Cullen number¶

An integer of the form C_n = n·2^n + 1, forming a named exponential sequence whose rare prime terms are Cullen primes.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3990
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Integer Sequences → Mathematics

Core Idea¶

A Cullen number is an integer C_n=n times 2 to the n plus one for a natural-number index n. The exponential factor produces rapid growth while congruences and factorization properties identify many composite terms; prime values are exceptional members of the sequence. 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 specific exponential integer sequence and its divisibility and primality questions. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the same natural index n appears both as multiplier and exponent in n·2^n+1 fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

How would you explain it like I'm…

Double, Multiply, Add One

Pick a counting number, like 3. Multiply 2 by itself 3 times to get 8, then multiply that by 3 to get 24, and add 1 to get 25. That's a Cullen number. The same number you picked is used twice, once for how many 2s and once to multiply.

n Times 2-to-the-n Plus 1

A Cullen number comes from a simple recipe: take a counting number n, multiply it by 2 raised to the power n, and add 1. So for n = 1 you get 1 × 2 + 1 = 3, for n = 2 you get 2 × 4 + 1 = 9, and for n = 3 you get 3 × 8 + 1 = 25. The same number n is used twice: once as the multiplier and once as the power of 2. Because of the doubling, these numbers grow really quickly. Mathematicians like asking which Cullen numbers are prime, and it turns out prime ones are rare.

Numbers of Form n·2ⁿ + 1

A Cullen number is an integer of the form C_n = n · 2^n + 1, where n is a natural number. The same n appears in two roles: as the multiplier in front and as the exponent on 2. That exponential factor makes the sequence grow very rapidly. Number theorists study which Cullen numbers are prime and how the others factor; many terms can be shown to be composite using divisibility patterns (congruences), and prime Cullen numbers are exceptional. A number like n · 2^m + 1 with m different from n is not a Cullen number, since the definition ties the multiplier and exponent together.

A Cullen number is an integer Cₙ = n·2ⁿ + 1 for a natural-number index n, so for n = 1, 2, 3, 4 the values are 3, 9, 25 and 65. The defining constraint is that the same natural index n serves both as multiplier and as exponent; sequences where these differ, such as k·2ⁿ + 1 for independent k, are neighboring but distinct objects. The exponential factor makes the sequence grow rapidly. Congruence arguments and factorization properties identify many composite terms, so prime Cullen numbers are exceptional members of the sequence. The concept is thus a specific exponential integer sequence around which divisibility and primality questions are organized, not number theory in general.

Scope of Application¶

Cullen number belongs to number theory and is useful where the analyst can specify a natural-number index n, exponentiation by two, multiplication and addition, the integer C_n, divisibility tests, recurrence relations, and primality status, then evaluate the same natural index n appears both as multiplier and exponent in n·2^n+1. The scope is broad within that domain but bounded by the need for the same natural index n appears both as multiplier and exponent in n·2^n+1. 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 same natural index n appears both as multiplier and exponent in n·2^n+1 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 Cullen 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 Cullen number. Cullen 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: a natural-number index n, exponentiation by two, multiplication and addition, the integer C_n, divisibility tests, recurrence relations, and primality status. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the same natural index n appears both as multiplier and exponent in n·2^n+1 independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse a natural-number index n, exponentiation by two, multiplication and addition, the integer C_n, divisibility tests, recurrence relations, and primality status, The exponential factor produces rapid growth while congruences and factorization properties identify many composite terms; prime values are exceptional members of the sequence., and type the carrier, state every parameter and convention in the definition, test that the same natural index n appears both as multiplier and exponent in n·2^n+1, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Cullen number Domain-specific

Parents (1) — outgoing typed relations

Cullen number is a kind of Recursion Prime

The family is an indexed integer construction with a derived recurrence.

Hierarchy path (1) — routes to 1 parentless root

Cullen number → is a kind of Recursion

Neighborhood in Abstraction Space¶

Cullen number sits in a crowded region of the domain-specific corpus (23^rd 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

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