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Research draft

prime number

vr.tr.prime-number · XCT.QTY

Let an agent handle primes mathematically, with tests, notable classes and records, and their role in cryptography.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent handle primes mathematically, with tests, notable classes and records, and their role in cryptography.

A natural number greater than 1 that has no positive divisors other than 1 and itself, such as 2, 3, 5, 7 and 11.

What it is for: Primes are the building blocks of integers and underpin much of number theory and public-key cryptography.

It can be test primality; factor composite numbers into primes; search for large primes, such as Mersenne primes; use them, in cryptographic keys.

Distinguishing features

Exactly two positive divisors

Infinitely many exist

2 is the only even prime

Primality tests may be probabilistic or proven

What it looks like

Numbers; notable primes are listed in tables and records.

Physical character

primes below 100: 25 count

largest known prime digits: over 41 million digits - a Mersenne prime found in 2024

How it is recognised

Divisible only by 1 and itself

Notation for special classes such as Mersenne primes 2^p - 1

Records such as the largest known prime

Related models

is a kind of - natural number

integer

is used in - RSA

cryptographic key

is studied by - number theory

mathematics

is confused with - not all odd numbers are prime

odd number

In practice

Families and kinds

Mersenne and double Mersenne primes

twin primes

palindromic primes

titanic and gigantic primes by size

provable versus probable primes

Identifiers

OEIS A000040 sequence of primes reference

Standards and regulation

FIPS 186 prime generation requirements for cryptography

Failure modes and hazards

Using probable primes where proof is required

Weak primes in cryptography

Off-by-one errors in counting primes

Also called

largest known prime numbertitanic primegigantic primeprovable primepalindromic primedouble Mersenne primeMersenne primerepunit primeWieferich primeFermat primeProth primeillegal primeprimes in arithmetic progressionwidely digitally delicate primecluster primeFibonacci primetwin primeprime factorsexy primeirregular primeTruncatable primeJacob prime numberPi primecuban primegood primeBi-twin chainsupersingular primeWilson primebalanced primeNewman–Shanks–Williams primeprimorial primeStrong primedelicate primesafe primeChen primeWolstenholme primePythagorean primeunique primeWagstaff primePillai prime

+18

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.prime-number

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Definition What is a prime.

The definition is exact.

Primality

Is it prime.

Primality

Primality result.

  1. Is this number prime, by which test? measurement
  2. Is the result proven or probabilistic? boundary

Factorisation

Prime factors.

Factors

Prime factorisation.

  1. What is the prime factorisation of this number? measurement
  2. How long does factoring take at this size? measurement
Classes Special primes.

Classes have their own properties.

Class

Mersenne, twin, palindromic.

Class

Class membership.

  1. Does it belong to a special class? definition
  2. What defines that class? definition

Records

Largest known.

Records

Record status.

  1. What is the largest known prime of this class? provenance
  2. Who verified it? provenance
Theory Distribution and conjectures.

Theorems and conjectures must be kept apart.

Distribution

Prime number theorem.

Distribution

How primes are distributed.

  1. How many primes are there below n, approximately? measurement
  2. Which theorem gives that? provenance

Conjectures

Open problems.

Conjectures

Open problems.

  1. Is this statement proven or a conjecture? boundary
  2. What is known partially? provenance
Cryptography Use in keys.

Primes secure communications.

Keys

Prime generation.

Key generation

Prime generation for keys.

  1. How are primes generated for keys? definition
  2. Which standard applies? provenance

Strength

Size and safety.

Strength

Security of prime sizes.

  1. Are these primes large enough for current security? boundary
  2. What does quantum computing change? boundary

What the second pass must settle

  • Should prime classes be separate entries?
  • How should record primes be tracked?
  • How should agents present unproven conjectures such as the twin prime conjecture?
Draft as JSON