{"title":"Beatty primes from fractional powers of almost-primes","authors":"Victor Zhenyu Guo , Jinjiang Li , Min Zhang","doi":"10.1016/j.indag.2023.04.004","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mrow><mi>α</mi><mo>></mo><mn>1</mn></mrow></math></span> be irrational and of finite type, <span><math><mrow><mi>β</mi><mo>∈</mo><mi>R</mi></mrow></math></span>. In this paper, it is proved that for <span><math><mrow><mi>R</mi><mo>⩾</mo><mn>13</mn></mrow></math></span> and any fixed <span><math><mrow><mi>c</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, there exist infinitely many primes in the intersection of Beatty sequence <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span> and <span><math><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>⌋</mo></mrow></math></span>, where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> is an explicit constant depending on <span><math><mi>R</mi></math></span> herein, <span><math><mi>n</mi></math></span> is a natural number with at most <span><math><mi>R</mi></math></span> prime factors, counted with multiplicity.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S001935772300040X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be irrational and of finite type, . In this paper, it is proved that for and any fixed , there exist infinitely many primes in the intersection of Beatty sequence and , where is an explicit constant depending on herein, is a natural number with at most prime factors, counted with multiplicity.
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.