← Back to catalogue
Published

double Mersenne number

vr.tr.double-mersenne-number · thing-q1318413

Let an agent explain double Mersenne numbers, relay known results and open problems from mathematical references, describe their relation to Mersenne primes and the Catalan-Mersenne conjecture, and distinguish them from ordinary Mersenne numbers.

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

Understand What a double Mersenne number is.

Definition

Definition.

Definition

Definition.

  1. What is a double Mersenne number, and how does it relate to Mersenne numbers and primes? definition
  2. Is the question about double Mersenne numbers, Mersenne numbers or Fermat numbers? boundary

Known

Known results.

Known

Known.

  1. Which double Mersenne numbers are known to be prime or composite, and how were factors found? provenance
  2. Which entry fits Mersenne prime? action
Problems Open problems.

Catalan

Catalan-Mersenne conjecture.

Catalan

Catalan.

  1. What is the Catalan-Mersenne sequence, and why is the primality of its fifth term unknown? provenance
  2. Which references are standard? provenance

MM(61)

Candidates.

MM(61)

MM(61).

  1. Why are MM(61) and larger candidates beyond current primality testing? provenance
  2. Which sources are cited? provenance
Compute Computation.

Tests

Primality tests.

Tests

Tests.

  1. How does the Lucas-Lehmer test apply, and why is it infeasible for huge double Mersenne numbers? provenance
  2. Which entry fits Lucas-Lehmer test? action

Factors

Factor searches.

Factors

Factors.

  1. How are factors of double Mersenne numbers searched for? provenance
  2. Which entry fits distributed computing in number theory? action
Context History and culture.

History

History.

History

History.

  1. How did Catalan, Lucas and later mathematicians study these numbers? provenance
  2. Which entry fits the history of number theory? action

Culture

Culture.

Culture

Culture.

  1. How have double Mersenne numbers appeared in popular mathematics and fiction? provenance
  2. Which entry fits mathematics in popular culture? action

Classifiers Filled

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

What it is Filled

A number of the form 2^(2^p - 1) - 1, denoted MM(p), where both p and 2^p - 1 are prime, making it a Mersenne number whose exponent is itself a Mersenne prime; the first four cases MM(2), MM(3), MM(5) and MM(7) are prime, while MM(13) is known to be composite. The related Catalan-Mersenne sequence consists of iterated Mersenne numbers whose primality beyond the fifth term is unknown; double Mersenne numbers are among the largest numbers studied for primality.

Why it exists Filled

Let an agent explain double Mersenne numbers, relay known results and open problems from mathematical references, describe their relation to Mersenne primes and the Catalan-Mersenne conjecture, and distinguish them from ordinary Mersenne numbers.

Distinguishing features Filled

  • The nested Mersenne form requires both p and 2^p - 1 to be prime in this sense.
  • Four double Mersenne primes are known.
  • These numbers are related to the Catalan-Mersenne sequence.
  • Some cases beyond MM(7), including MM(13), are known to be composite, while other cases remain unresolved.

What robots and AI may and may not do Filled

Must not

  • Present the conjecture about further primes as settled.
  • Claim a primality result for a large case without a verifiable proof or certificate.
  • Misattribute a factorisation or search result to the wrong contributor.
  • Present distributed search progress as a completed proof.

Only with a human decision

  • Publishing a claimed primality or factorisation result under a person's name.

May

  • Explain what double Mersenne numbers are and report the known primality status of the small cases.
  • Distinguish proven results from open conjectures about these numbers.

Moral aspects Filled

  • Claims in computational number theory must be verifiable, or the record is corrupted.
  • Volunteer searchers deserve credit for the results their computation produced.

Who is affected

  • Mathematicians and students
  • Volunteer distributed computing participants

Owners Filled

Steward

Nobody owns them; mathematical and integer sequence databases record the known status.

Master systems

  • Published integer sequence and factorisation databases

Links to other meta-models Filled

parent

  • Q10338607 - registry parent class

related

  • Mersenne number - in registry terms
  • Mersenne prime - exponents in known cases
  • Catalan-Mersenne conjecture - about iterated Mersenne numbers
  • primality testing - and factor searches

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.double-mersenne-number
  • Wikidata: Q1318413 (https://www.wikidata.org/wiki/Q1318413)
  • OEIS: A077586 double Mersenne numbers

Direct properties not applicable Not applicable

  • known double Mersenne primes: 4 count - MM(2), MM(3), MM(5), MM(7)
  • MM(7): 2^127 - 1 value - about 1.7e38
  • MM(13) digits: about 2466 count - known composite

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Mersenne number with Mersenne exponent
  • MM(2) = 7, MM(3) = 127, MM(5), MM(7) prime; MM(13) composite
  • Ordinary Mersenne numbers are 2^n - 1; Fermat numbers are 2^(2^n) + 1
  • Not a visible object; enormous integers.

Capabilities and actions required Filled

  • explain the definition
  • relay known results
  • describe open problems
  • distinguish from Mersenne numbers

Hazards and failure modes required Filled

  • Assuming MM(p) is prime whenever M(p) is
  • Confusing with Mersenne numbers or Fermat numbers
  • Overstating what is proved

Standards and interfaces required Filled

  • No regulation; standard mathematical definitions

Context of use required Filled

  • Not applicable; a mathematical concept.
  • double Mersenne primes
  • composite double Mersenne numbers such as MM(13), MM(17), MM(19), MM(31)
  • the Catalan-Mersenne sequence
  • candidates such as MM(61) with unknown status

Sources Filled

  1. Wikidata item Q1318413: double Mersenne number - identity and sense of the item
  2. Wikipedia: Double Mersenne number - general description of the item

Open questions

  • Should the Catalan-Mersenne conjecture be a separate entry?
  • How should mathematical references be linked?
  • How should the status of MM(61) be tracked?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q1318413/spec.json