---

Claim. Every covering system with pairwise distinct moduli, each of the form $p-1$ for a prime $p\ge5$, contains a modulus $p-1$ with $p>877$; equivalently, the $149$ moduli $p-1$ with $5\le p\le877$ do not cover the integers. This is Theorem 1.1 (the range constraint) of the seven-page research note On covering systems with moduli of the form $p-1$, signed "Research note" and dated July 11, 2026, in the repository linked above, a partial result toward Problem 273: it neither constructs such a system nor excludes one. Theorem 1.2 (the budget constraint) adds that every such covering system has $\sum1/(p-1)\ge1+\exp(-3.363054\times10^{21})$, derived from the framework of Filaseta and Kalogirou with the observation that the Pierpont primes of the pool contribute about $0.706$ to the reciprocal sum. Proposition 2.1 (the parity reduction) notes that all the moduli are even, so such a covering system exists exactly when two covering systems with disjoint sets of distinct moduli from $\{(p-1)/2:p\ge5\}$ exist. The repository holds the note's TeX source, the scripts and rational certificates behind the two theorems, and a Lean file that checks the two halving equivalences and the parity constraint behind Proposition 2.1 and its covering half (an even-modulus system covers the integers exactly when both parity halves do), leaving the step from distinct moduli to disjoint halves to the repository's notes and checking nothing of Theorem 1.1 or 1.2; the note says it was produced in a human-directed workflow with AI assistants in executor and validator roles, and the thread post credits unnamed AI assistants. The note's author is pseudonymous (the site user ideal_ombrer), which names this page.

Covers. The nonexistence of such a covering system whose moduli all have $p\le877$, a partial step toward a negative answer. The budget bound of Theorem 1.2 settles no instance. Whether any such covering system exists remains open on the note's own account. The later deposit of Zeraoulia on Zeraoulia's claim page bounds the least common multiple of the moduli instead.

Depends on. Nothing in this wiki.

Standing. Claimed. The page is dated by the note's own date and by the commit of the linked revision, both 2026-07-11; the repository was created on 2026-07-10 UTC with an earlier compilation of the note, and the thread post announcing it followed on 2026-07-12. The note is not refereed, no outside review is recorded, and the Lean file covers only part of the parity reduction, which this corpus has not built or audited, so no evidence kind is listed. The site's label is OPEN, and the claim is partial and derives nothing for the problem's standing.