{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.double-mersenne-number","code":"thing-q1318413","url":"https://ver.cy/models/thing/q1318413/","name":"double Mersenne 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 number of the form 2^(2^p - 1) - 1, denoted MM(p), where both p and 2^p - 1 are prime, making it a Mersenne number whose exponent is itself a Mersenne prime; the first four cases MM(2), MM(3), MM(5) and MM(7) are prime, while MM(13) is known to be composite. The related Catalan-Mersenne sequence consists of iterated Mersenne numbers whose primality beyond the fifth term is unknown; double Mersenne numbers are among the largest numbers studied for primality.","purpose":"Let an agent explain double Mersenne numbers, relay known results and open problems from mathematical references, describe their relation to Mersenne primes and the Catalan-Mersenne conjecture, and distinguish them from ordinary Mersenne numbers.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["The nested Mersenne form requires both p and 2^p - 1 to be prime in this sense.","Four double Mersenne primes are known.","These numbers are related to the Catalan-Mersenne sequence.","Some cases beyond MM(7), including MM(13), are known to be composite, while other cases remain unresolved."],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a double Mersenne number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a double Mersenne number, and how does it relate to Mersenne numbers and primes?","kind":"definition"},{"text":"Is the question about double Mersenne numbers, Mersenne numbers or Fermat numbers?","kind":"boundary"}]}]},{"id":"known","name":"Known","description":"Known results.","findings":[{"id":"known-finding","name":"Known","description":"Known.","questions":[{"text":"Which double Mersenne numbers are known to be prime or composite, and how were factors found?","kind":"provenance"},{"text":"Which entry fits Mersenne prime?","kind":"action"}]}]}]},{"id":"problems","name":"Problems","description":"Open problems.","layers":[{"id":"catalan","name":"Catalan","description":"Catalan-Mersenne conjecture.","findings":[{"id":"catalan-finding","name":"Catalan","description":"Catalan.","questions":[{"text":"What is the Catalan-Mersenne sequence, and why is the primality of its fifth term unknown?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"mm61","name":"MM(61)","description":"Candidates.","findings":[{"id":"mm61-finding","name":"MM(61)","description":"MM(61).","questions":[{"text":"Why are MM(61) and larger candidates beyond current primality testing?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"compute","name":"Compute","description":"Computation.","layers":[{"id":"tests","name":"Tests","description":"Primality tests.","findings":[{"id":"tests-finding","name":"Tests","description":"Tests.","questions":[{"text":"How does the Lucas-Lehmer test apply, and why is it infeasible for huge double Mersenne numbers?","kind":"provenance"},{"text":"Which entry fits Lucas-Lehmer test?","kind":"action"}]}]},{"id":"factors","name":"Factors","description":"Factor searches.","findings":[{"id":"factors-finding","name":"Factors","description":"Factors.","questions":[{"text":"How are factors of double Mersenne numbers searched for?","kind":"provenance"},{"text":"Which entry fits distributed computing in number theory?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and culture.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Catalan, Lucas and later mathematicians study these numbers?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"culture","name":"Culture","description":"Culture.","findings":[{"id":"culture-finding","name":"Culture","description":"Culture.","questions":[{"text":"How have double Mersenne numbers appeared in popular mathematics and fiction?","kind":"provenance"},{"text":"Which entry fits mathematics in popular culture?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain what double Mersenne numbers are and report the known primality status of the small cases.","Distinguish proven results from open conjectures about these numbers."],"mustNot":["Present the conjecture about further primes as settled.","Claim a primality result for a large case without a verifiable proof or certificate.","Misattribute a factorisation or search result to the wrong contributor.","Present distributed search progress as a completed proof."],"requiresHuman":["Publishing a claimed primality or factorisation result under a person's name."]},"ethics":{"considerations":["Claims in computational number theory must be verifiable, or the record is corrupted.","Volunteer searchers deserve credit for the results their computation produced."],"affectedParties":["Mathematicians and students","Volunteer distributed computing participants"]},"owners":{"steward":"Nobody owns them; mathematical and integer sequence databases record the known status.","roles":[],"masterSystems":["Published integer sequence and factorisation databases"]},"relations":[{"target":"Q10338607","type":"parent","note":"registry parent class"},{"target":"Mersenne number","type":"related","note":"in registry terms"},{"target":"Mersenne prime","type":"related","note":"exponents in known cases"},{"target":"Catalan-Mersenne conjecture","type":"related","note":"about iterated Mersenne numbers"},{"target":"primality testing","type":"related","note":"and factor searches"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.double-mersenne-number","Wikidata: Q1318413 (https://www.wikidata.org/wiki/Q1318413)","OEIS: A077586 double Mersenne numbers"]},"properties":{"applicability":"not-applicable","items":["known double Mersenne primes: 4 count - MM(2), MM(3), MM(5), MM(7)","MM(7): 2^127 - 1 value - about 1.7e38","MM(13) digits: about 2466 count - known composite"]},"recognition":{"applicability":"optional","items":["Mersenne number with Mersenne exponent","MM(2) = 7, MM(3) = 127, MM(5), MM(7) prime; MM(13) composite","Ordinary Mersenne numbers are 2^n - 1; Fermat numbers are 2^(2^n) + 1","Not a visible object; enormous integers."]},"capabilities":{"applicability":"required","items":["explain the definition","relay known results","describe open problems","distinguish from Mersenne numbers"]},"hazards":{"applicability":"required","items":["Assuming MM(p) is prime whenever M(p) is","Confusing with Mersenne numbers or Fermat numbers","Overstating what is proved"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","double Mersenne primes","composite double Mersenne numbers such as MM(13), MM(17), MM(19), MM(31)","the Catalan-Mersenne sequence","candidates such as MM(61) with unknown status"]}},"sources":[{"title":"Wikidata item Q1318413: double Mersenne number","url":"https://www.wikidata.org/wiki/Q1318413","note":"identity and sense of the item"},{"title":"Wikipedia: Double Mersenne number","url":"https://en.wikipedia.org/wiki/Double_Mersenne_number","note":"general description of the item"}],"openQuestions":["Should the Catalan-Mersenne conjecture be a separate entry?","How should mathematical references be linked?","How should the status of MM(61) be tracked?"],"resources":{"spec":"/models/things/publications/thing-q1318413/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1318413/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T02:44:35Z","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 2026-10-07T10:10:25+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}}