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Claim. Write $\mu(n_1,\ldots,n_r)$ for the minimum density of the integers left uncovered by one class $a_i\pmod{n_i}$ per modulus, the quantity whose complement is the first question of Problem 278. Cambie's note argues that this first question has no general answer of the kind the problem seems to ask for, in three parts. For the distinct moduli $\{3\}\cup\{3p:p\in P\}$, $P$ a finite set of primes other than $3$, the minimum is

$$\mu=\frac13\min_{P=P_1\sqcup P_2} \Bigl(\prod_{p\in P_1}\bigl(1-\tfrac1p\bigr) +\prod_{p\in P_2}\bigl(1-\tfrac1p\bigr)\Bigr),$$

a balanced-partition problem on the weights $-\log(1-1/p)$, and moduli $\{qp:p\in P\}$ with a common prime $q$ give a $q$-part analogue, so different common-divisor patterns lead to different formulas. For primes chosen in a short interval, the subset-sum order of these weights reproduces the order type of Chvátal's hard knapsack instances, which obstructs the comparison-based recursive algorithms of Chvátal's class on distinct moduli. For lists of moduli in which a modulus may repeat, deciding whether $\mu=0$ is NP-hard, because it is the Exact Pinwheel Covering problem that Kawamura, Kobayashi and Kusano proved NP-hard, so an exact polynomial-time evaluation of $\mu$ would give $\mathrm P=\mathrm{NP}$. The second question is recorded in the note as settled by Simpson (claim page). The first version of the note (carded as Cambie 2025, with its results on the card) carries the partition formulas and the knapsack construction; the second version of 2026-08-17 restates them with the scope qualifications above and adds the NP-hardness theorem, whose observation the note's acknowledgment credits to GPT Pro Sol 5.6, the system the site's claim names GPT pro 5.6 Sol. The site's proof-claims tab carries the second version as a proof claim submitted by Cambie on 2026-08-17; it had no comments on 2026-10-06.

Submission note. Posted to erdosproblems.com as a proof claim by Stijn Cambie (account StijnC) on 17 August 2026, giving "GPT pro 5.6 Sol" as the AI used:

Multiple arguments are given why the problem cannot be expected to have a computational efficient formula. Compared with the v1, the equivalence with the NP-hard pinwheel covering problem [Arxiv: Kawamura, Kobayashi and Kusano '25] has been added, which seems an even stronger argument (the latter observed by GPT pro 5.6).

Covers. The first question for two structured families, for which the exact minimum uncovered density is determined as a balanced-partition minimum: the distinct moduli $\{3\}\cup\{3p:p\in P\}$, by the displayed formula (Proposition 2 of the note's second version), and the moduli $\{qp:p\in P\}$ with a common prime $q$, by its $q$-part analogue, formula (3) there, with a $(q-1)$-part variant when $q$ itself is among the moduli. The knapsack order and the NP-hardness theorem settle no instance: they argue that no efficient general formula should be expected, and the note itself states what it leaves open, that it does not prove that no closed formula exists, since a formula could contain an optimization such as the balanced partition, and that the NP-hardness theorem does not cover the pairwise-distinct moduli of the problem's statement. For general moduli the value of the maximum covered density is not determined; Onishi's claim page offers an exact characterization and algorithm, which Cambie's own comment on that claim calls a complementary attempt reaching an essentially opposite answer, leaving the thread's closure to the site's curator.

Depends on. Nothing in this wiki: the inputs are Chvátal's theorem on hard knapsack problems and the NP-hardness theorem of Kawamura, Kobayashi and Kusano, cited in the note and not held in the library.

Standing. Claimed: the note is an arXiv preprint with no journal publication, referee report or curator acceptance located through 2026-10-06; the site labels the problem OPEN (page last edited 20 January 2026). The author announced the first version on the problem's discussion thread on 2025-08-26, which with the arXiv v1 date of 2025-08-25 gives the page its date; the proof claim was submitted on 2026-08-17 with the second version.