{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.lucas-number","code":"thing-q2503280","url":"https://ver.cy/models/thing/q2503280/","name":"Lucas 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 member of the integer sequence beginning two, one, three, four, seven, eleven, eighteen, in which each term is the sum of the two preceding terms, named after Edouard Lucas and closely related to the Fibonacci numbers, sharing the golden ratio as the limit of successive quotients and satisfying identities that link the two sequences; Lucas numbers appear in number theory, primality testing and combinatorics.","purpose":"Let an agent explain Lucas numbers and their definition and properties, describe their relation to Fibonacci numbers and the golden ratio, support computation and proofs, and support learning in number theory.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Same recurrence as Fibonacci","Different initial values","Golden ratio limit","Closed form"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What Lucas numbers are.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"How are Lucas numbers defined, and what are the first terms and the closed form?","kind":"definition"},{"text":"Are Lucas numbers or general Lucas sequences meant?","kind":"boundary"}]}]},{"id":"fibonacci","name":"Fibonacci","description":"Relation to Fibonacci.","findings":[{"id":"fibonacci-finding","name":"Fibonacci","description":"Fibonacci.","questions":[{"text":"What identities connect Lucas and Fibonacci numbers, and why do both approach the golden ratio?","kind":"definition"},{"text":"Which entry fits Fibonacci numbers?","kind":"action"}]}]}]},{"id":"compute","name":"Compute","description":"Computation and proofs.","layers":[{"id":"compute","name":"Compute","description":"Computing.","findings":[{"id":"compute-finding","name":"Compute","description":"Computation.","questions":[{"text":"How can Lucas numbers be computed efficiently?","kind":"action"},{"text":"Which entry fits recurrence relations?","kind":"action"}]}]},{"id":"prove","name":"Prove","description":"Proving identities.","findings":[{"id":"prove-finding","name":"Prove","description":"Proof.","questions":[{"text":"How can a given Lucas identity be proved by induction or matrices?","kind":"action"},{"text":"Which entry fits proof techniques?","kind":"action"}]}]}]},{"id":"apply","name":"Apply","description":"Applications.","layers":[{"id":"primality","name":"Primality","description":"Primality testing.","findings":[{"id":"primality-finding","name":"Primality","description":"Primality.","questions":[{"text":"How are Lucas numbers and sequences used in primality tests?","kind":"provenance"},{"text":"Which entry fits primality testing?","kind":"action"}]}]},{"id":"combinatorics","name":"Combinatorics","description":"Combinatorics.","findings":[{"id":"combinatorics-finding","name":"Combinatorics","description":"Combinatorics.","questions":[{"text":"What do Lucas numbers count in combinatorics?","kind":"provenance"},{"text":"Which findings are current?","kind":"boundary"}]}]}]},{"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":"Who was Edouard Lucas, and how did the sequence get its name?","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 Lucas numbers be taught alongside Fibonacci numbers?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain Lucas numbers and their relation to the Fibonacci numbers and the golden ratio.","Compute Lucas numbers, stating the indexing used.","Prove or check identities involving them."],"mustNot":["Mix up Lucas and Fibonacci numbers in an answer.","Use an indexing convention without stating it.","Present an unverified identity as proven.","Present homework answers as the student's own work when that would be dishonest."],"requiresHuman":["Relying on a Lucas-number result in a published proof or software."]},"ethics":{"considerations":["Small indexing errors propagate into proofs and code.","Help with learning should build understanding, not replace it."],"affectedParties":["Students","Mathematicians","Programmers"]},"owners":{"steward":"Nobody: a mathematical object held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"category"},{"target":"Carmichael number","type":"related","note":"another number-theoretic class"},{"target":"irrational","type":"related","note":"the golden ratio as limit"},{"target":"remainder","type":"related","note":"divisibility properties"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.lucas-number","Wikidata: Q2503280 (https://www.wikidata.org/wiki/Q2503280)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Sequence two, one, three, four, seven, eleven","Each term the sum of the previous two","Fibonacci numbers start zero, one; Pell numbers use a different recurrence","Not physical; a sequence of integers."]},"capabilities":{"applicability":"required","items":["explain definition and properties","describe relation to Fibonacci numbers","support computation and proofs","support learning"]},"hazards":{"applicability":"required","items":["Confusing Lucas numbers with Fibonacci numbers","Confusing Lucas numbers with Lucas sequences generally","Indexing errors"]},"interfaces":{"applicability":"required","items":["Mathematical notation conventions","Integer sequence catalogues","No regulation of the concept"]},"context":{"applicability":"required","items":["Fibonacci-like sequence starting two, one.","Lucas numbers proper","Lucas primes","Lucas sequences generalising the recurrence","identities linking Lucas and Fibonacci numbers","Lucas numbers in primality tests"]}},"sources":[],"openQuestions":["Should Lucas sequences be a separate entry?","How should sequence catalogues be linked?","How should open problems be tracked?"],"resources":{"spec":"/models/things/publications/thing-q2503280/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q2503280/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}}