{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.perfect-power","code":"thing-q1972803","url":"https://ver.cy/models/thing/q1972803/","name":"perfect power","alternateNames":["power of five","power of four","power of six","cube number","power of three"],"kind":"thing","status":"published","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A positive integer that can be written as m to the power k with integers m greater than 1 and k at least 2, such as squares (4, 9), cubes (8, 27) and higher powers; Catalan conjecture, proved by Mihailescu, shows 8 and 9 are the only consecutive perfect powers.","purpose":"Let an agent handle perfect powers by definition, testing, examples and notable theorems.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Integer power of an integer","Includes squares and cubes","Rare among large numbers","Subject of deep theorems"],"structure":{"bundles":[{"id":"test","name":"Test","description":"Is it a perfect power.","layers":[{"id":"check","name":"Check","description":"Roots.","findings":[{"id":"check-finding","name":"Check","description":"Perfect power test.","questions":[{"text":"Is this number a perfect power, and of which base and exponent?","kind":"boundary"},{"text":"How is it tested exactly?","kind":"action"}]}]},{"id":"code","name":"Code","description":"Implementation.","findings":[{"id":"code-finding","name":"Code","description":"Implementation.","questions":[{"text":"How can integer roots be computed without floating point errors?","kind":"action"},{"text":"Which libraries help?","kind":"provenance"}]}]}]},{"id":"examples","name":"Examples","description":"Sequences.","layers":[{"id":"list","name":"List","description":"Early terms.","findings":[{"id":"list-finding","name":"List","description":"List.","questions":[{"text":"What are the first perfect powers?","kind":"provenance"},{"text":"How dense are they among integers?","kind":"measurement"}]}]},{"id":"kinds","name":"Kinds","description":"Squares and cubes.","findings":[{"id":"kinds-finding","name":"Kinds","description":"Kinds.","questions":[{"text":"Is it a square, cube or higher power?","kind":"definition"},{"text":"Can it be several at once, such as 64?","kind":"definition"}]}]}]},{"id":"theorems","name":"Theorems","description":"Results.","layers":[{"id":"catalan","name":"Catalan","description":"Consecutive powers.","findings":[{"id":"catalan-finding","name":"Catalan","description":"Catalan conjecture.","questions":[{"text":"What does the Catalan conjecture, now a theorem, state?","kind":"provenance"},{"text":"Who proved it and when?","kind":"provenance"}]}]},{"id":"open","name":"Open","description":"Open problems.","findings":[{"id":"open-finding","name":"Open","description":"Open problems.","questions":[{"text":"Which related problems remain open?","kind":"provenance"},{"text":"Is their status stated accurately?","kind":"boundary"}]}]}]},{"id":"learning","name":"Learning","description":"Teaching.","layers":[{"id":"patterns","name":"Patterns","description":"Visual models.","findings":[{"id":"patterns-finding","name":"Patterns","description":"Patterns.","questions":[{"text":"How can squares and cubes be shown with arrays and blocks?","kind":"action"},{"text":"Which exercises help?","kind":"action"}]}]},{"id":"notation","name":"Notation","description":"Exponents.","findings":[{"id":"notation-finding","name":"Notation","description":"Exponent notation.","questions":[{"text":"How is exponent notation written and read?","kind":"definition"},{"text":"What common mistakes occur?","kind":"boundary"}]}]}]}]},"agentConduct":{"may":["Test whether an integer is a perfect power using exact integer arithmetic.","List perfect powers and their bases and exponents for teaching or research."],"mustNot":["Rely on floating point roots alone to decide that a large number is a perfect power.","Confuse perfect powers with powers of a fixed base.","Present an unproved conjecture about powers as a theorem."],"requiresHuman":["Publishing a claimed new result about perfect powers as established."]},"ethics":{"considerations":["Correct mathematics matters where perfect power tests feed into cryptographic and number theory software."],"affectedParties":["Students and teachers","Users of number theory software"]},"owners":{"steward":"Nobody owns the concept; it belongs to mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q28920052","type":"parent","note":"registry parent class"},{"target":"non-negative integer","type":"related","note":"category"},{"target":"square number","type":"related","note":"kind"},{"target":"Catalan conjecture","type":"related","note":"theorem"},{"target":"number theory","type":"related","note":"field"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.perfect-power","Wikidata: Q1972803 (https://www.wikidata.org/wiki/Q1972803)","OEIS: A001597 integer sequence"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Forms such as m squared or m cubed","Squares and cubes are special cases","Powers of a fixed base such as 2 form a different sequence","Integers such as 16, 27, 32 and 64."]},"capabilities":{"applicability":"required","items":["test whether a number is a perfect power","list squares, cubes and higher powers","explain related theorems","implement exact tests in code"]},"hazards":{"applicability":"required","items":["Floating point errors in root tests","Confusing perfect powers with powers of a base"]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Number theory and computation.","square numbers","cube numbers","fourth and higher powers","powers of two, three and five"]}},"sources":[{"title":"Wikidata item Q1972803: perfect power","url":"https://www.wikidata.org/wiki/Q1972803","note":"identity and sense of the item"},{"title":"Wikipedia: Perfect power","url":"https://en.wikipedia.org/wiki/Perfect_power","note":"general description of the item"}],"openQuestions":["Should cubes be a separate entry?","How should OEIS be linked?","How should related conjectures be tracked?"],"resources":{"spec":"/models/things/publications/thing-q1972803/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1972803/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T11:32:52Z","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":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:28+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No specific standard, law or official classification applies to perfect powers as an abstract mathematical concept."},"score":1.0}}