Congruence m-Cover¶
A finite family of integer residue classes with moduli greater than one that covers every integer at least m times, for m at least two.
Core Idea¶
A congruence m-cover is a finite indexed family of integer residue classes with each modulus greater than one, together with an integer threshold \(m\geq2\).[1] Every integer must lie in at least \(m\) of the classes.[1] An exact m-cover requires exactly \(m\) memberships at every integer.[1] The family is indexed: if the same class appears twice, those are two memberships in the count.[1]
The restriction on moduli here follows the bounded Covering System entry. Sun's more general m-cover definition also allows modulus one; this entry uses only moduli greater than one so that every instance is a strict kind of the canonical covering-system type.[1]
Scope of Application¶
This is a number-theoretic type for constructions that need multiple applicable congruence rules at every integer.[1] Guerrini, Imbert and Winterhalter give an exact two-cover whose residue table provides two congruence choices at every integer in a scalar-multiplication construction.[2] The cover establishes the number of available classes; it does not, by itself, establish any performance or security result.
Clarity¶
The difference from an ordinary cover is a lower bound on the number of hits.[1] The even/odd classes \(0\pmod2\) and \(1\pmod2\) cover every integer once, so they are a covering system but do not qualify here when \(m\geq2\).[1] A two-cover can have three or more hits at some integers; it is exact only when all hit counts equal two.[1]
This is also distinct from a set-cover optimization problem or a sequence-specific Covering Set. It has no finite-universe objective and no prime-divisor certificate for a particular sequence. Its objects are integer congruence classes, and its defining test concerns all integers.
Manages Complexity¶
Let \(L\) be the least common multiple of the moduli. Membership in each class repeats after \(L\), so the number of hits repeats too.[3] Count hits for each residue \(0,\ldots,L-1\); if every count is at least \(m\), the family is an m-cover of all integers.[3] If every count equals \(m\), it is exact.[3] A failed residue disproves the universal claim.
Abstract Reasoning¶
The period test gives both a proof and a counterexample method.[3] One residue with fewer than \(m\) witnesses defeats an alleged m-cover; a complete residue table with exactly \(m\) witnesses per row establishes an exact m-cover.[3] The reciprocal-modulus sum is the average hit count over one period.[3] It must be at least \(m\) for an m-cover and equals \(m\) for an exact m-cover, but its value alone does not show that every residue reaches the threshold.[3]
Knowledge Transfer¶
The useful distinction is between at least one applicable rule and several applicable rules. A construction requiring choice among classes can use an m-cover; a construction requiring only coverage may use an ordinary cover. In either case, the application must prove separately that the result attached to a chosen class follows from that class's congruence condition.
Examples¶
Nineteen classes. Sun reproduces Choi's nineteen-class exact two-cover.[3] Its classes have moduli 2, 3, 6, 10, 15 and 30; an independent count of the displayed list gives exactly two hits for each residue modulo 30.[3] The statement here concerns exact multiplicity, not the separate question of decomposing this family into two one-covers.
Ten classes. Guerrini, Imbert and Winterhalter display \(1\pmod2;\ 2,3\pmod4;\ 0,2,4\pmod6;\ 0,1,4,5\pmod8\).[2] Their Figure 1 and Table 2 show two hits on each of the 24 residues in a complete period.[2] This is a structurally different exact two-cover: it has ten classes and a different modulus pattern.
A nonexact variant. Add \(0\pmod4\) to that ten-class family. The two-hit lower bound survives, while multiples of four acquire a third hit. It remains a two-cover but is no longer exact.[2]
Relationships to Other Abstractions¶
Current abstraction Congruence m-Cover Domain-specific
Parents (1) — outgoing typed relations
-
Congruence m-Cover is a kind of Covering System Domain-specific
Every congruence m-cover is a covering system with the same finite residue-class family and universal coverage, with at least m hits per integer.
Hierarchy path (1) — routes to 1 parentless root
- Congruence m-Cover → is a kind of Covering System → presupposes Periodicity → is a kind of Invariance
Neighborhood in Abstraction Space¶
Congruence m-Cover sits in a sparse region of the domain-specific corpus (93rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2769 abstractions)
Nearest neighbors
- Covering System — 0.81
- Covering Set — 0.79
- Frobenius Formula — 0.79
- Refactorable number — 0.78
- Primefree Sequence — 0.78
Computed from structural-signature embeddings · 2026-10-25
Not to Be Confused With¶
A count of two hits per residue proves exactness but does not prove that the family cannot be split into two one-covers.[3] That is a separate structural claim. The direct hierarchy parent is Covering System, because the same congruence-family and all-integers obligations are retained and the lower bound increases from one to at least two.
References¶
[1] Zhi-Wei Sun, “On m-Covers and m-Systems,” Bulletin of the Australian Mathematical Society 81 (2010), 223–235, pp. 223–224, https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7DB2FD192E54DE078343FDC6F83721C/S0004972709000641a.pdf/on-m-covers-and-m-systems.pdf. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i
[2] Eleonora Guerrini, Laurent Imbert and Théo Winterhalter, “Randomizing Scalar Multiplication Using Exact Covering Systems of Congruences,” IACR ePrint 2015/475 (2015), §2.1, Fig. 1 and Table 2, https://eprint.iacr.org/2015/475.pdf. registry ↩a ↩b ↩c ↩d
[3] Zhi-Wei Sun, “Zero-Sum Problems for Abelian p-Groups and Covers of the Integers by Residue Classes,” Israel Journal of Mathematics 170 (2009), 235–252, arXiv:math/0305369v8, PDF p. 4, https://arxiv.org/pdf/math/0305369. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j