2015.181.1.6, received December 22, 2013; Crossref dates the issue to January 2015. Reviewed: the site's curator, Thomas Bloom, labels the problem disproved and credits Hough, building on [FFKPY07], with the answer no and the bound $10^{16}$ (page last edited 5 April 2026; its thread holds only the 2025 correction of the bound from $10^{18}$ to $10^{16}$, and its proof-claim tab is empty), and the community database records the problem disproved. The library's compilation of the complete proof and its finite numerical certificate is author-recorded coverage by this project and awards nothing here. Not listed as formalized: the site's label reads DISPROVED (LEAN), but the catalog's statement (google-deepmind/formal-conjectures, ErdosProblems/2.lean) expresses the negative answer with a sorry body at the revision of 4 September 2026, as the problem page's Formalization section records, and the Lean development behind the site's label formalizes the proof of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, linked on their claim page, not Hough's; this corpus has built neither file.

Not covered. The largest attainable minimum modulus. The later proof of Balister, Bollobás, Morris, Sahasrabudhe and Tiba lowers the bound to $616000$, on its own claim page; the best construction known to the problem page, Owens's 2014 thesis, has minimum modulus $42$.