{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.achilles-number","code":"thing-q3232444","url":"https://ver.cy/models/thing/q3232444/","name":"Achilles number","alternateNames":[],"kind":"thing","status":"published","version":"0.2.0-wave.3","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 is powerful, meaning every prime factor appears at least squared, but is not a perfect power, such as 72, 108, 200, 288, 392, 432, 500, 648, 675 and 800; Achilles numbers are named for the hero who was powerful but imperfect, and they include strong Achilles numbers whose Euler totient is also an Achilles number.","purpose":"Let an agent define Achilles numbers, distinguish them from powerful numbers and perfect powers, give examples, and help test whether a number qualifies.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Every prime factor at least squared","Not a perfect power","Named after Achilles","Strong subclass via totient"],"structure":{"bundles":[{"id":"define","name":"Define","description":"What Achilles numbers are.","layers":[{"id":"definition","name":"Definition","description":"Definition and examples.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is an Achilles number, and what are the first examples?","kind":"definition"},{"text":"Is the number powerful, and is it a perfect power?","kind":"boundary"}]}]},{"id":"relations","name":"Relations","description":"Related classes.","findings":[{"id":"relations-finding","name":"Relations","description":"Relations.","questions":[{"text":"How do Achilles numbers relate to powerful numbers and perfect powers?","kind":"definition"},{"text":"Which entry fits powerful number?","kind":"action"}]}]}]},{"id":"compute","name":"Compute","description":"Testing and generating.","layers":[{"id":"test","name":"Test","description":"Testing a number.","findings":[{"id":"test-finding","name":"Test","description":"Test.","questions":[{"text":"How is a number tested for the Achilles property by factorisation and exponent gcd?","kind":"action"},{"text":"Is the number in question an Achilles number?","kind":"measurement"}]}]},{"id":"strong","name":"Strong","description":"Strong Achilles numbers.","findings":[{"id":"strong-finding","name":"Strong","description":"Strong.","questions":[{"text":"What are strong Achilles numbers, and how are they found?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"theory","name":"Theory","description":"Properties.","layers":[{"id":"density","name":"Density","description":"Density and structure.","findings":[{"id":"density-finding","name":"Density","description":"Density.","questions":[{"text":"How are Achilles numbers distributed, and what structure do they have?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"related","name":"Related","description":"Related problems.","findings":[{"id":"related-finding","name":"Related","description":"Related.","questions":[{"text":"How do powerful numbers figure in problems such as consecutive powerful numbers?","kind":"provenance"},{"text":"Which entry fits the specific problem?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"Naming and status.","layers":[{"id":"naming","name":"Naming","description":"Naming.","findings":[{"id":"naming-finding","name":"Naming","description":"Naming.","questions":[{"text":"Why are they named after Achilles, and who introduced the term?","kind":"provenance"},{"text":"Which entry fits recreational mathematics?","kind":"action"}]}]},{"id":"status","name":"Status","description":"Mathematical status.","findings":[{"id":"status-finding","name":"Status","description":"Status.","questions":[{"text":"How do mathematicians regard the class, with positions attributed?","kind":"provenance"},{"text":"Is the presentation neutral?","kind":"boundary"}]}]}]}]},"agentConduct":{"may":["Define Achilles numbers and check whether a number is one, showing why.","Explain the link to powerful numbers and perfect powers."],"mustNot":["Forget to exclude perfect powers.","Misidentify powerful numbers.","Overstate the mathematical significance of Achilles numbers.","Confuse the mathematical term with the mythological figure."],"requiresHuman":["Publishing new claimed results."]},"ethics":{"considerations":["Accurate definitions matter for students.","Mathematical play deserves honesty about significance."],"affectedParties":["Students","Puzzle enthusiasts"]},"owners":{"steward":"Nobody: a mathematical concept held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q3191171","type":"parent","note":"registry parent class"},{"target":"powerful number","type":"related","note":"in registry terms"},{"target":"perfect power","type":"related","note":"the excluded case"},{"target":"Euler totient function","type":"related","note":"in the strong subclass"},{"target":"recreational mathematics","type":"related","note":"the field"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.achilles-number","Wikidata: Q3232444 (https://www.wikidata.org/wiki/Q3232444)","OEIS: A052486 Achilles numbers"]},"properties":{"applicability":"not-applicable","items":["first terms: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800 sequence"]},"recognition":{"applicability":"optional","items":["Powerful but not a perfect power","Sequence 72, 108, 200, 288, 392","Perfect powers such as 64 are powerful but excluded; squarefree numbers are not powerful","Not a visible object; integers of a special form."]},"capabilities":{"applicability":"required","items":["define the class and its conditions","distinguish from powerful numbers and perfect powers","give examples","test numbers"]},"hazards":{"applicability":"required","items":["Forgetting to exclude perfect powers","Misidentifying powerful numbers","Overstating mathematical significance"]},"interfaces":{"applicability":"required","items":["No regulation; the OEIS is the reference database"]},"context":{"applicability":"required","items":["A recreational and number-theoretic classification.","Achilles numbers","strong Achilles numbers","powerful numbers as the parent class","perfect powers as the excluded class"]}},"sources":[{"title":"Wikidata item Q3232444: Achilles number","url":"https://www.wikidata.org/wiki/Q3232444","note":"identity and sense of the item"},{"title":"Wikipedia: Achilles number","url":"https://en.wikipedia.org/wiki/Achilles_number","note":"general description of the item"}],"openQuestions":["Should powerful number be the primary entry?","How should the OEIS be linked?","How should the naming story be sourced?"],"resources":{"spec":"/models/things/publications/thing-q3232444/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q3232444/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T23:55:53Z","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":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:51:58+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}