Covering Set¶
Certify that every term of a modularly periodic integer sequence has a divisor from one finite prime set by covering every index class with a periodic divisibility congruence.
Core Idea¶
A covering set, in the number-theoretic sense used here, is a finite set of primes that acts as a certificate for an infinite integer sequence: every term of the sequence is divisible by at least one prime in the set. The certificate becomes finite because divisibility of an exponential or recurrence sequence modulo a fixed prime is periodic. Each prime therefore covers one or more residue classes of term indices, and the union of those index classes must cover every index in scope.
Scope of Application¶
The home domain is elementary and computational number theory of exponentially growing or periodically reducible integer sequences.
- Sierpiński sequences: certify that \(k2^n+1\) is composite for every index by a finite collection of small prime divisors.
- Riesel sequences: apply the corresponding construction to \(k2^n-1\), with sign-adjusted modular compatibility.
- Other bases: replace 2 by \(b>1\); the period becomes the multiplicative order of \(b\) modulo each covering prime, provided \(p\nmid b\).
- Primefree linear recurrences: exploit periodicity of recurrence states modulo primes so zero-position classes cover every sequence index.
Clarity¶
A clear covering-set claim should be published as a certificate table rather than as a bare list. For every row record the prime \(p\), the index condition \(n\equiv r\pmod m\), the sequence-parameter congruence used, and the resulting divisibility statement. Then state a common modulus \(M=\operatorname{lcm}(m_1,\ldots,m_s)\) and verify all residues \(0,\ldots,M-1\). This exposes both possible defects: a bad row and a gap between rows.
Manages Complexity¶
The covering set compresses an infinite universal claim into three finite objects: a bounded prime list, a bounded residue table, and a bounded common-period coverage check. Instead of factoring ever larger terms, the reasoner works in small residue rings. The certificate is auditable, portable between proof and computation, and cheap to recheck.
Abstract Reasoning¶
The key reasoning move is local-to-global through periodicity and coverage. Each prime supplies a local theorem on one arithmetic progression of indices. Completeness of the union upgrades those local theorems to a global quantifier over all indices. The proof is not induction: it does not propagate truth from \(n\) to \(n+1\). It quotients the infinite index set by a finite common period and checks representatives.
Knowledge Transfer¶
The full abstraction transfers literally within number theory wherever an indexed integer sequence has finite modular state and selected primes cover all zero-position classes. The objects change—exponentials, sign variants, Lucas-type recurrences—but the proof obligations do not: compute periods, bind index classes to primes, cover the domain, and rule out trivial equality.
Useful knowledge transfers between Sierpiński/Riesel work and primefree recurrences. A researcher who sees a modular cycle can search its zero positions; who sees several zero-position progressions can ask whether they cover; who has a cover can solve parameter congruences rather than test parameters blindly.
Relationships to Other Abstractions¶
Current abstraction Covering Set Domain-specific
Parents (2) — more general patterns this builds on
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Covering Set is a kind of Coverage / Reachability Prime
Covering Set instantiates Coverage / Reachability in a strict number-theoretic form: the required targets are all allowed indices, and each must be reached by at least one prime-divisibility rule.
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Covering Set presupposes Periodicity Prime
Covering Set instantiates Coverage / Reachability in a strict number-theoretic form: the required targets are all allowed indices, and each must be reached by at least one prime-divisibility rule.
Hierarchy paths (3) — routes to 3 parentless roots
- Covering Set → Coverage / Reachability → Completeness
- Covering Set → Periodicity → Invariance
- Covering Set → Coverage / Reachability → Surjectivity → Function (Mapping)
Neighborhood in Abstraction Space¶
Covering Set sits in a sparse region of the domain-specific corpus (68th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Algebraic Structure & Reciprocity Theorems (5 abstractions)
Nearest neighbors
- Primefree Sequence — 0.86
- Wilf Equivalence — 0.85
- Quartic reciprocity — 0.85
- Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain — 0.85
- Field of fractions — 0.84
Computed from structural-signature embeddings · 2026-09-08