{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.euclid-number","code":"thing-q728223","url":"https://ver.cy/models/thing/q728223/","name":"Euclid 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 E_n = p_n# + 1, one more than the product of the first n prime numbers, named after Euclid because his proof of the infinitude of primes considers such products, giving 3, 7, 31, 211, 2311, 30031 and so on; the first few Euclid numbers are prime but 30031 is composite, it is unknown whether infinitely many Euclid numbers are prime, and they are related to primorials and to the primorial primes.","purpose":"Let an agent explain Euclid numbers and their relation to Euclid proof, relay known primality results and open problems from number theory references, and distinguish Euclid numbers from primorial primes and Euclid-Mullin sequences.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Primorial plus one","Euclid proof link","Mostly composite","Open primality question"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a Euclid number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a Euclid number, and how does it relate to Euclid proof that there are infinitely many primes?","kind":"definition"},{"text":"Is the question about Euclid numbers, primorial primes or the Euclid-Mullin sequence?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the first Euclid numbers, and which are prime or composite?","kind":"measurement"},{"text":"Which entry fits the specific number?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Results.","layers":[{"id":"results","name":"Results","description":"Known results.","findings":[{"id":"results-finding","name":"Results","description":"Results.","questions":[{"text":"What is known about the primality and factors of Euclid numbers?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"open","name":"Open","description":"Open problems.","findings":[{"id":"open-finding","name":"Open","description":"Open.","questions":[{"text":"Is it known whether infinitely many Euclid numbers are prime or composite?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"compare","name":"Compare","description":"Related sequences.","layers":[{"id":"primorial","name":"Primorial","description":"Primorials and primorial primes.","findings":[{"id":"primorial-finding","name":"Primorial","description":"Primorial.","questions":[{"text":"How do primorials and primorial primes relate to Euclid numbers?","kind":"provenance"},{"text":"Which entry fits primorial?","kind":"action"}]}]},{"id":"mullin","name":"Mullin","description":"Euclid-Mullin sequence.","findings":[{"id":"mullin-finding","name":"Mullin","description":"Mullin.","questions":[{"text":"What is the Euclid-Mullin sequence, and how does it differ?","kind":"provenance"},{"text":"Which entry fits Euclid-Mullin sequence?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and teaching.","layers":[{"id":"history","name":"History","description":"Euclid proof.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How does Euclid proof work, and how is it often misstated?","kind":"provenance"},{"text":"Which entry fits Euclid theorem?","kind":"action"}]}]},{"id":"computation","name":"Computation","description":"Computation.","findings":[{"id":"computation-finding","name":"Computation","description":"Computation.","questions":[{"text":"How are large Euclid numbers tested for primality?","kind":"action"},{"text":"Which entry fits primality test?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Compute Euclid numbers and state which are known to be prime or composite, citing sources.","Explain the role of such products in Euclid's proof of the infinitude of primes.","Distinguish Euclid numbers from primorial primes and the Euclid-Mullin sequence."],"mustNot":["State that every Euclid number is prime.","Claim that Euclid's proof constructs new primes directly as Euclid numbers.","Claim new primality results without verifiable computation or a published source."],"requiresHuman":["Announcing a new mathematical result about Euclid numbers to the public."]},"ethics":{"considerations":["The common misreading of Euclid's proof spreads a false lesson to students."],"affectedParties":["Students and teachers","Mathematicians"]},"owners":{"steward":"Nobody: a mathematical concept 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":"primorial","type":"related","note":"plus one"},{"target":"Euclid","type":"related","note":"and his proof of infinite primes"},{"target":"primorial prime","type":"related","note":"when prime"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.euclid-number","Wikidata: Q728223 (https://www.wikidata.org/wiki/Q728223)","OEIS: A006862 Euclid numbers"]},"properties":{"applicability":"not-applicable","items":["first Euclid numbers: 3, 7, 31, 211, 2311, 30031 sequence - OEIS A006862","first composite Euclid number: 30031 value - 59 times 509","known prime Euclid numbers: for n = 1 to 5, 11, 75, 171, 172, 384 and more list - per OEIS"]},"recognition":{"applicability":"optional","items":["Primorial plus one","E_1 = 3, E_2 = 7, E_3 = 31, E_4 = 211, E_5 = 2311, E_6 = 30031","Primorial primes include p# - 1 forms; the Euclid-Mullin sequence uses smallest prime factors","Not a visible object; integers one more than primorials."]},"capabilities":{"applicability":"required","items":["explain the definition","relay known results","describe open problems","distinguish related sequences"]},"hazards":{"applicability":"required","items":["Assuming all Euclid numbers are prime","Misreading Euclid proof as producing primes directly","Confusing with primorial primes or the Euclid-Mullin sequence"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","Euclid numbers p_n# + 1","primorial primes of the form p# plus or minus 1","Euclid-Mullin sequence","Kummer numbers p_n# - 1"]}},"sources":[{"title":"Wikidata item Q728223: Euclid number","url":"https://www.wikidata.org/wiki/Q728223","note":"identity and sense of the item"},{"title":"Wikipedia: Euclid number","url":"https://en.wikipedia.org/wiki/Euclid_number","note":"general description of the item"}],"openQuestions":["Should primorial and primorial prime be the primary linked entries?","How should number theory references be linked?","How should the record of prime Euclid numbers be kept current?"],"resources":{"spec":"/models/things/publications/thing-q728223/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q728223/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T04:06:09Z","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}}