{"title":"斐波那契素数,形式为 2 - k 及以上的素数","authors":"Jon Grantham , Andrew Granville","doi":"10.1016/j.jnt.2024.02.002","DOIUrl":null,"url":null,"abstract":"<div><p>We speculate on the distribution of primes in exponentially growing, linear recurrence sequences <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, or else there exists a constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> (which we can give good approximations to) such that there are <span><math><mo>∼</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>u</mi></mrow></msub><mi>log</mi><mo></mo><mi>N</mi></math></span> primes <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>n</mi><mo>≤</mo><mi>N</mi></math></span>, as <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span>. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"261 ","pages":"Pages 190-219"},"PeriodicalIF":0.6000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fibonacci primes, primes of the form 2n − k and beyond\",\"authors\":\"Jon Grantham , Andrew Granville\",\"doi\":\"10.1016/j.jnt.2024.02.002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We speculate on the distribution of primes in exponentially growing, linear recurrence sequences <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, or else there exists a constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> (which we can give good approximations to) such that there are <span><math><mo>∼</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>u</mi></mrow></msub><mi>log</mi><mo></mo><mi>N</mi></math></span> primes <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>n</mi><mo>≤</mo><mi>N</mi></math></span>, as <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span>. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo></math></span>.</p></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"261 \",\"pages\":\"Pages 190-219\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-03-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24000544\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000544","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Fibonacci primes, primes of the form 2n − k and beyond
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes , or else there exists a constant (which we can give good approximations to) such that there are primes with , as . We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the .
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.