Problem 273

../

claims/: The 2 claim pages of Problem 273, one per claimant's result; the problem's standing derives from them.

---

Statement. Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?

Formulation. A covering system here is a finite family of congruence classes with pairwise distinct moduli. That is the source's reading: the formal-conjectures definition of a strict covering system notes that it is the notion Erdős used in [ErGr80], and erdos_273 asks for one. Without distinct moduli, the four classes modulo $4=5-1$ cover the integers. Without finiteness, the classes $z_i\pmod{p_i-1}$ cover them, for an enumeration $(z_i)$ of the integers and distinct primes $p_i\ge5$. The site's remark that Selfridge found an example from divisors of $360$ when $p=3$ is allowed is notable only under this reading, since the two classes modulo $2=3-1$ already cover.

Status. Open, the site's label (OPEN; page last edited 01 October 2025). The site's proof-claims tab carries a partial proof claim by Rafik Zeraoulia (submitted 2026-07-27, with OpenAI GPT-5.6 Thinking as the assisting system the claim names) that any such covering system with distinct moduli has moduli whose least common multiple is at least $393120$, by an exact computer-assisted sieve, recorded as claimed on its claim page. The discussion thread carries a research note of July 2026 by the pseudonymous user ideal_ombrer, whose Theorem 1.1 asserts that every such covering system uses a modulus $p-1$ with $p>877$, recorded as claimed on its claim page, and later posts, without a manuscript, asserting larger bounds on the least common multiple, described on Zeraoulia's claim page.

Source. erdosproblems.com/273, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #273, https://www.erdosproblems.com/273.

References.

• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980). Not a site reference; added for the covering-system convention the Formulation cites.

Formalization. Statement in formal-conjectures at the revision current on 2026-10-06, which states the question as erdos_273 with answer(sorry) and a sorry body and carries no formal_proof attribute; its variant for primes $p\ge3$ is marked solved with answer true, also with a sorry body.

Current assessment

The site formulation asks whether a covering system exists all of whose moduli have the form $p-1$ for primes $p\ge5$; the Formulation above records the distinct-moduli reading, the one the source and both claim pages use, and the standing answers the site's wording in that reading. The site labels the problem OPEN (page last edited 1 October 2025); no proof claim of the problem itself is recorded.

Partial results. Two claims restrict where such a covering system can live without deciding whether one exists. The research note of July 2026 by the pseudonymous user ideal_ombrer asserts that every such covering system uses a modulus $p-1$ with $p>877$, and that its reciprocal sum is at least $1+\exp(-3.363054\times10^{21})$; it is recorded as a claimed partial result on its claim page: unrefereed, with no outside review, and its Lean file covers only part of the parity reduction. Zeraoulia's deposit of 26 July 2026 asserts that the least common multiple of the moduli is at least $393120$, by an exact computer-assisted sieve; it is recorded as a claimed partial result on its claim page: unrefereed, with no named outside review, and its verification script has not been rerun by this corpus. Posts of 2026-09-24 and 2026-09-28 on the site's discussion thread by a second pseudonymous user report a re-implementation of Zeraoulia's certificate and exact searches excluding every least common multiple below $4\times10^6$ and then below $8\times10^6$, with code but no manuscript, so they have no claim page.

Search scope: the site's page and commentary (last edited 1 October 2025), its proof-claims tab (no comments on Zeraoulia's claim as of 2026-10-07) and its discussion thread (posts through 2026-09-28), and the formal-conjectures statement file at the revision current on 2026-10-06, which carries no formal_proof attribute. Remaining gap: whether any such covering system exists; neither claim constructs or excludes one, and no proof has been compiled or independently reviewed by this corpus.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• balister_2018_erdos_covering_problem_density_uncovered_set
• balister_2018_erdos_covering_problem_density_uncovered_set / theorem_1_2
• balister_2018_erdos_covering_problem_density_uncovered_set / theorem_1_4
• balister_2018_erdos_covering_problem_density_uncovered_set / theorem_8_1
• filaseta_2024_covering_systems_sum_reciprocals_moduli_close
• filaseta_2024_covering_systems_sum_reciprocals_moduli_close / theorem_1
• filaseta_2024_covering_systems_sum_reciprocals_moduli_close / theorem_2