{"id":"https://openalex.org/W7214054474","doi":"https://doi.org/10.48550/arxiv.2609.24347","title":"The lower bound of shifted primes with large prime factors","display_name":"The lower bound of shifted primes with large prime factors","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214054474","doi":"https://doi.org/10.48550/arxiv.2609.24347"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.24347","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24347","pdf_url":"https://arxiv.org/pdf/2609.24347","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.24347","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5141183994","display_name":"Zhiyuan Yang","orcid":"https://orcid.org/0009-0009-2394-0680","observed_orcids":["https://orcid.org/0009-0009-2394-0680"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Yang, Zhiyuan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5141183994"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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":1.0,"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":1.0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/shifted-primes","display_name":"shifted primes","score":0.8289999961853027},{"id":"https://openalex.org/keywords/largest-prime-factor","display_name":"largest prime factor","score":0.949999988079071},{"id":"https://openalex.org/keywords/asymptotic-density","display_name":"asymptotic density","score":0.9309999942779541},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.6589999794960022},{"id":"https://openalex.org/keywords/prime-gaps","display_name":"prime gaps","score":0.4699999988079071},{"id":"https://openalex.org/keywords/prime-k-tuples","display_name":"prime k-tuples","score":0.5070000290870667},{"id":"https://openalex.org/keywords/prime-counting-function","display_name":"prime counting function","score":0.3100000023841858}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7095999717712402},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6110000014305115},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5260999798774719},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.48249998688697815},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34389999508857727},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.3433000147342682},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.2994000017642975},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.29600000381469727},{"id":"https://openalex.org/C156491182","wikidata":"https://www.wikidata.org/wiki/Q752723","display_name":"Natural density","level":2,"score":0.2906999886035919},{"id":"https://openalex.org/C163540240","wikidata":"https://www.wikidata.org/wiki/Q936614","display_name":"Almost prime","level":4,"score":0.26170000433921814}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.24347","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24347","pdf_url":"https://arxiv.org/pdf/2609.24347","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.24347","doi":"https://doi.org/10.48550/arxiv.2609.24347","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.24347","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.24347","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24347","pdf_url":"https://arxiv.org/pdf/2609.24347","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214054474.pdf","grobid_xml":"https://content.openalex.org/works/W7214054474.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2760892069","https://openalex.org/W4414970984","https://openalex.org/W4402521891","https://openalex.org/W2885913640","https://openalex.org/W4313730411","https://openalex.org/W2513124401","https://openalex.org/W1588000126","https://openalex.org/W7213342621","https://openalex.org/W2902223536","https://openalex.org/W7137984878"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"consider":[4],"the":[5,15],"asymptotic":[6],"density":[7],"of":[8,19],"$T_c(x):=\\#\\{p\\leq":[9],"x:P^+(p-1)\\geq":[10],"p^c\\}$,":[11],"where":[12],"$P^+(n)$":[13],"denote":[14],"largest":[16],"prime":[17],"factor":[18],"$n$.":[20],"We":[21],"show":[22],"that":[23],"for":[24,26,29,44],"$x\\rightarrow\\infty$,":[25],"any":[27],"$00$":[28],"$11/320.0436$.":[30],"This":[31],"improves":[32],"a":[33],"previous":[34],"result":[35],"by":[36],"Liu-Wu-Xi":[37],"(2020),":[38],"who":[39],"showed":[40],"\\begin{align*}":[41],"\\mathop{\\lim\\inf}_{x\\rightarrow\\infty}\\frac{T_c(x)}{\u03c0(x)}&\\geq\\max\\left(1-4\u03c1\\left(\\frac{1}{c}\\right),1-c\\right),":[42],"\\end{align*}":[43],"$0":[45]},"counts_by_year":[],"updated_date":"2026-11-01T11:40:01.658920","created_date":"2026-09-23T00:00:00"}