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Source. Pp. 79–80 (physical pp. 3–4 of the scan).

T. Cochrane and G. Myerson, Covering congruences in higher dimensions, Rocky Mountain J. Math. 26 (1996), no. 1, 77–81, doi:10.1216/rmjm/1181072104; the edition read and its page mapping are named on the source card. This page records contextual statements and proof limits. It is not a complete proof page for the constructions named here.

Other composite covers reported by the source

• The paper says (p. 79) that James Dewar constructed a covering system in which every modulus is divisible by $4$. It cites Dewar's 1971 conference article but does not reproduce the construction.
• Besides the twenty-class cover in Lemma 2, the authors report (p. 80) that John Selfridge supplied a $21$-class composite cover whose moduli divide $1440$ and do not exceed $96$, and a $25$-class composite cover whose moduli divide $3456$, do not exceed $576$, and have no prime divisors other than $2$ and $3$. The residue classes and proofs are not printed, so these are source-attributed statements only.

Questions as posed in 1996

The paper asks (p. 80): "Are there homogeneous covers for $\mathbf{Z}^n$, $n > 2$, that are not trivial extensions or simple transformations of homogeneous covers for $\mathbf{Z}^2$? Are there any particularly simple or elegant ones?" It also asks, calling a subgroup $H$ of $\mathbf Z^n$ Type 1 when it comes from a single linear congruence (equivalently, $\mathbf Z^n/H$ is cyclic) and Type 2 otherwise, whether, for $n\ge2$, the group $\mathbf Z^n$ can be written as a finite union of subgroups, no two of the same index, with some or all of them of Type 2; a union of Type 1 subgroups padded with redundant Type 2 subgroups does not count. Finally it asks whether there are homogeneous covers that do not come from composite covers of $\mathbf Z$.

These are recorded as historical questions, not current open problems. A dated metadata check on 2026-09-05 located Andrzej Schinzel's 1997 paper, On homogeneous covering congruences, Rocky Mountain J. Math. 27, 335–342, and Boping Jin and Gerry Myerson's 2005 paper, Homogeneous covering congruences and subgroup covers, J. Number Theory 110, 120–135, DOI 10.1016/j.jnt.2003.11.012. The latter publisher abstract says that it answers the earlier questions. Neither later paper's proof has been inspected or compiled here, so no question-by-question modern status is asserted.

Scope

The complete proofs retained from this source are the two lemmas, main theorem, and their subgroup, matrix, and higher-dimensional consequences. The contextual items above are not counted among those complete proofs.

Bears on. Historical context for higher-dimensional covering systems, without changing the status of Problem 2 or Problem 7.