{"title":"素数k元组的渐近密度及Hardy和Littlewood的一个猜想","authors":"L. T'oth","doi":"10.12921/cmst.2019.0000033","DOIUrl":null,"url":null,"abstract":"In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the \"Skewes number\" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find \"Skewes numbers\" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.","PeriodicalId":10561,"journal":{"name":"computational methods in science and technology","volume":"51 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood\",\"authors\":\"L. T'oth\",\"doi\":\"10.12921/cmst.2019.0000033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the \\\"Skewes number\\\" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find \\\"Skewes numbers\\\" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.\",\"PeriodicalId\":10561,\"journal\":{\"name\":\"computational methods in science and technology\",\"volume\":\"51 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-10-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"computational methods in science and technology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.12921/cmst.2019.0000033\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"computational methods in science and technology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.12921/cmst.2019.0000033","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood
In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find "Skewes numbers" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.
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