{"id":"https://openalex.org/W7225322313","doi":"https://doi.org/10.5281/zenodo.23971255","title":"Proof of Bala's totient-period conjectures: OEIS A258899 and A258903, COMMENTS (3 and 8 August 2025)","display_name":"Proof of Bala's totient-period conjectures: OEIS A258899 and A258903, COMMENTS (3 and 8 August 2025)","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225322313","doi":"https://doi.org/10.5281/zenodo.23971255"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23971255","doi":"https://doi.org/10.5281/zenodo.23971255","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23971255","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":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23971255","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160916179","display_name":"Haobo Ma","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ma, Haobo","raw_affiliation_strings":["ChronoAI Pte Ltd, Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ChronoAI Pte Ltd, Singapore","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5118983640","display_name":"Wenlin Zhang","orcid":"https://orcid.org/0009-0005-2414-7968","observed_orcids":["https://orcid.org/0009-0005-2414-7968"]},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Zhang, Wenlin","raw_affiliation_strings":["National University of Singapore; ChronoAI Pte Ltd, Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"National University of Singapore; ChronoAI Pte Ltd, Singapore","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"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/T11166","display_name":"Analytic Number Theory Research","score":0.8827999830245972,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.8827999830245972,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.04360000044107437,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.02070000022649765,"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/eventual-periodicity","display_name":"eventual periodicity","score":0.7870000004768372},{"id":"https://openalex.org/keywords/euler-s-totient-function","display_name":"Euler totient function","score":0.9620000123977661},{"id":"https://openalex.org/keywords/modular-periodicity","display_name":"modular periodicity","score":0.6240000128746033},{"id":"https://openalex.org/keywords/integer-weights","display_name":"integer weights","score":0.7559999823570251}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7569000124931335},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6886000037193298},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6344000101089478},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6291999816894531},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5997999906539917},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5723000168800354},{"id":"https://openalex.org/C198944804","wikidata":"https://www.wikidata.org/wiki/Q190026","display_name":"Euler's totient function","level":3,"score":0.5090000033378601},{"id":"https://openalex.org/C2781291010","wikidata":"https://www.wikidata.org/wiki/Q178580","display_name":"Period (music)","level":2,"score":0.46950000524520874},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.46639999747276306},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.4657000005245209},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46320000290870667},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.36469998955726624},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3116999864578247},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.29919999837875366},{"id":"https://openalex.org/C6026789","wikidata":"https://www.wikidata.org/wiki/Q948010","display_name":"Primitive root modulo n","level":3,"score":0.28610000014305115},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.27570000290870667},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.26910001039505005},{"id":"https://openalex.org/C61994974","wikidata":"https://www.wikidata.org/wiki/Q2297602","display_name":"Integer sequence","level":3,"score":0.2632000148296356}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23971255","doi":"https://doi.org/10.5281/zenodo.23971255","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23971255","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":"Preprint"},{"id":"doi:10.5281/zenodo.23971256","doi":"https://doi.org/10.5281/zenodo.23971256","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23971256","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":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23971255","doi":"https://doi.org/10.5281/zenodo.23971255","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23971255","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":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7212032386","https://openalex.org/W7204270587","https://openalex.org/W7203670576","https://openalex.org/W7170040297","https://openalex.org/W2779095362","https://openalex.org/W6930308387","https://openalex.org/W7140325302","https://openalex.org/W7203570837","https://openalex.org/W6939765541","https://openalex.org/W7221434867"],"abstract_inverted_index":{"The":[0,39],"conjectures":[1],"in":[2,10,17,55],"Peter":[3],"Bala's":[4],"COMMENTS":[5],"of":[6],"3":[7],"August":[8,15,59],"2025":[9,16],"OEIS":[11],"A258899":[12],"and":[13,61],"8":[14,58],"A258903":[18],"are":[19],"proved:":[20],"modulo":[21],"every":[22],"positive":[23],"integer":[24,72],"m,":[25],"each":[26],"sequence":[27],"is":[28,43],"eventually":[29],"periodic":[30],"with":[31],"least":[32],"eventual":[33,68],"period":[34,69],"dividing":[35],"Euler's":[36],"totient":[37],"phi(m).":[38],"explicit":[40],"sufficient":[41],"onset":[42],"2*m+4.":[44],"A":[45],"weighted":[46],"divisor-factorial":[47],"tail":[48],"identity":[49],"also":[50],"proves":[51],"the":[52,63],"constant-tail":[53],"conjecture":[54],"A121860":[56],"(COMMENTS,":[57],"2025),":[60],"gives":[62],"Carmichael":[64],"exponent":[65],"as":[66],"an":[67],"for":[70],"all":[71],"weights.":[73]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-11-06T00:00:00"}
