{ "model": { "rank": 8009, "code": "thing-q117927", "model_id": "vr.tr.amicable-numbers", "name": "amicable numbers", "purpose": "Let an agent define amicable numbers, verify and find pairs, explain known rules and results, and present open questions and history.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q117927/", "tier": 2, "score": 52, "payload": { "layer": "wikidata", "aliases": [], "aliasCount": 0, "merged": 0, "knownIn": 52, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q117927", "url": "https://www.wikidata.org/wiki/Q117927", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T21:45:44Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q117927", "registryId": "vr.tr.amicable-numbers", "name": "amicable numbers", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q117927/", "model": { "registry_id": "vr.tr.amicable-numbers", "name": "amicable numbers", "purpose": "Let an agent define amicable numbers, verify and find pairs, explain known rules and results, and present open questions and history.", "definition": "Two different natural numbers such that the sum of the proper divisors of each equals the other, the smallest pair being 220 and 284, known since antiquity, with many pairs found by Thabit ibn Qurra rule, Euler and computer searches; amicable numbers are a case of sociable numbers of period two and remain a topic of open questions in number theory.", "what_it_is_for": "A classical topic in number theory.", "affordances": [ "define and verify amicable pairs", "explain rules for generating pairs", "present known results and open problems", "describe history" ], "distinguishing_features": [ "Divisor sum relation", "Pairs", "Sociable numbers of period two", "Open problems" ], "appearance": "Not physical; pairs of integers.", "visual_identification": [ "Each is the sum of the proper divisors of the other", "Smallest pair 220 and 284", "A perfect number equals its own divisor sum" ], "physical_properties": [], "families_and_kinds": [ "amicable pairs", "regular and irregular pairs", "pairs from the Thabit rule", "sociable numbers of longer periods (related)", "quasi-amicable and other variants" ], "related_models": [ { "relation": "is a kind of", "target": "sociable number", "why": "category" }, { "relation": "is related to", "target": "perfect number", "why": "related concept" }, { "relation": "is related to", "target": "constant", "why": "contrast with fixed values" }, { "relation": "is related to", "target": "complex conjugate", "why": "another paired concept in mathematics" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical notation conventions" ], "failure_modes_and_hazards": [ "Confusing with friendly or perfect numbers", "Miscounting divisors", "Overstating what is proven" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What they are.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and examples.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What are amicable numbers, and why are 220 and 284 amicable?", "kind": "definition" }, { "text": "How do they differ from perfect and friendly numbers?", "kind": "definition" } ] } ] }, { "id": "verify", "name": "Verify", "description": "Checking a pair.", "findings": [ { "id": "verify-finding", "name": "Verify", "description": "Verification.", "questions": [ { "text": "Is this pair of numbers amicable, computed from their divisor sums?", "kind": "action" }, { "text": "How can pairs be found by computer?", "kind": "action" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Rules and results.", "rationale": "Mathematics.", "layers": [ { "id": "rules", "name": "Rules", "description": "Generating rules.", "findings": [ { "id": "rules-finding", "name": "Rules", "description": "Rules.", "questions": [ { "text": "How do the rules of Thabit ibn Qurra and Euler generate amicable pairs?", "kind": "definition" }, { "text": "Why do they not produce all pairs?", "kind": "definition" } ] } ] }, { "id": "open", "name": "Open", "description": "Open problems.", "findings": [ { "id": "open-finding", "name": "Open", "description": "Open problems.", "questions": [ { "text": "Are there infinitely many amicable pairs, and is there a pair with one odd and one even number? What is known?", "kind": "provenance" }, { "text": "How many pairs are known?", "kind": "provenance" } ] } ] } ] }, { "id": "history", "name": "History", "description": "History.", "rationale": "Attribution.", "layers": [ { "id": "ancient", "name": "Ancient", "description": "Antiquity and medieval work.", "findings": [ { "id": "ancient-finding", "name": "Ancient", "description": "History.", "questions": [ { "text": "How were amicable numbers known to the Pythagoreans and studied by medieval Arabic mathematicians?", "kind": "provenance" }, { "text": "What did Fermat, Descartes and Euler contribute?", "kind": "provenance" } ] } ] }, { "id": "modern", "name": "Modern", "description": "Computer era.", "findings": [ { "id": "modern-finding", "name": "Modern", "description": "Modern era.", "questions": [ { "text": "How have computer searches expanded the known pairs?", "kind": "provenance" }, { "text": "Which databases list them?", "kind": "provenance" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching and culture.", "rationale": "Education.", "layers": [ { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can amicable numbers be used to teach divisors and programming?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "culture", "name": "Culture", "description": "Cultural references.", "findings": [ { "id": "culture-finding", "name": "Culture", "description": "Culture.", "questions": [ { "text": "How have amicable numbers appeared in folklore, talismans and literature?", "kind": "provenance" }, { "text": "Which entry fits sociable numbers?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should sociable numbers be a single entry?", "How should databases of pairs be linked?", "How should open problems be tracked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "unit, quantity or measure", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a amicable numbers, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent define amicable numbers, verify and find pairs, explain known rules and results, and present open questions and history.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Described in 52 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a amicable numbers, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see: a unit is a convention. What is seen is an instrument reading in it." ], "relatedModels": [], "standing": "Described in 52 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one amicable numbers from another, and a amicable numbers from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a amicable numbers sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a amicable numbers from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a amicable numbers passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a amicable numbers between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a amicable numbers, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a amicable numbers is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definition-and-realisation", "name": "Definition, dimension and realisation", "description": "What amicable numbers is defined as and how it is realised in practice.", "rationale": "A unit is not the quantity it measures, and its definition has a history that older data still depends on.", "layers": [ { "id": "definition", "name": "Definition and dimension", "description": "The defining relation and the quantity it measures.", "findings": [ { "id": "definition-record", "name": "Definition, body and date", "description": "What defines it now, and what defined it before.", "questions": [ { "id": "definition-record-q01", "text": "How is amicable numbers defined, by which body, and since when?", "kind": "definition" }, { "id": "definition-record-q02", "text": "What was the previous definition, and what data still depends on it?", "kind": "provenance" } ] } ] }, { "id": "conversion", "name": "Conversion and compatibility", "description": "Relations to other units, exact or approximate.", "findings": [ { "id": "conversion-record", "name": "Conversions and their exactness", "description": "Factors, and whether each is exact by definition.", "questions": [ { "id": "conversion-record-q01", "text": "What conversions relate amicable numbers to other units, and which are exact by definition?", "kind": "measurement" }, { "id": "conversion-record-q02", "text": "Where does rounding a conversion change a decision, and how should that be recorded?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a amicable numbers came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a amicable numbers and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a amicable numbers is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a amicable numbers, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a amicable numbers, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a amicable numbers is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 4, "layers": 8, "findings": 10, "questions": 28 } } } }