{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.proth-number","code":"thing-q593418","url":"https://ver.cy/models/thing/q593418/","name":"Proth 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 of the form k times 2 to the power n plus 1, where k is odd and k is less than 2 to the power n, such as 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81 and 97; Proth numbers are named after Francois Proth, whose 1878 theorem gives a simple primality test for them, and Proth primes include many of the largest known primes found by distributed computing.","purpose":"Let an agent define Proth numbers and Proth primes, explain Proth theorem and its use as a primality test, relate them to Fermat and Cullen numbers, and help test or list examples.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Special form","Simple primality test","Source of large primes","Connection to Fermat numbers"],"structure":{"bundles":[{"id":"define","name":"Define","description":"What Proth numbers are.","layers":[{"id":"definition","name":"Definition","description":"Definition and examples.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a Proth number, and what are the first examples?","kind":"definition"},{"text":"Does the number in question satisfy the form and the condition on k?","kind":"boundary"}]}]},{"id":"primes","name":"Primes","description":"Proth primes.","findings":[{"id":"primes-finding","name":"Primes","description":"Primes.","questions":[{"text":"What are Proth primes, and which large primes have this form?","kind":"definition"},{"text":"Which entry fits Proth prime?","kind":"action"}]}]}]},{"id":"test","name":"Test","description":"Primality testing.","layers":[{"id":"theorem","name":"Theorem","description":"Proth theorem.","findings":[{"id":"theorem-finding","name":"Theorem","description":"Theorem.","questions":[{"text":"What does Proth theorem state, and how is it applied as a test?","kind":"action"},{"text":"Is the Proth number in question prime?","kind":"measurement"}]}]},{"id":"algorithms","name":"Algorithms","description":"Algorithms.","findings":[{"id":"algorithms-finding","name":"Algorithms","description":"Algorithms.","questions":[{"text":"How is the test implemented efficiently for large numbers?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"theory","name":"Theory","description":"Related forms.","layers":[{"id":"fermat","name":"Fermat","description":"Fermat numbers.","findings":[{"id":"fermat-finding","name":"Fermat","description":"Fermat.","questions":[{"text":"How are Fermat numbers Proth numbers, and what does Pepin test have to do with Proth theorem?","kind":"provenance"},{"text":"Which entry fits Fermat number?","kind":"action"}]}]},{"id":"sierpinski","name":"Sierpinski","description":"Sierpinski and related problems.","findings":[{"id":"sierpinski-finding","name":"Sierpinski","description":"Sierpinski.","questions":[{"text":"How do Proth numbers relate to Sierpinski numbers and the Seventeen or Bust project?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"context","name":"Context","description":"History and computing.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"Who was Francois Proth, and how was his theorem received?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"computing","name":"Computing","description":"Distributed prime searches.","findings":[{"id":"computing-finding","name":"Computing","description":"Computing.","questions":[{"text":"How do distributed projects search for Proth primes?","kind":"provenance"},{"text":"Which entry fits distributed computing?","kind":"action"}]}]}]}]},"agentConduct":{"may":["State the definition of a Proth number including the condition that k is less than 2 to the n.","Check whether a number is a Proth number and show the working.","Explain Proth's theorem and its use for testing primes."],"mustNot":["Omit the condition that k is odd and less than 2 to the n.","Confuse Proth numbers with Proth primes.","Report unverified results for large cases as confirmed.","Claim new prime discoveries without verification."],"requiresHuman":["Announcing a newly found large Proth prime."]},"ethics":{"considerations":["Correct credit for discoveries matters to mathematicians and volunteer projects.","Accurate definitions keep students from learning errors."],"affectedParties":["Students","Mathematicians","Volunteer computing participants"]},"owners":{"steward":"Nobody: a mathematical concept held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"in registry terms"},{"target":"Proth theorem","type":"related","note":"the primality test"},{"target":"Fermat number","type":"related","note":"as a special case"},{"target":"Francois Proth","type":"related","note":"the mathematician"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.proth-number","Wikidata: Q593418 (https://www.wikidata.org/wiki/Q593418)","OEIS: A080075 Proth numbers"]},"properties":{"applicability":"not-applicable","items":["first terms: 3, 5, 9, 13, 17, 25, 33, 41, 49, 57 sequence","Proth theorem: 1878 year"]},"recognition":{"applicability":"optional","items":["k times 2 to the n plus 1 with odd k less than 2 to the n","Sequence 3, 5, 9, 13, 17, 25, 33","Fermat numbers are a special case; Sierpinski numbers are related","Not a visible object; integers of a special form."]},"capabilities":{"applicability":"required","items":["define the numbers and primes","explain Proth theorem","relate to other special forms","help test or list examples"]},"hazards":{"applicability":"required","items":["Omitting the condition k less than 2 to the n","Confusing Proth numbers with Proth primes","Computation errors for large cases"]},"interfaces":{"applicability":"required","items":["No regulation; the OEIS and prime databases are references"]},"context":{"applicability":"required","items":["Primality testing and prime searching.","Proth numbers","Proth primes","Fermat numbers as Proth numbers with k equal to 1","related forms such as Cullen and Sierpinski numbers"]}},"sources":[{"title":"Wikidata item Q593418: Proth number","url":"https://www.wikidata.org/wiki/Q593418","note":"identity and sense of the item"},{"title":"Wikipedia: Proth number","url":"https://en.wikipedia.org/wiki/Proth_number","note":"general description of the item"}],"openQuestions":["Should Proth prime be a separate entry?","How should prime databases be linked?","How should records of largest Proth primes be kept current?"],"resources":{"spec":"/models/things/publications/thing-q593418/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q593418/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T23:41:50Z","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 2026-10-06T19:52:01+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}}