Research in Number Theory最新文献

{"title":"Asymptotic Fermat for signatures (r, r, p) using the modular approach","authors":"Diana Mocanu","doi":"10.1007/s40993-023-00474-6","DOIUrl":"https://doi.org/10.1007/s40993-023-00474-6","url":null,"abstract":"Abstract Let K be a totally real field, and $$rge 5$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $$(x,y,z) in mathcal {O}_K^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> of the signature ( r , r , p ) equation $$x^r+y^r=z^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $$K(zeta _r)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . When $$K=mathbb {Q}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> we get that for most of the primes $$r<150$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo><</mml:mo> <mml:mn>150</mml:mn> </mml:mrow> </mml:math> with $$r not equiv 1 mod 8$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≢</mml:mo> <mml:mn>1</mml:mn> <mml:mspace /> <mml:mo>mod</mml:mo> <mml:mspace /> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> there are no non-trivial, primitive integer solutions ( x , y , z ) with 2| z for signatures ( r , r , p ) when p is sufficiently large. Similar results hold for quadratic fields, for example when $$K=mathbb {Q}(sqrt{2})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> there are no non-trivial, primitive solutions $$(x,y,z)in mathcal {O}_K^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135193836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields","authors":"Neelam Kandhil, Rashi Lunia, Jyothsnaa Sivaraman","doi":"10.1007/s40993-023-00472-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00472-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On equivariant class formulas for Anderson modules","authors":"Tiphaine Beaumont","doi":"10.1007/s40993-023-00473-7","DOIUrl":"https://doi.org/10.1007/s40993-023-00473-7","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Class numbers of certain families of totally real biquadratic fields and a result of Mollin","authors":"Nimish Kumar Mahapatra","doi":"10.1007/s40993-023-00475-5","DOIUrl":"https://doi.org/10.1007/s40993-023-00475-5","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136136081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence conditions for p-adic continued fractions","authors":"N. Murru, G. Romeo, Giordano Santilli","doi":"10.1007/s40993-023-00470-w","DOIUrl":"https://doi.org/10.1007/s40993-023-00470-w","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81659236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic and arithmetical properties of Mahler infinite products generated by the second degree polynomials","authors":"D. Duverney, T. Kurosawa","doi":"10.1007/s40993-023-00469-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00469-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84302574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharper bounds for the error term in the prime number theorem","authors":"Andrew Fiori, H. Kadiri, Joshua Swidinsky","doi":"10.1007/s40993-023-00454-w","DOIUrl":"https://doi.org/10.1007/s40993-023-00454-w","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"47 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88661793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reciprocal polynomials and curves with many points over a finite field","authors":"Rohit Gupta, E. A. Mendoza, Luciane Quoos","doi":"10.1007/s40993-023-00464-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00464-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86354547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Automorphic products that are singular modulo primes","authors":"Haowu Wang, Brandon Williams","doi":"10.1007/s40993-023-00495-1","DOIUrl":"https://doi.org/10.1007/s40993-023-00495-1","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139354695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the metacyclic 2-groups whose abelianizations are of type (2,2n)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(2,","authors":"B. Aaboun, A. Zekhnini","doi":"10.1007/s40993-023-00461-x","DOIUrl":"https://doi.org/10.1007/s40993-023-00461-x","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74710132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}