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Cullen number

vr.tr.cullen-number · thing-q532389

Let an agent define Cullen numbers precisely, list small examples and the known Cullen primes, relate them to Woodall and generalised Cullen numbers, and point to sequence references and open problems.

Thing Registry Cross-cutting context XCT.QTY
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Define The definition.

Definition

Definition.

Definition

Definition.

  1. What is a Cullen number, and how does it differ from a Woodall number? definition
  2. Is the question about Cullen, Woodall or generalised Cullen numbers? boundary

Examples

Examples.

Examples

Examples.

  1. What are the first Cullen numbers, and how are they computed? measurement
  2. Which entry fits integer sequences? action
Primes Cullen primes.

Known

Known primes.

Known

Known.

  1. Which Cullen primes are known, and where is the current list maintained? provenance
  2. Is the list current? boundary

Divisibility

Divisibility results.

Divisibility

Divisibility.

  1. What divisibility properties make almost all Cullen numbers composite? provenance
  2. Which references are standard? provenance
Compute Computation and search.

Search

Prime searches.

Search

Search.

  1. How are Cullen primes searched for, and which projects do it? provenance
  2. Which entry fits distributed computing projects? action

Generalise

Generalisations.

Generalise

Generalise.

  1. What generalised Cullen numbers exist for other bases? provenance
  2. Should a generalisation be a separate entry? boundary
Context History and teaching.

History

Cullen and Cunningham.

History

History.

  1. Who were Cullen and Cunningham, and how did the numbers get their name? provenance
  2. Which entry fits the person? action

Teach

Teaching.

Teach

Teaching.

  1. How are Cullen numbers used in teaching and puzzles? action
  2. Which misconceptions arise? provenance

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.QTY

What it is Filled

A natural number of the form n times 2 to the power n, plus 1, named after James Cullen who studied them in 1905; almost all Cullen numbers are composite, only a few Cullen primes are known, and the related Woodall numbers have the form n times 2 to the power n, minus 1.

Why it exists Filled

Let an agent define Cullen numbers precisely, list small examples and the known Cullen primes, relate them to Woodall and generalised Cullen numbers, and point to sequence references and open problems.

Distinguishing features Filled

  • Defined by n times 2 to the n, plus 1
  • Almost all composite
  • Few known primes
  • Paired with Woodall numbers

What robots and AI may and may not do Filled

Must not

  • Announce a new Cullen prime without independent verification.
  • Confuse Cullen numbers with Woodall numbers or with generalised forms.
  • Use other people's computing resources for prime searches without permission.

Only with a human decision

  • Announcing a new record to the mathematical community.

May

  • Compute Cullen numbers and report known Cullen primes from published records.
  • Explain their relation to Woodall numbers.

Moral aspects Filled

  • Credit for discoveries in distributed searches belongs to the people who ran them.
  • False record claims mislead others.

Who is affected

  • Researchers and volunteers in prime searches
  • Students

Owners Filled

Steward

Nobody owns the concept; it belongs to mathematics.

Links to other meta-models Filled

parent

  • Q16317911 - registry parent class

related

  • positive integer - a property of integers
  • Woodall number - the minus-one counterpart
  • On-Line Encyclopedia of Integer Sequences - sequence A002064
  • number theory - the field

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.cullen-number
  • Wikidata: Q532389 (https://www.wikidata.org/wiki/Q532389)
  • OEIS: A002064 Cullen numbers
  • OEIS: A005849 indices of Cullen primes

Direct properties not applicable Not applicable

  • first terms: 1, 3, 9, 25, 65, 161, 385 integers - for n from 0
  • smallest Cullen prime beyond trivial: n equal to 1 and n equal to 141 indices - Cullen primes are rare

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Form n times 2 to the n, plus 1
  • First terms 1, 3, 9, 25, 65, 161
  • Woodall numbers subtract 1 instead of adding
  • Not visible; a property of integers.

Capabilities and actions required Filled

  • compute Cullen numbers for given n
  • list known Cullen primes
  • distinguish Cullen from Woodall and generalised forms
  • point to sequence references

Hazards and failure modes required Filled

  • Confusing Cullen with Woodall numbers
  • Quoting outdated lists of known primes
  • Presenting conjectures as theorems

Standards and interfaces required Filled

  • No regulation; definitions follow the literature and the OEIS

Context of use required Filled

  • Not for anything; an object of number theory.
  • Cullen numbers
  • Cullen primes
  • generalised Cullen numbers with other bases
  • Woodall numbers as the related family

Sources Filled

  1. Wikidata item Q532389: Cullen number - identity and sense of the item
  2. Wikipedia: Cullen number - general description of the item

Open questions

  • Should Cullen primes be a separate entry?
  • How should prime search projects be linked?
  • Which conjectures, such as infinitude of Cullen primes, should be recorded?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q532389/spec.json