{"id":"https://openalex.org/W7213606277","doi":"https://doi.org/10.48550/arxiv.2609.20208","title":"A characterization of Sophie Germain primes - II","display_name":"A characterization of Sophie Germain primes - II","publication_year":2026,"publication_date":"2026-07-25","ids":{"openalex":"https://openalex.org/W7213606277","doi":"https://doi.org/10.48550/arxiv.2609.20208"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.20208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20208","pdf_url":"https://arxiv.org/pdf/2609.20208","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","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.20208","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008510143","display_name":"Paolo Leonetti","orcid":"https://orcid.org/0000-0001-7819-5301","observed_orcids":["https://orcid.org/0000-0001-7819-5301"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Leonetti, Paolo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5008510143"],"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":0.9932000041007996,"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":0.9932000041007996,"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":0.0015999999595806003,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0013000000035390258,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/sophie-germain-primes","display_name":"Sophie Germain primes","score":0.9950000047683716},{"id":"https://openalex.org/keywords/complete-residue-systems","display_name":"complete residue systems","score":0.9580000042915344},{"id":"https://openalex.org/keywords/permutations","display_name":"permutations","score":0.6269999742507935},{"id":"https://openalex.org/keywords/even-integers","display_name":"even integers","score":0.7559999823570251},{"id":"https://openalex.org/keywords/square-free-integers","display_name":"square-free integers","score":0.7070000171661377},{"id":"https://openalex.org/keywords/modular-arithmetic","display_name":"modular arithmetic","score":0.5960000157356262}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.734499990940094},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.6322000026702881},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5619999766349792},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5090000033378601},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.5041000247001648},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.49709999561309814},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.448199987411499},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3012999892234802},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.2888999879360199},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.28760001063346863},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2678000032901764}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.20208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20208","pdf_url":"https://arxiv.org/pdf/2609.20208","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.2609.20208","doi":"https://doi.org/10.48550/arxiv.2609.20208","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.20208","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.20208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20208","pdf_url":"https://arxiv.org/pdf/2609.20208","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213606277.pdf","grobid_xml":"https://content.openalex.org/works/W7213606277.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2513707977","https://openalex.org/W2070012809","https://openalex.org/W2250105436","https://openalex.org/W2772124206","https://openalex.org/W7155574072","https://openalex.org/W2380806311","https://openalex.org/W4401178087","https://openalex.org/W1503962236","https://openalex.org/W2057027794","https://openalex.org/W2312433923"],"abstract_inverted_index":{"Let":[0],"$n\\ge":[1],"4$":[2],"be":[3],"an":[4,48],"even":[5],"integer.":[6],"We":[7],"show":[8],"that":[9],"$\\{1^{\u03c3(1)},2^{\u03c3(2)},\\ldots,n^{\u03c3(n)}\\}$":[10],"is":[11,29,33],"a":[12],"complete":[13],"residue":[14],"system":[15],"modulo":[16],"$n$":[17],"for":[18,54],"some":[19],"permutation":[20],"$\u03c3$":[21],"of":[22],"$\\{1,2,\\ldots,n\\}$":[23],"if":[24,27],"and":[25,31],"only":[26],"$p:=n/2$":[28],"prime":[30],"$p-1$":[32],"squarefree.":[34],"This":[35],"complements":[36],"the":[37],"results":[38],"in":[39],"[Int.":[40],"J.":[41],"Number":[42],"Theory":[43],"\\textbf{14}":[44],"(2018),":[45],"653--660],":[46],"where":[47],"analogue":[49],"characterization":[50],"has":[51],"been":[52],"proved":[53],"odd":[55],"integers.":[56]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-19T00:00:00"}
