{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.smooth-number","code":"thing-q1529876","url":"https://ver.cy/models/thing/q1529876/","name":"smooth number","alternateNames":["regular number"],"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 whose prime factors are all at most some bound B, called B-smooth; for example 5-smooth or regular numbers have only the factors 2, 3 and 5; smooth numbers are central to factoring and discrete logarithm algorithms, to fast Fourier transform sizes and to ancient Babylonian arithmetic.","purpose":"Let an agent define smooth numbers precisely, test smoothness for a given bound, explain their distribution and their role in algorithms, and relate regular numbers to music, computing and history.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Bounded largest prime factor","Density described by the Dickman function","Key to sieve-based algorithms","Regular numbers as the classic case"],"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 does B-smooth mean, and what are regular numbers?","kind":"definition"},{"text":"Is the question about smooth, rough or squarefree numbers?","kind":"boundary"}]}]},{"id":"test","name":"Test","description":"Testing.","findings":[{"id":"test-finding","name":"Test","description":"Test.","questions":[{"text":"How is smoothness tested for a given number and bound?","kind":"action"},{"text":"What is the result for the number in question?","kind":"measurement"}]}]}]},{"id":"distribution","name":"Distribution","description":"How common they are.","layers":[{"id":"density","name":"Density","description":"Density.","findings":[{"id":"density-finding","name":"Density","description":"Density.","questions":[{"text":"How many smooth numbers are there up to x, and what does the Dickman function describe?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"results","name":"Results","description":"Known results.","findings":[{"id":"results-finding","name":"Results","description":"Results.","questions":[{"text":"What is proven versus heuristic about smooth number distribution?","kind":"provenance"},{"text":"What is proven versus conjectured?","kind":"boundary"}]}]}]},{"id":"algorithms","name":"Algorithms","description":"Uses in computation.","layers":[{"id":"factoring","name":"Factoring","description":"Factoring and logarithms.","findings":[{"id":"factoring-finding","name":"Factoring","description":"Factoring.","questions":[{"text":"How do the quadratic sieve, number field sieve and index calculus use smooth numbers?","kind":"provenance"},{"text":"Which entry fits integer factorization?","kind":"action"}]}]},{"id":"other","name":"Other","description":"Other uses.","findings":[{"id":"other-finding","name":"Other","description":"Other.","questions":[{"text":"How are regular numbers used in FFT sizes, music theory and Babylonian tables?","kind":"provenance"},{"text":"Which entry fits the fast Fourier transform?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and teaching.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did regular numbers appear in Babylonian mathematics and in later theory?","kind":"provenance"},{"text":"Which entry fits Babylonian mathematics?","kind":"action"}]}]},{"id":"teach","name":"Teach","description":"Teaching.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How are smooth numbers taught in number theory and cryptography courses?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Test whether a number is B-smooth and state the bound used.","Explain the role of smooth numbers in factoring, discrete logarithms and transform sizes."],"mustNot":["Call a number smooth without stating the bound.","Confuse smooth numbers with rough or squarefree numbers.","Present heuristic estimates of how many smooth numbers exist as proven results.","Overstate what smooth-number methods imply for the security of deployed cryptography."],"requiresHuman":["Choosing cryptographic key sizes on the basis of smoothness estimates."]},"ethics":{"considerations":["Results about smooth numbers bear on how secure some cryptographic systems are."],"affectedParties":["Learners of number theory","People relying on cryptography"]},"owners":{"steward":"Nobody owns the concept; it belongs to the shared body of mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"a property of integers"},{"target":"integer factorization","type":"related","note":"sieve algorithms"},{"target":"rough number","type":"related","note":"the opposite property"},{"target":"On-Line Encyclopedia of Integer Sequences","type":"related","note":"sequence A051037 for regular numbers"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.smooth-number","Wikidata: Q1529876 (https://www.wikidata.org/wiki/Q1529876)","OEIS: A051037 regular numbers"]},"properties":{"applicability":"not-applicable","items":["regular numbers below 60: 1 to 60 with factors 2, 3, 5 only integers - used in Babylonian reciprocal tables"]},"recognition":{"applicability":"optional","items":["All prime factors at most B","Regular numbers are 5-smooth: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12","Rough numbers have only large prime factors, the opposite property","Not visible; a property of integers."]},"capabilities":{"applicability":"required","items":["test whether a number is B-smooth","explain the definition and examples","explain the role in factoring algorithms","relate regular numbers to their uses"]},"hazards":{"applicability":"required","items":["Omitting the bound B when calling a number smooth","Confusing smooth with rough or with squarefree","Presenting heuristic distribution results as exact"]},"interfaces":{"applicability":"required","items":["No regulation; definitions follow the literature"]},"context":{"applicability":"required","items":["Not for anything in itself; a property used in algorithms and computation.","B-smooth numbers for a bound B","regular or 5-smooth numbers","7-smooth humble numbers","powersmooth numbers","smooth numbers in algorithm analysis"]}},"sources":[{"title":"Wikidata item Q1529876: smooth number","url":"https://www.wikidata.org/wiki/Q1529876","note":"identity and sense of the item"},{"title":"Wikipedia: Smooth number","url":"https://en.wikipedia.org/wiki/Smooth_number","note":"general description of the item"}],"openQuestions":["Should regular numbers be a separate entry?","How should OEIS entries be linked?","How should the Dickman function be represented?"],"resources":{"spec":"/models/things/publications/thing-q1529876/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1529876/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T20:28:17Z","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 2026-10-06T19:51:54+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}}