{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.automorphic-number","code":"thing-q2313686","url":"https://ver.cy/models/thing/q2313686/","name":"automorphic 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 whose square ends in the same digits as the number itself in a given base, such as 5, 6, 25, 76, 376 and 625 in base ten, arising from the solutions of n^2 = n modulo powers of the base and related to the 10-adic idempotents; automorphic numbers form two infinite sequences in base ten extending the digits ...90625 and ...09376, and generalise to trimorphic numbers whose cubes end in the same digits.","purpose":"Let an agent explain automorphic numbers, relay their definition, examples and construction from mathematical references, describe the connection to p-adic numbers, and distinguish them from related recreational number classes.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Self-reproducing square","Base dependence","Two infinite sequences","p-adic idempotents"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What an automorphic number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is an automorphic number, and what are the first examples in base ten?","kind":"definition"},{"text":"Is the question about automorphic numbers or a related digit-based class?","kind":"boundary"}]}]},{"id":"construction","name":"Construction","description":"Construction.","findings":[{"id":"construction-finding","name":"Construction","description":"Construction.","questions":[{"text":"How are longer automorphic numbers built digit by digit, and why are there exactly two sequences?","kind":"action"},{"text":"Which entry fits the specific construction?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Theory.","layers":[{"id":"padic","name":"p-adic","description":"p-adic view.","findings":[{"id":"padic-finding","name":"p-adic","description":"p-adic.","questions":[{"text":"How do automorphic numbers correspond to idempotents in the 10-adic integers?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"bases","name":"Bases","description":"Other bases.","findings":[{"id":"bases-finding","name":"Bases","description":"Bases.","questions":[{"text":"How do automorphic numbers behave in bases that are prime powers or products of primes?","kind":"provenance"},{"text":"Which entry fits number base?","kind":"action"}]}]}]},{"id":"explore","name":"Explore","description":"Related classes.","layers":[{"id":"trimorphic","name":"Trimorphic","description":"Trimorphic numbers.","findings":[{"id":"trimorphic-finding","name":"Trimorphic","description":"Trimorphic.","questions":[{"text":"What are trimorphic numbers, and how do they relate?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"recreational","name":"Recreational","description":"Recreational mathematics.","findings":[{"id":"recreational-finding","name":"Recreational","description":"Recreational.","questions":[{"text":"How do automorphic numbers fit into recreational number theory?","kind":"provenance"},{"text":"Which entry fits recreational mathematics?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and computation.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"When were automorphic numbers studied, and by whom?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"computation","name":"Computation","description":"Computation.","findings":[{"id":"computation-finding","name":"Computation","description":"Computation.","questions":[{"text":"How can automorphic numbers be computed programmatically?","kind":"action"},{"text":"Which entry fits computational number theory?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain what automorphic numbers are and list the known small ones in a stated base.","Check or generate automorphic numbers in a given base and show the modular condition used."],"mustNot":["State that a number is automorphic without saying in which base.","Present a computational search result as a proof of a general pattern.","Claim significance for these numbers in numerology, prediction or finance."],"requiresHuman":["Publishing a claimed new result about these numbers under a person's name."]},"ethics":{"considerations":["Number curiosities are often dressed up as mystical or predictive, which misleads people.","Credit for computational results should go to those who produced them."],"affectedParties":["Students and learners","Recreational mathematicians"]},"owners":{"steward":"Nobody owns the concept; integer sequence databases record the known values.","roles":[],"masterSystems":["Published integer sequence databases"]},"relations":[{"target":"Q28920052","type":"parent","note":"registry parent class"},{"target":"non-negative integer","type":"related","note":"in registry terms"},{"target":"p-adic number","type":"related","note":"through 10-adic idempotents"},{"target":"trimorphic number","type":"related","note":"using cubes"},{"target":"OEIS","type":"related","note":"as A003226"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.automorphic-number","Wikidata: Q2313686 (https://www.wikidata.org/wiki/Q2313686)","OEIS: A003226 automorphic numbers"]},"properties":{"applicability":"not-applicable","items":["first base ten examples: 0, 1, 5, 6, 25, 76, 376, 625 sequence - OEIS A003226","base ten sequences: 2 count - for n > 1 digits"]},"recognition":{"applicability":"optional","items":["Numbers whose square ends in the number","Base ten sequences ending 90625 and 09376","Trimorphic numbers use cubes; narcissistic numbers use digit powers","Not a visible object; integers with a self-reproducing square."]},"capabilities":{"applicability":"required","items":["explain the definition","relay examples and construction","describe the p-adic connection","distinguish related classes"]},"hazards":{"applicability":"required","items":["Assuming base independence","Confusing with other digit-based classes","Miscounting sequences"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","automorphic numbers in base ten","automorphic numbers in other bases","trimorphic numbers","circular and other digit-based number classes"]}},"sources":[{"title":"Wikidata item Q2313686: automorphic number","url":"https://www.wikidata.org/wiki/Q2313686","note":"identity and sense of the item"},{"title":"Wikipedia: Automorphic number","url":"https://en.wikipedia.org/wiki/Automorphic_number","note":"general description of the item"}],"openQuestions":["Should trimorphic number be a separate entry?","How should OEIS and references be linked?","How should the p-adic interpretation be recorded?"],"resources":{"spec":"/models/things/publications/thing-q2313686/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q2313686/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T02:44:35Z","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 2026-10-06T19:51:57+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}}