Woodall number¶

A natural number of the form n times two to the n minus one.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7504
Origin domain: number theory
Subdomain: number theory

Core Idea¶

Indexing convention for natural n must be stated, primality of a Woodall number is a separate property and generalized Woodall families alter the multiplier or base. Each index n is mapped through the exponential-arithmetic formula, producing a sparse integer sequence whose modular factors and primality can be studied from n and exponent structure. 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¶

Woodall 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 natural index n, formula W-n equals n times 2-to-n minus 1, indexing start, initial sequence values, relation to Cullen numbers, divisibility and congruence properties, Woodall-prime condition and known versus unknown primality status are explicit. The scope is broad within that domain but bounded by the need for the natural index n, formula W-n equals n times 2-to-n minus 1, indexing start, initial sequence values, relation to Cullen numbers, divisibility and congruence properties, Woodall-prime condition and known versus unknown primality status are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the natural index n, formula W-n equals n times 2-to-n minus 1, indexing start, initial sequence values, relation to Cullen numbers, divisibility and congruence properties, Woodall-prime condition and known versus unknown primality status 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.

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 Woodall number. Woodall 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 natural index n, formula W-n equals n times 2-to-n minus 1, indexing start, initial sequence values, relation to Cullen numbers, divisibility and congruence properties, Woodall-prime condition and known versus unknown primality status 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, Each index n is mapped through the exponential-arithmetic formula, producing a sparse integer sequence whose modular factors and primality can be studied from n and exponent structure., and type the carrier, state every parameter and convention in the definition, test that the natural index n, formula W-n equals n times 2-to-n minus 1, indexing start, initial sequence values, relation to Cullen numbers, divisibility and congruence properties, Woodall-prime condition and known versus unknown primality status are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Woodall number Domain-specific

Parents (1) — more general patterns this builds on

Woodall number is a kind of Formalization Prime

The proposed strict upward parent is prime:formalization.

Hierarchy paths (2) — routes to 2 parentless roots

Woodall number → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

Woodall number sits in a crowded region of the domain-specific corpus (17^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

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