{"id":"https://openalex.org/W7220722940","doi":"https://doi.org/10.48550/arxiv.2610.05661","title":"Amicable numbers","display_name":"Amicable numbers","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220722940","doi":"https://doi.org/10.48550/arxiv.2610.05661"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.05661","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05661","pdf_url":"https://arxiv.org/pdf/2610.05661","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/2610.05661","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5093939278","display_name":"Paul Pollack","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Pollack, Paul","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5093939278"],"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":0.9994999766349792,"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":0.9994999766349792,"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":0.00019999999494757503,"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/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"}}],"keywords":[{"id":"https://openalex.org/keywords/amicable-numbers","display_name":"amicable numbers","score":0.9430000185966492},{"id":"https://openalex.org/keywords/amicable-pairs","display_name":"amicable pairs","score":0.9169999957084656},{"id":"https://openalex.org/keywords/astra","display_name":"ASTRA","score":0.31700000166893005},{"id":"https://openalex.org/keywords/chatgpt","display_name":"ChatGPT","score":0.8119999766349792},{"id":"https://openalex.org/keywords/upper-bounds","display_name":"upper bounds","score":0.5120000243186951}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6312999725341797},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.5526000261306763},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5325000286102295},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39309999346733093},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3693000078201294},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.32690000534057617},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.28349998593330383},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.26919999718666077},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.24709999561309814}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.05661","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05661","pdf_url":"https://arxiv.org/pdf/2610.05661","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.2610.05661","doi":"https://doi.org/10.48550/arxiv.2610.05661","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.05661","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":"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":"pmh:oai:arXiv.org:2610.05661","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05661","pdf_url":"https://arxiv.org/pdf/2610.05661","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/W7220722940.pdf","grobid_xml":"https://content.openalex.org/works/W7220722940.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2181317310","https://openalex.org/W2327708564","https://openalex.org/W2744560236","https://openalex.org/W7099851730","https://openalex.org/W2173329866","https://openalex.org/W2097648229","https://openalex.org/W4236760430","https://openalex.org/W2332526475","https://openalex.org/W6990060065","https://openalex.org/W7096086337"],"abstract_inverted_index":{"Let":[0,27],"$s(n):=\\sum_{d\\mid":[1],"n,\\,d":[2],"<":[3],"n}":[4],"d$":[5],"denote":[6],"the":[7,10,30,63,67],"sum":[8],"of":[9,13,66],"proper":[11],"divisors":[12],"$n$.":[14],"Distinct":[15],"positive":[16,31],"integers":[17,32],"$n,m$":[18],"form":[19],"an":[20,39],"amicable":[21,40],"pair":[22],"if":[23],"$s(n)=m$":[24],"and":[25],"$s(m)=n$.":[26],"$A(x)$":[28],"count":[29],"not":[33],"exceeding":[34],"$x$":[35],"that":[36,51],"belong":[37],"to":[38],"pair.":[41],"We":[42],"report":[43],"on":[44],"a":[45],"proof,":[46],"found":[47],"by":[48],"ChatGPT":[49],"Astra,":[50],"$$":[52],"A(x)\\le":[53],"x\\exp\\{-(1/2+o(1))\\log":[54],"x\\,\\log_3":[55],"x/\\log_2":[56],"x\\}$$":[57],"as":[58],"$x\\to\\infty$.":[59],"Here":[60],"$\\log_k$":[61],"denotes":[62],"$k$th":[64],"iterate":[65],"natural":[68],"logarithm.":[69]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-07T00:00:00"}