Covering congruences in higher dimensions

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higher_dimensional_extension: Extends the homogeneous congruence and matrix covers from two coordinates to every integer dimension at least two.

lemma_1: Adds one vertical congruence for each prime divisor and lifts every composite residue class to obtain a homogeneous cover of Z squared.

lemma_2: Verifies the explicit twenty congruence classes branch by branch over the odd and even integers.

odd_even_composite_cover: Combines an explicit five-class cover of the odd integers with the doubled form of one external large-minimum covering system.

other_constructions_and_questions: Records source-level pointers that lack printed constructions and treats the paper's closing questions as historical rather than currently open.

subgroup_matrix_corollaries: Converts each primitive homogeneous congruence into a proper subgroup and an explicit full-rank matrix with the same index.

theorem: Constructs primitive homogeneous congruences with distinct moduli that cover every ordered pair of integers.

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Todd Cochrane and Gerry Myerson, Covering congruences in higher dimensions, Rocky Mountain Journal of Mathematics 26 (1996), no. 1, 77–81.

Source and version

The copy read for this card is the complete five-page scan at Cochrane's public PDF URL, downloaded from that URL (657308 bytes). All five physical pages were read visually. No notice is printed on the five scanned pages; the publisher's article page on Project Euclid could not be read on 2026-10-02 (DOI 10.1216/rmjm/1181072104, a bot challenge), its issue listing shows only an Open Access icon and names no license, and the Crossref record carries no license field; a bare Open Access icon names no license, every other right reserved.

The scan itself has no printed journal page numbers. The bibliographic pages 77–81 and volume and issue data come from Cochrane's author publication list, not from pagination visible in the scan; Crossref gives the same volume, issue and year. The scan is an author-hosted copy of the published work; it is not described as a publisher download or asserted byte-identical to a separate journal PDF.

The result pages cite the journal pages. The article's five pages 77–81 correspond in order to the five physical pages of the scan, so physical page $k$ is journal page $76+k$; the scan's running heads (author names on the even pages 78 and 80, the short title on 79 and 81) agree with that order. Each result page also gives the physical page.

Read status

Claims checked: every statement on the result pages was read clause by clause against the page images of the scan, and a second reader rechecked each statement, label and page against the scan. The proofs on the result pages are written here; the second reader also checked their arithmetic (the residue tables of Lemma 2, the odd and even lifts of the second construction, and the lattice-basis identity and determinant of the matrix corollary).

Complete proof chain

The paper calls a one-dimensional cover composite when all its distinct moduli are composite. Its Lemma 1 lifts any such cover to $\mathbb Z^2$: prime divisors of the product of the moduli handle nonunits in the second coordinate, and an inverse modulo that product reduces the unit case to the original cover.

Lemma 2 gives the required self-contained input, a twenty-class example attributed to John Selfridge. Its proof partitions every odd and even residue branch. All moduli divide $720$, are distinct, and are composite. Combining the lemmas proves the main theorem: twenty lifted classes and the three prime moduli $2,3,5$ give a primitive homogeneous cover with twenty-three distinct moduli.

The subgroup and matrix corollaries prove explicitly that each congruence kernel has index $m$ and construct a two-row lattice basis of determinant $-m$. The higher-dimensional extension adds zero coefficients, or an identity block to the matrices, for every dimension at least two.

A second composite-cover construction is complete relative to the paper's exact external input: one distinct cover whose moduli are all greater than $12$, cited there to Guy's Section F13. It combines an explicit odd lift with the doubled external cover of the even integers. This is one finite large-minimum example, not a claim that the minimum modulus can be arbitrarily large.

Context and limits

The context page records Dewar's construction, two additional Selfridge examples, and the paper's closing questions only at the scope printed here. Their constructions are absent from this source. Later Schinzel and Jin–Myerson papers are identified as leads; their proofs are not reviewed, so the 1996 questions are not relabeled as current open questions or marked resolved one by one.

The complete reconstructed content consists of Lemmas 1 and 2, the main theorem, the subgroup/matrix deduction, the higher-dimensional extension, and the odd/even construction relative to its stated external existence input. No proof from Guy, Dewar, Schinzel, Jin–Myerson, Fabrykowski, or Porubský is silently included.

No present-day priority, optimality, or formal-verification claim is made.

Bears on

• Problem 2: the second construction (p. 79) takes as input one covering system with distinct moduli all greater than $12$, which the paper does not construct but cites to Section F13 of Guy's 1981 Unsolved Problems in Number Theory. The paper proves nothing about the minimum modulus of a covering system with distinct moduli; its own explicit cover in Lemma 2 (pp. 79–80) has smallest modulus $4$.
• Problem 7: context only. Every cover of $\mathbb Z$ that the paper writes out has an even modulus (the introductory cover on p. 77 contains $2$, the covers on pp. 79–80 contain $4$), and the paper says nothing about covering systems with all moduli odd.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.