Cunningham number¶

An integer of the form b^n−1 or b^n+1 with integer base b that is not itself a perfect power, organizing prominent exponential families for factorization and primality study.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4000
Origin domain: number theory
Subdomain: exponential integer sequences

Core Idea¶

A Cunningham number is an integer C±(b,n)=b^n±1 where b and n are integers in the stated range and b is not a perfect power, so duplicate representations from exponent absorption are normalized away. Exponential algebra creates systematic factors when exponents or signs satisfy divisibility conditions. Factoring tables collect primitive and inherited factors, while special subfamilies include Mersenne and Fermat numbers. 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.

Scope of Application¶

Cunningham number belongs to number theory and is useful where the analyst can specify integers b and n, a sign choice, the exponential expression b^n±1, and a normalization excluding perfect-power bases, then evaluate the integer has the exact normalized form b^n plus or minus one under the declared base, exponent, and sign convention. The scope is broad within that domain but bounded by the need for the integer has the exact normalized form b^n plus or minus one under the declared base, exponent, and sign convention. 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 integer has the exact normalized form b^n plus or minus one under the declared base, exponent, and sign convention 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 Cunningham 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 Cunningham number. Cunningham 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: integers b and n, a sign choice, the exponential expression b^n±1, and a normalization excluding perfect-power bases. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the integer has the exact normalized form b^n plus or minus one under the declared base, exponent, and sign convention independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse integers b and n, a sign choice, the exponential expression b^n±1, and a normalization excluding perfect-power bases, Exponential algebra creates systematic factors when exponents or signs satisfy divisibility conditions. Factoring tables collect primitive and inherited factors, while special subfamilies include Mersenne and Fermat numbers., and type the carrier, state every parameter and convention in the definition, test that the integer has the exact normalized form b^n plus or minus one under the declared base, exponent, and sign convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Cunningham number Domain-specific

Parents (1) — more general patterns this builds on

Cunningham number is a kind of Classification Prime

The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

Cunningham number → Classification

Neighborhood in Abstraction Space¶

Cunningham number sits in a moderately populated region (48^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Number Theory & Reciprocity (28 abstractions)

Nearest neighbors

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