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.
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.
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. 2. 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) — more general patterns this builds on
-
Cullen number is a kind of Recursion Prime
The proposed strict upward parent is
prime:recursion.
Hierarchy path (1) — routes to 1 parentless root
- Cullen number → Recursion
Neighborhood in Abstraction Space¶
Cullen number sits in a moderately populated region (40th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Number-Theoretic Sequences & Classes (37 abstractions)
Nearest neighbors
- Unusual number — 0.90
- Composite number — 0.90
- Highly powerful number — 0.90
- Niven's constant — 0.89
- Modular exponentiation — 0.89
Computed from structural-signature embeddings · 2026-09-08