Perrin number¶

A doubly infinite integer sequence generated from initial values 3, 0 and 2 by adding terms two and three positions earlier.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6051
Origin domain: number theory
Subdomain: number theory

Core Idea¶

Indexing and extension to negative indices must be declared; divisibility by n is a necessary but not sufficient primality condition because Perrin pseudoprimes exist. The fixed linear recurrence propagates initial values forward and backward, its characteristic polynomial gives closed forms and modular recurrence behavior yields primality-related congruences. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of number theory. It is the domain-specific identity fixed by the index domain, initial values, recurrence and reverse recurrence, characteristic polynomial and roots, generating function or matrix form, modular divisibility claim and pseudoprime exceptions are explicit.

Scope of Application¶

Perrin number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the index domain, initial values, recurrence and reverse recurrence, characteristic polynomial and roots, generating function or matrix form, modular divisibility claim and pseudoprime exceptions are explicit. The scope is broad within that domain but bounded by the need for the index domain, initial values, recurrence and reverse recurrence, characteristic polynomial and roots, generating function or matrix form, modular divisibility claim and pseudoprime exceptions are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the index domain, initial values, recurrence and reverse recurrence, characteristic polynomial and roots, generating function or matrix form, modular divisibility claim and pseudoprime exceptions are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Perrin number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Perrin number. Perrin number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the index domain, initial values, recurrence and reverse recurrence, characteristic polynomial and roots, generating function or matrix form, modular divisibility claim and pseudoprime exceptions are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The fixed linear recurrence propagates initial values forward and backward, its characteristic polynomial gives closed forms and modular recurrence behavior yields primality-related congruences., and type the carrier, state every parameter and convention in the definition, test that the index domain, initial values, recurrence and reverse recurrence, characteristic polynomial and roots, generating function or matrix form, modular divisibility claim and pseudoprime exceptions are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Perrin number Domain-specific

Parents (1) — more general patterns this builds on

Perrin number is a kind of Recurrence Prime

The proposed strict upward parent is prime:recurrence.

Hierarchy path (1) — routes to 1 parentless root

Perrin number → Recurrence

Neighborhood in Abstraction Space¶

Perrin number sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Number Theory & Reciprocity (28 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08