{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.carmichael-number","code":"thing-q849530","url":"https://ver.cy/models/thing/q849530/","name":"Carmichael number","alternateNames":[],"kind":"thing","status":"research-draft","version":"0.2.0-wave.2","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"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.","purpose":"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.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Absolute Fermat pseudoprime","Korselt criterion","Square-free with at least three primes","Infinitely many"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What Carmichael numbers are.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"How are Carmichael numbers defined, and what does the Korselt criterion say?","kind":"definition"},{"text":"Which entry fits Fermat little theorem?","kind":"action"}]}]},{"id":"examples","name":"Examples","description":"Examples and structure.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the smallest Carmichael numbers, and what structure do they share?","kind":"definition"},{"text":"Which property is meant?","kind":"boundary"}]}]}]},{"id":"apply","name":"Apply","description":"Primality testing.","layers":[{"id":"testing","name":"Testing","description":"Fermat and stronger tests.","findings":[{"id":"testing-finding","name":"Testing","description":"Testing.","questions":[{"text":"Why do Carmichael numbers defeat the Fermat test, and how do stronger tests handle them?","kind":"provenance"},{"text":"Which entry fits primality testing?","kind":"action"}]}]},{"id":"check","name":"Check","description":"Checking a number.","findings":[{"id":"check-finding","name":"Check","description":"Checking.","questions":[{"text":"Is this number a Carmichael number, and how can it be verified?","kind":"action"},{"text":"Which entry fits factorisation?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Theory and results.","layers":[{"id":"infinitude","name":"Infinitude","description":"Infinitely many.","findings":[{"id":"infinitude-finding","name":"Infinitude","description":"Infinitude.","questions":[{"text":"How was it proved that there are infinitely many Carmichael numbers?","kind":"provenance"},{"text":"Which findings are current?","kind":"boundary"}]}]},{"id":"generalisations","name":"Generalisations","description":"Generalisations.","findings":[{"id":"generalisations-finding","name":"Generalisations","description":"Generalisations.","questions":[{"text":"What generalisations and related pseudoprime classes exist?","kind":"provenance"},{"text":"Which entry fits pseudoprimes?","kind":"action"}]}]}]},{"id":"learn","name":"Learn","description":"History and teaching.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How were Carmichael numbers discovered and named?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"teach","name":"Teach","description":"Teaching.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How can Carmichael numbers be taught in number theory?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]}]}]},"agentConduct":{"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."],"mustNot":["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."],"requiresHuman":["Choosing a primality test for cryptographic keys."]},"ethics":{"considerations":["Weak primality testing can leave cryptographic systems open.","Help with learning should build understanding, not replace it."],"affectedParties":["Users of cryptographic systems","Students","Software developers"]},"owners":{"steward":"Nobody: a mathematical object held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q1406394","type":"parent","note":"registry parent class"},{"target":"Q4346700","type":"parent","note":"registry parent class"},{"target":"Fermat pseudoprime","type":"related","note":"category"},{"target":"Knodel number","type":"related","note":"category"},{"target":"Lucas number","type":"related","note":"another number-theoretic class"},{"target":"remainder","type":"related","note":"congruences underlying the definition"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.carmichael-number","Wikidata: Q849530 (https://www.wikidata.org/wiki/Q849530)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["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":{"applicability":"required","items":["explain definition and criterion","describe significance","support computation","support learning"]},"hazards":{"applicability":"required","items":["Trusting Fermat tests alone for primality","Confusing with ordinary pseudoprimes","Computation errors"]},"interfaces":{"applicability":"required","items":["Mathematical notation conventions","Integer sequence catalogues","No regulation of the concept"]},"context":{"applicability":"required","items":["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":[],"openQuestions":["Should pseudoprimes be a separate entry?","How should sequence catalogues be linked?","How should open problems be tracked?"],"resources":{"spec":"/models/things/publications/thing-q849530/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q849530/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T05:47:46Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"missing"},"notes":{"interaction.properties":"Plane XCT: no invented physical properties.","sources":"Written from model knowledge without web access; claims are unverified."},"score":0.938}}