{"id":"https://openalex.org/W7225617189","doi":"https://doi.org/10.5281/zenodo.24092050","title":"Note_Kaprekar_Bounds","display_name":"Note_Kaprekar_Bounds","publication_year":2026,"publication_date":"2026-11-09","ids":{"openalex":"https://openalex.org/W7225617189","doi":"https://doi.org/10.5281/zenodo.24092050"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24092050","doi":"https://doi.org/10.5281/zenodo.24092050","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24092050","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24092050","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5161345811","display_name":"Micha\u00ebl Malingr\u00e9au","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Micha\u00ebl Malingr\u00e9au","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5161345811"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.9330999851226807,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.9330999851226807,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13806","display_name":"Mathematical and Computational Methods","score":0.01899999938905239,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.018799999728798866,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/convergence-probability","display_name":"convergence probability","score":0.6549999713897705},{"id":"https://openalex.org/keywords/discrete-dynamical-systems","display_name":"discrete dynamical systems","score":0.8930000066757202},{"id":"https://openalex.org/keywords/combinatorial-enumeration","display_name":"combinatorial enumeration","score":0.871999979019165}],"concepts":[{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6765000224113464},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.579800009727478},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5095000267028809},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.36149999499320984},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.34689998626708984},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3427000045776367},{"id":"https://openalex.org/C2779193601","wikidata":"https://www.wikidata.org/wiki/Q20026918","display_name":"Mathematical theory","level":2,"score":0.328000009059906},{"id":"https://openalex.org/C76969082","wikidata":"https://www.wikidata.org/wiki/Q486902","display_name":"Mathematical model","level":2,"score":0.2985000014305115},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.2791999876499176},{"id":"https://openalex.org/C130327152","wikidata":"https://www.wikidata.org/wiki/Q748349","display_name":"Mathematical structure","level":2,"score":0.27880001068115234},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27140000462532043},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.26600000262260437}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24092050","doi":"https://doi.org/10.5281/zenodo.24092050","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24092050","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.5281/zenodo.24092051","doi":"https://doi.org/10.5281/zenodo.24092051","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24092051","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24092050","doi":"https://doi.org/10.5281/zenodo.24092050","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24092050","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7164055685","https://openalex.org/W3094540970","https://openalex.org/W2765947207","https://openalex.org/W7162936294","https://openalex.org/W7220295246","https://openalex.org/W1978871266","https://openalex.org/W4252791950","https://openalex.org/W7160700953","https://openalex.org/W7159665725","https://openalex.org/W7166852987"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"investigate":[2],"the":[3,7,23,56,65],"convergence":[4,31],"properties":[5,54],"of":[6,14,55,87],"four-digit":[8],"Kaprekar":[9],"process":[10],"across":[11],"infinite":[12],"families":[13],"numerical":[15,69],"bases.":[16],"Building":[17],"upon":[18],"previously":[19,47],"established":[20],"results":[21],"in":[22,76],"mathematical":[24,44],"literature,":[25],"we":[26],"derive":[27],"general":[28],"bounds":[29],"on":[30],"probabilities":[32],"using":[33],"analytical":[34],"methods":[35],"and":[36,51,71],"exact":[37],"combinatorial":[38],"counting":[39],"techniques.":[40],"Our":[41],"findings":[42,89],"provide":[43],"evidence":[45],"challenging":[46],"published":[48],"asymptotic":[49],"conjectures":[50],"reveal":[52],"structural":[53],"underlying":[57],"discrete":[58],"dynamical":[59],"system.":[60],"The":[61,85],"complete":[62],"manuscript,":[63],"including":[64],"explicit":[66],"formulas,":[67],"proofs,":[68],"verification,":[70],"detailed":[72],"results,":[73],"is":[74],"preserved":[75],"this":[77],"dated,":[78],"restricted-access":[79],"version":[80],"pending":[81],"independent":[82],"scientific":[83],"evaluation.":[84],"originality":[86],"these":[88],"relative":[90],"to":[91,95],"subsequent":[92],"research":[93],"remains":[94],"be":[96],"independently":[97],"assessed.":[98]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}
