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

v2 · 2026-09-19 · History

Domain-specific #

4000

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Number Theory, Exponential Integer Sequences → Mathematics

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.

How would you explain it like I'm…

One More or Less Than a Power

Take a number, like 2, and multiply it by itself a bunch of times: 2, 4, 8, 16. Then add one or take away one, to get 17 or 15. Numbers made like that are Cunningham numbers. Mathematicians love trying to break them into smaller numbers that multiply together.

Numbers One Off a Power

A Cunningham number is a number you get by taking a base number b, raising it to a power n (multiplying b by itself n times), and then adding 1 or subtracting 1. So 2⁴ + 1 = 17 and 3² − 1 = 8 are examples. To avoid counting the same number twice, the base can't itself already be a power, like 4 (which is 2 × 2). Mathematicians keep big tables of how these numbers split into prime factors. Some famous number families, like Mersenne numbers (2 to a power, minus 1), are special kinds of Cunningham numbers.

Normalized bⁿ ± 1 Integers

A Cunningham number is an integer of the form bⁿ + 1 or bⁿ − 1, where b and n are integers in a stated range and b is not itself a perfect power. That last condition removes duplicate representations: 4³ − 1 is the same as 2⁶ − 1, so only base 2 is used. Because of the algebra of exponents, these numbers have predictable factors when the exponent or sign meets certain divisibility conditions. For example, bⁿ − 1 is always divisible by b − 1. Factoring tables sort out which factors are 'inherited' from smaller exponents and which are new ('primitive'). Mersenne numbers (2ⁿ − 1) and Fermat numbers (2 raised to a power of 2, plus 1) are special subfamilies.

 

A Cunningham number is an integer C±(b, n) = bⁿ ± 1 where b and n lie in a declared range and b is not a perfect power, a normalization that removes duplicate representations arising when an exponent is absorbed into the base. Algebraic identities for powers produce systematic factors when the exponent or sign meets divisibility conditions; for example, if d divides n then b^d − 1 divides bⁿ − 1. Factorization tables therefore distinguish primitive factors, appearing for the first time at a given exponent, from factors inherited from smaller exponents. Special subfamilies include the Mersenne numbers 2ⁿ − 1 and the Fermat numbers. The concept is thus the normalized near-power integers together with their factorization structure across bases, exponents and signs; an integer that is merely close to a power, or written in an unnormalized form, does not meet the definition.

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

1. 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) — outgoing typed relations

• Cunningham number is a kind of Classification Prime

  The identity classifies integers through one normalized exponential form.

Hierarchy path (1) — routes to 1 parentless root

• Cunningham number → is a kind of Classification

Neighborhood in Abstraction Space

Cunningham number sits in a crowded region of the domain-specific corpus (29^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Classifications & Arithmetic Functions (45 abstractions)

Nearest neighbors

• Super-Poulet number — 0.89
• Square number — 0.88
• Woodall number — 0.88
• Nonhypotenuse number — 0.88
• Unusual number — 0.88

Computed from structural-signature embeddings · 2026-10-25