{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.cullen-number","code":"thing-q532389","url":"https://ver.cy/models/thing/q532389/","name":"Cullen 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 natural number of the form n times 2 to the power n, plus 1, named after James Cullen who studied them in 1905; almost all Cullen numbers are composite, only a few Cullen primes are known, and the related Woodall numbers have the form n times 2 to the power n, minus 1.","purpose":"Let an agent define Cullen numbers precisely, list small examples and the known Cullen primes, relate them to Woodall and generalised Cullen numbers, and point to sequence references and open problems.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Defined by n times 2 to the n, plus 1","Almost all composite","Few known primes","Paired with Woodall numbers"],"structure":{"bundles":[{"id":"define","name":"Define","description":"The definition.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a Cullen number, and how does it differ from a Woodall number?","kind":"definition"},{"text":"Is the question about Cullen, Woodall or generalised Cullen numbers?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the first Cullen numbers, and how are they computed?","kind":"measurement"},{"text":"Which entry fits integer sequences?","kind":"action"}]}]}]},{"id":"primes","name":"Primes","description":"Cullen primes.","layers":[{"id":"known","name":"Known","description":"Known primes.","findings":[{"id":"known-finding","name":"Known","description":"Known.","questions":[{"text":"Which Cullen primes are known, and where is the current list maintained?","kind":"provenance"},{"text":"Is the list current?","kind":"boundary"}]}]},{"id":"divisibility","name":"Divisibility","description":"Divisibility results.","findings":[{"id":"divisibility-finding","name":"Divisibility","description":"Divisibility.","questions":[{"text":"What divisibility properties make almost all Cullen numbers composite?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"compute","name":"Compute","description":"Computation and search.","layers":[{"id":"search","name":"Search","description":"Prime searches.","findings":[{"id":"search-finding","name":"Search","description":"Search.","questions":[{"text":"How are Cullen primes searched for, and which projects do it?","kind":"provenance"},{"text":"Which entry fits distributed computing projects?","kind":"action"}]}]},{"id":"generalise","name":"Generalise","description":"Generalisations.","findings":[{"id":"generalise-finding","name":"Generalise","description":"Generalise.","questions":[{"text":"What generalised Cullen numbers exist for other bases?","kind":"provenance"},{"text":"Should a generalisation be a separate entry?","kind":"boundary"}]}]}]},{"id":"context","name":"Context","description":"History and teaching.","layers":[{"id":"history","name":"History","description":"Cullen and Cunningham.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"Who were Cullen and Cunningham, and how did the numbers get their name?","kind":"provenance"},{"text":"Which entry fits the person?","kind":"action"}]}]},{"id":"teach","name":"Teach","description":"Teaching.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How are Cullen numbers used in teaching and puzzles?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Compute Cullen numbers and report known Cullen primes from published records.","Explain their relation to Woodall numbers."],"mustNot":["Announce a new Cullen prime without independent verification.","Confuse Cullen numbers with Woodall numbers or with generalised forms.","Use other people's computing resources for prime searches without permission."],"requiresHuman":["Announcing a new record to the mathematical community."]},"ethics":{"considerations":["Credit for discoveries in distributed searches belongs to the people who ran them.","False record claims mislead others."],"affectedParties":["Researchers and volunteers in prime searches","Students"]},"owners":{"steward":"Nobody owns the concept; it belongs to mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"a property of integers"},{"target":"Woodall number","type":"related","note":"the minus-one counterpart"},{"target":"On-Line Encyclopedia of Integer Sequences","type":"related","note":"sequence A002064"},{"target":"number theory","type":"related","note":"the field"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.cullen-number","Wikidata: Q532389 (https://www.wikidata.org/wiki/Q532389)","OEIS: A002064 Cullen numbers","OEIS: A005849 indices of Cullen primes"]},"properties":{"applicability":"not-applicable","items":["first terms: 1, 3, 9, 25, 65, 161, 385 integers - for n from 0","smallest Cullen prime beyond trivial: n equal to 1 and n equal to 141 indices - Cullen primes are rare"]},"recognition":{"applicability":"optional","items":["Form n times 2 to the n, plus 1","First terms 1, 3, 9, 25, 65, 161","Woodall numbers subtract 1 instead of adding","Not visible; a property of integers."]},"capabilities":{"applicability":"required","items":["compute Cullen numbers for given n","list known Cullen primes","distinguish Cullen from Woodall and generalised forms","point to sequence references"]},"hazards":{"applicability":"required","items":["Confusing Cullen with Woodall numbers","Quoting outdated lists of known primes","Presenting conjectures as theorems"]},"interfaces":{"applicability":"required","items":["No regulation; definitions follow the literature and the OEIS"]},"context":{"applicability":"required","items":["Not for anything; an object of number theory.","Cullen numbers","Cullen primes","generalised Cullen numbers with other bases","Woodall numbers as the related family"]}},"sources":[{"title":"Wikidata item Q532389: Cullen number","url":"https://www.wikidata.org/wiki/Q532389","note":"identity and sense of the item"},{"title":"Wikipedia: Cullen number","url":"https://en.wikipedia.org/wiki/Cullen_number","note":"general description of the item"}],"openQuestions":["Should Cullen primes be a separate entry?","How should prime search projects be linked?","Which conjectures, such as infinitude of Cullen primes, should be recorded?"],"resources":{"spec":"/models/things/publications/thing-q532389/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q532389/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T20:13:51Z","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 2026-10-06T19:52:00+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}}