Carmichael number
Let an agent explain Carmichael numbers and their definition and characterisation, describe their significance for primality testing, support computation and proofs, and support learning in number theory.
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.
Researched by: Claude
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What Carmichael numbers are.
Definition
Definition.
Definition
Definition.
- How are Carmichael numbers defined, and what does the Korselt criterion say? definition
- Which entry fits Fermat little theorem? action
Examples
Examples and structure.
Examples
Examples.
- What are the smallest Carmichael numbers, and what structure do they share? definition
- Which property is meant? boundary
Apply Primality testing.
Testing
Fermat and stronger tests.
Testing
Testing.
- Why do Carmichael numbers defeat the Fermat test, and how do stronger tests handle them? provenance
- Which entry fits primality testing? action
Check
Checking a number.
Check
Checking.
- Is this number a Carmichael number, and how can it be verified? action
- Which entry fits factorisation? action
Theory Theory and results.
Infinitude
Infinitely many.
Infinitude
Infinitude.
- How was it proved that there are infinitely many Carmichael numbers? provenance
- Which findings are current? boundary
Generalisations
Generalisations.
Generalisations
Generalisations.
- What generalisations and related pseudoprime classes exist? provenance
- Which entry fits pseudoprimes? action
Learn History and teaching.
History
History.
History
History.
- How were Carmichael numbers discovered and named? provenance
- Which references are standard? provenance
Teach
Teaching.
Teach
Teaching.
- How can Carmichael numbers be taught in number theory? action
- 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 composite positive integer that satisfies the congruence of Fermat little theorem for every base coprime to it, so that it passes the Fermat primality test for all such bases despite not being prime, the smallest being five hundred and sixty-one, characterised by the Korselt criterion as square-free numbers each of whose prime factors minus one divides the number minus one; Carmichael numbers are infinite in number and matter for primality testing and cryptography.
Why it exists Filled
Let an agent explain Carmichael numbers and their definition and characterisation, describe their significance for primality testing, support computation and proofs, and support learning in number theory.
Distinguishing features Filled
- Absolute Fermat pseudoprime
- Korselt criterion
- Square-free with at least three primes
- Infinitely many
What robots and AI may and may not do Filled
Must not
- Rely on a Fermat test alone to declare a number prime in security software.
- Confuse Carmichael numbers with other pseudoprimes.
- Present an unverified computation as checked.
- Present homework answers as the student's own work when that would be dishonest.
Only with a human decision
- Choosing a primality test for cryptographic keys.
May
- Explain Carmichael numbers and the Korselt criterion.
- Check whether a given number is a Carmichael number.
- Explain why Fermat tests alone cannot certify primality.
Moral aspects Filled
- Weak primality testing can leave cryptographic systems open.
- Help with learning should build understanding, not replace it.
Who is affected
- Users of cryptographic systems
- Students
- Software developers
Owners Filled
Steward
Nobody: a mathematical object held in common.
Links to other meta-models Filled
parent
- Q1406394 - registry parent class
- Q4346700 - registry parent class
related
- Fermat pseudoprime - category
- Knodel number - category
- Lucas number - another number-theoretic class
- remainder - congruences underlying the definition
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.carmichael-number
- Wikidata: Q849530 (https://www.wikidata.org/wiki/Q849530)
Direct properties not applicable Not applicable
Not applicable
Plane XCT: no invented physical properties.
Recognition optional Filled
- Composite passing Fermat test for all coprime bases
- Five hundred and sixty-one is the smallest
- Fermat pseudoprimes pass for some bases; primes pass because they are prime
- Not physical; integers.
Capabilities and actions required Filled
- explain definition and criterion
- describe significance
- support computation
- support learning
Hazards and failure modes required Filled
- Trusting Fermat tests alone for primality
- Confusing with ordinary pseudoprimes
- Computation errors
Standards and interfaces required Filled
- Mathematical notation conventions
- Integer sequence catalogues
- No regulation of the concept
Context of use required Filled
- Composites passing Fermat tests for all bases.
- Carmichael numbers with three prime factors
- Carmichael numbers with many factors
- Chernick constructions
- generalisations to other pseudoprimes
- Carmichael numbers in cryptographic testing
Sources Missing, in the backlog
Not described yet. This gap is in the card backlog.
Note: Written from model knowledge without web access; claims are unverified.
Open questions
- Should pseudoprimes be a separate entry?
- How should sequence catalogues be linked?
- How should open problems be tracked?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q849530/spec.json