{"title":"从几乎质数的分数次幂得到漂亮的质数","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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-05-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S001935772300040X\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S001935772300040X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Beatty primes from fractional powers of almost-primes
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.
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