{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.taxicab-number","code":"thing-q1462591","url":"https://ver.cy/models/thing/q1462591/","name":"taxicab 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":"The smallest positive integer that can be expressed as a sum of two positive cubes in n distinct ways, written Ta(n); the first taxicab numbers are 2 for n equal to 1, 1729 for n equal to 2, and 87539319 for n equal to 3, and the name recalls the anecdote of Hardy and Ramanujan about the taxi numbered 1729.","purpose":"Let an agent define taxicab numbers precisely, list the known values and their representations, distinguish them from related notions such as cabtaxi numbers, and relate them to the Hardy-Ramanujan anecdote and to Diophantine problems.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Defined by sums of two positive cubes","Smallest with n representations","Only a few values known exactly","Named for the Hardy-Ramanujan anecdote"],"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 exactly is Ta(n), and what counts as a distinct representation?","kind":"definition"},{"text":"Is the question about taxicab numbers, cabtaxi numbers or a generalisation?","kind":"boundary"}]}]},{"id":"values","name":"Values","description":"Known values.","findings":[{"id":"values-finding","name":"Values","description":"Values.","questions":[{"text":"What are the known taxicab numbers and their representations?","kind":"measurement"},{"text":"Which values are exact and which are bounds?","kind":"boundary"}]}]}]},{"id":"compute","name":"Compute","description":"Computation.","layers":[{"id":"verify","name":"Verify","description":"Verification.","findings":[{"id":"verify-finding","name":"Verify","description":"Verify.","questions":[{"text":"How is a claimed representation verified, and how are candidates searched?","kind":"action"},{"text":"Which entry fits Diophantine equations?","kind":"action"}]}]},{"id":"bounds","name":"Bounds","description":"Bounds.","findings":[{"id":"bounds-finding","name":"Bounds","description":"Bounds.","questions":[{"text":"How were bounds for higher taxicab numbers found, with attribution?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"theory","name":"Theory","description":"Mathematics behind it.","layers":[{"id":"cubes","name":"Cubes","description":"Sums of two cubes.","findings":[{"id":"cubes-finding","name":"Cubes","description":"Cubes.","questions":[{"text":"What is known about representing numbers as sums of two cubes, and why do multiple representations exist?","kind":"provenance"},{"text":"Which entry fits sums of powers?","kind":"action"}]}]},{"id":"generalise","name":"Generalise","description":"Generalisations.","findings":[{"id":"generalise-finding","name":"Generalise","description":"Generalise.","questions":[{"text":"What generalisations to other powers and more terms exist?","kind":"provenance"},{"text":"Should a generalisation be a separate entry?","kind":"boundary"}]}]}]},{"id":"context","name":"Context","description":"History and culture.","layers":[{"id":"anecdote","name":"Anecdote","description":"Hardy and Ramanujan.","findings":[{"id":"anecdote-finding","name":"Anecdote","description":"Anecdote.","questions":[{"text":"What is the Hardy-Ramanujan anecdote, and how is it documented?","kind":"provenance"},{"text":"Which entry fits Ramanujan?","kind":"action"}]}]},{"id":"teach","name":"Teach","description":"Teaching.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How are taxicab numbers used in teaching and popular mathematics?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain taxicab numbers and the known values from published sources.","Verify that a number has the stated representations as sums of cubes."],"mustNot":["Present an upper bound as a proven taxicab number when it is not proven minimal.","Misstate the story of 1729 or attribute it to the wrong people.","Confuse taxicab numbers with cabtaxi numbers or with taxicab geometry."],"requiresHuman":["Correcting a published value."]},"ethics":{"considerations":["Accurate records of mathematical results protect credit and trust.","Popular anecdotes are often repeated with errors."],"affectedParties":["Students","Mathematicians"]},"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":"1729","type":"related","note":"the famous instance"},{"target":"cabtaxi number","type":"related","note":"a related sequence allowing negative cubes"},{"target":"On-Line Encyclopedia of Integer Sequences","type":"related","note":"sequence A011541"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.taxicab-number","Wikidata: Q1462591 (https://www.wikidata.org/wiki/Q1462591)","OEIS: A011541 taxicab numbers"]},"properties":{"applicability":"not-applicable","items":["Ta(2): 1729 integer","Ta(3): 87539319 integer","known values: Ta(1) to Ta(6) count - later values have upper bounds only"]},"recognition":{"applicability":"optional","items":["Smallest number with n representations as a sum of two positive cubes","Ta(2) equals 1729 equals 1 cubed plus 12 cubed equals 9 cubed plus 10 cubed","Cabtaxi numbers allow negative cubes and are a different sequence","Not visible; a property of integers."]},"capabilities":{"applicability":"required","items":["state the definition and known values","verify a representation as a sum of two cubes","distinguish taxicab from cabtaxi and generalised numbers","point to sequence references"]},"hazards":{"applicability":"required","items":["Quoting upper bounds as exact values","Confusing taxicab and cabtaxi numbers","Misstating the anecdote"]},"interfaces":{"applicability":"required","items":["No regulation; definitions follow the literature and the OEIS"]},"context":{"applicability":"required","items":["Not for anything; a curiosity in number theory.","taxicab numbers Ta(n)","cabtaxi numbers allowing negative cubes","generalised taxicab numbers for other powers and more terms","upper bounds for unknown values"]}},"sources":[{"title":"Wikidata item Q1462591: taxicab number","url":"https://www.wikidata.org/wiki/Q1462591","note":"identity and sense of the item"},{"title":"Wikipedia: Taxicab number","url":"https://en.wikipedia.org/wiki/Taxicab_number","note":"general description of the item"}],"openQuestions":["Which values beyond Ta(6) can be recorded as bounds?","How should OEIS entries be linked?","Should 1729 be a separate entry?"],"resources":{"spec":"/models/things/publications/thing-q1462591/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1462591/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 2026-10-06T19:51:54+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}}