{"id":"https://openalex.org/W7216550722","doi":"https://doi.org/10.48550/arxiv.2610.00765","title":"Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields","display_name":"Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7216550722","doi":"https://doi.org/10.48550/arxiv.2610.00765"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.00765","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00765","pdf_url":"https://arxiv.org/pdf/2610.00765","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/2610.00765","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5114485837","display_name":"Mateo Matijasevick","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Matijasevick, Mateo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5154987241","display_name":"Santiago Rodr\u00edguez","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rodr\u00edguez, Santiago","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5028092559","display_name":"Gregorio Salazar","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Salazar, Gregorio","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"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":0.5160999894142151,"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":0.5160999894142151,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.4487000107765198,"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"}},{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.009800000116229057,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/perfect-powers","display_name":"perfect powers","score":0.8690000176429749},{"id":"https://openalex.org/keywords/perfect-squares","display_name":"perfect squares","score":0.6930000185966492},{"id":"https://openalex.org/keywords/inert-primes","display_name":"inert primes","score":0.6359999775886536},{"id":"https://openalex.org/keywords/quadratic-fields","display_name":"quadratic fields","score":0.9869999885559082},{"id":"https://openalex.org/keywords/gaussian-integers","display_name":"Gaussian integers","score":0.7749999761581421},{"id":"https://openalex.org/keywords/diophantine-equations","display_name":"Diophantine equations","score":0.5590000152587891},{"id":"https://openalex.org/keywords/quadratic-forms","display_name":"quadratic forms","score":0.4020000100135803},{"id":"https://openalex.org/keywords/real-quadratic-fields","display_name":"real quadratic fields","score":0.6809999942779541}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8317000269889832},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.7382000088691711},{"id":"https://openalex.org/C12817185","wikidata":"https://www.wikidata.org/wiki/Q878259","display_name":"Quadratic residue","level":2,"score":0.6561999917030334},{"id":"https://openalex.org/C120561534","wikidata":"https://www.wikidata.org/wiki/Q724975","display_name":"Gaussian integer","level":2,"score":0.6233999729156494},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5819000005722046},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5389000177383423},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.532800018787384},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.5321999788284302},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5293999910354614},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5024999976158142},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.49720001220703125},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4932999908924103},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.4810999929904938},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.4790000021457672},{"id":"https://openalex.org/C173808221","wikidata":"https://www.wikidata.org/wiki/Q50705","display_name":"Square number","level":2,"score":0.46880000829696655},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.45719999074935913},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.42899999022483826},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.4244999885559082},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4099999964237213},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.4047999978065491},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.387800008058548},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.3513999879360199},{"id":"https://openalex.org/C95136341","wikidata":"https://www.wikidata.org/wiki/Q625519","display_name":"Quadratic field","level":4,"score":0.34599998593330383},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.3296999931335449},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.32710000872612},{"id":"https://openalex.org/C171223647","wikidata":"https://www.wikidata.org/wiki/Q207264","display_name":"Fermat number","level":3,"score":0.32179999351501465},{"id":"https://openalex.org/C188139237","wikidata":"https://www.wikidata.org/wiki/Q50706","display_name":"Square-free integer","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.2842999994754791},{"id":"https://openalex.org/C80695182","wikidata":"https://www.wikidata.org/wiki/Q864134","display_name":"Binary quadratic form","level":4,"score":0.2842000126838684},{"id":"https://openalex.org/C21440807","wikidata":"https://www.wikidata.org/wiki/Q4968811","display_name":"Isotropic quadratic form","level":5,"score":0.2768000066280365},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C2778975207","wikidata":"https://www.wikidata.org/wiki/Q7078836","display_name":"Of the form","level":2,"score":0.2720000147819519},{"id":"https://openalex.org/C138888516","wikidata":"https://www.wikidata.org/wiki/Q2266843","display_name":"Infinite product","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.2628999948501587},{"id":"https://openalex.org/C165656649","wikidata":"https://www.wikidata.org/wiki/Q748339","display_name":"Legendre symbol","level":5,"score":0.26010000705718994},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.2597000002861023},{"id":"https://openalex.org/C102607178","wikidata":"https://www.wikidata.org/wiki/Q102761","display_name":"Least common multiple","level":2,"score":0.2590999901294708}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.00765","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00765","pdf_url":"https://arxiv.org/pdf/2610.00765","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.2610.00765","doi":"https://doi.org/10.48550/arxiv.2610.00765","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.00765","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:2610.00765","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00765","pdf_url":"https://arxiv.org/pdf/2610.00765","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/W7216550722.pdf","grobid_xml":"https://content.openalex.org/works/W7216550722.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2291745595","https://openalex.org/W1999545419","https://openalex.org/W4312433147","https://openalex.org/W2008064996","https://openalex.org/W2391034784","https://openalex.org/W1537464248","https://openalex.org/W4385606337","https://openalex.org/W2354386379","https://openalex.org/W2963840906","https://openalex.org/W1988021340"],"abstract_inverted_index":{"Let":[0],"$k$":[1,119,145],"be":[2],"a":[3,27,35,81,126],"positive":[4,11],"integer,":[5],"and":[6,15,42,46,50,66,112,129,172],"consider":[7],"the":[8,67,75,120,138,141,153,156,159,165],"sequences":[9],"of":[10,54,105,144,158],"rationals":[12],"with":[13],"$x_0\\in\\N$":[14],"$x_{n+1}=k/(x_0+x_1+\\dots+x_n)$.":[16],"Write":[17],"$x_n=a_n/b_n$":[18],"in":[19,62,150],"lowest":[20],"terms.":[21],"We":[22],"show":[23],"that":[24,32,63,77,97],"there":[25],"is":[26,34,56,124,131,162],"rational":[28,127],"constant":[29],"$c>0$":[30],"such":[31,40],"$c\\,a_n+b_n$":[33],"perfect":[36],"power":[37],"for":[38,94],"every":[39,43,51,173],"sequence":[41,101],"$n\\ge2$":[44],"if":[45,48],"only":[47],"$k=h^2$":[49],"prime":[52,103],"factor":[53,104],"$h$":[55,106],"congruent":[57],"to":[58,114],"$3$":[59],"modulo":[60],"$4$;":[61],"case":[64],"$c=2/h$":[65],"powers":[68],"are":[69],"squares.":[70],"The":[71],"proof":[72],"rests":[73],"on":[74],"observation":[76],"$t_n=(x_0+\\dots+x_n)/h$":[78],"satisfies":[79],"$t_{n+1}=t_n+1/t_n$,":[80],"recursion":[82],"under":[83],"which":[84,169],"reduced":[85],"fractions":[86],"never":[87],"cancel.":[88],"This":[89],"yields":[90],"an":[91,133,147],"exact":[92],"formula":[93],"$c\\,a_n+b_n$,":[95],"shows":[96],"along":[98],"any":[99],"single":[100],"each":[102],"spoils":[107],"at":[108],"most":[109],"one":[110],"term,":[111],"leads":[113],"two":[115],"generalizations.":[116],"For":[117,152],"arbitrary":[118],"invariant":[121],"$b_n^2-\\frac4k":[122],"a_n^2$":[123],"always":[125,132],"square,":[128],"it":[130],"integer":[134],"square":[135,142],"exactly":[136],"when":[137],"primes":[139],"dividing":[140],"part":[143],"satisfy":[146],"inertness":[148],"condition":[149],"$\\Q(\\sqrt{-k})$.":[151],"recursions":[154],"$x_{n+1}=h^2/(x_0+\\dots+x_n+n\u03bch)$":[155],"role":[157],"Gaussian":[160],"integers":[161],"played":[163],"by":[164],"quadratic":[166,175],"fields":[167],"$\\Q(\\sqrt{\u03bc^2-4})$,":[168],"include":[170],"$\\Q(\\sqrt{-3})$":[171],"real":[174],"field.":[176]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-03T00:00:00"}