{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.congruent-number","code":"thing-q325978","url":"https://ver.cy/models/thing/q325978/","name":"congruent 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 congruent number is a positive integer that is the area of a right triangle with rational side lengths, such as 5, 6, 7, 20, 21 and 24. The congruent number problem has roots in ancient Arabic and medieval mathematics and remains open in general. A positive integer n is congruent if and only if the elliptic curve y^2 = x^3 - n^2 x has positive rank over the rational numbers, an unconditional equivalence. For squarefree positive integers, Tunnell's theorem gives an unconditional necessary criterion for congruence; sufficiency of that criterion depends on the Birch and Swinnerton-Dyer conjecture.","purpose":"Let an agent explain congruent numbers and the congruent number problem, relay known results, Tunnell theorem and the link to elliptic curves from mathematical references, and distinguish congruent numbers from the unrelated notion of congruence.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Rational right triangle area","Ancient origin","Elliptic curve connection","Open problem"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a congruent number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a congruent number, and why is 5 congruent but 1, 2 and 3 not?","kind":"definition"},{"text":"Is the question about congruent numbers or about congruence modulo n?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the small congruent numbers, and what rational triangles realise them?","kind":"measurement"},{"text":"Which entry fits rational right triangle?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Theory.","layers":[{"id":"elliptic","name":"Elliptic","description":"Elliptic curves.","findings":[{"id":"elliptic-finding","name":"Elliptic","description":"Elliptic.","questions":[{"text":"How does n being congruent correspond to the elliptic curve y^2 = x^3 - n^2 x having positive rank?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"tunnell","name":"Tunnell","description":"Tunnell theorem.","findings":[{"id":"tunnell-finding","name":"Tunnell","description":"Tunnell.","questions":[{"text":"What does Tunnell theorem say, and why does it depend on the Birch and Swinnerton-Dyer conjecture?","kind":"provenance"},{"text":"Which entry fits Birch and Swinnerton-Dyer conjecture?","kind":"action"}]}]}]},{"id":"problem","name":"Problem","description":"The open problem.","layers":[{"id":"status","name":"Status","description":"Status.","findings":[{"id":"status-finding","name":"Status","description":"Status.","questions":[{"text":"What is known and unknown about classifying congruent numbers?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"computation","name":"Computation","description":"Computation.","findings":[{"id":"computation-finding","name":"Computation","description":"Computation.","questions":[{"text":"How have large computations enumerated congruent numbers?","kind":"provenance"},{"text":"Which entry fits computational number theory?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did the problem arise in Arabic manuscripts and Fibonacci work, and how did Fermat show 1 is not congruent?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"culture","name":"Culture","description":"Popular mathematics.","findings":[{"id":"culture-finding","name":"Culture","description":"Culture.","questions":[{"text":"How is the congruent number problem presented in popular mathematics?","kind":"provenance"},{"text":"Which entry fits unsolved problems in mathematics?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Test, list and explain congruent numbers with stated methods and sources.","Make clear which results depend on unproved conjectures."],"mustNot":["Present a result that depends on an unproved conjecture as unconditional.","Claim a number is or is not congruent without a check or a source.","Misattribute historical or modern results to the wrong mathematicians."],"requiresHuman":["Announcing a claimed proof of an open problem in this area."]},"ethics":{"considerations":["Credit for results in number theory belongs to those who proved them.","False claims about open problems mislead students and waste reviewers' time."],"affectedParties":["Mathematicians","Students"]},"owners":{"steward":"Nobody owns the concept; it is held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"in registry terms"},{"target":"Tunnell theorem","type":"related","note":"conditionally"},{"target":"elliptic curve","type":"related","note":"y^2 = x^3 - n^2 x"},{"target":"OEIS","type":"related","note":"as A003273"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.congruent-number","Wikidata: Q325978 (https://www.wikidata.org/wiki/Q325978)","OEIS: A003273 congruent numbers"]},"properties":{"applicability":"not-applicable","items":["smallest congruent number: 5 value - triangle 3/2, 20/3, 41/6","congruent numbers below 100: 37 count - OEIS A003273","Tunnell theorem: 1983 year"]},"recognition":{"applicability":"optional","items":["Area of a rational right triangle","Congruent numbers 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24","Congruence modulo n is a different concept; Pythagorean triples give integer-sided triangles","Not a visible object; integers and rational triangles."]},"capabilities":{"applicability":"required","items":["explain the definition","relay known results","describe the elliptic curve link","distinguish from congruence"]},"hazards":{"applicability":"required","items":["Confusing with modular congruence","Overstating what is proved unconditionally","Missing the squarefree reduction"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","congruent numbers","non-congruent numbers such as 1, 2, 3 and 4","squarefree congruent numbers as the essential cases","residue class results such as numbers congruent to 5, 6 or 7 mod 8 under BSD"]}},"sources":[{"title":"Wikidata item Q325978: congruent number","url":"https://www.wikidata.org/wiki/Q325978","note":"identity and sense of the item"},{"title":"Wikipedia: Congruent number","url":"https://en.wikipedia.org/wiki/Congruent_number","note":"general description of the item"}],"openQuestions":["Should the congruent number problem be a separate entry?","How should mathematical references be linked?","How should computational results such as the 2009 trillion-number computation be recorded?"],"resources":{"spec":"/models/things/publications/thing-q325978/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q325978/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T03:05:07Z","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 2026-10-07T10:10:30+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}}