{"title":"n^2+1的最大质因数","authors":"Jori Merikoski","doi":"10.4171/jems/1216","DOIUrl":null,"url":null,"abstract":"We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret\\`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.5000,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the largest prime factor of $n^2+1$\",\"authors\":\"Jori Merikoski\",\"doi\":\"10.4171/jems/1216\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret\\\\`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$\",\"PeriodicalId\":50003,\"journal\":{\"name\":\"Journal of the European Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2022-02-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the European Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/jems/1216\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the European Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jems/1216","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了n^2+1$的最大素数因子通常无限大于n^{1.279}$。这改善了de la Bret\ ' eche和Drappeau(2019)的结果，他们用1.2182美元代替1.279美元获得了这个结果。证明中的主要新成分是一种新的II型估计，并通过哈曼筛子法使用这种估计。为了证明II型估计，我们使用了Kloosterman和的线性形式上的Dehouillers和Iwaniec的界。我们还证明了在Selberg特征值猜想的条件下，指数$1.279$可以增加到$1.312.$
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret\`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
期刊介绍：
The Journal of the European Mathematical Society (JEMS) is the official journal of the EMS.
The Society, founded in 1990, works at promoting joint scientific efforts between the many different structures that characterize European mathematics. JEMS will publish research articles in all active areas of pure and applied mathematics. These will be selected by a distinguished, international board of editors for their outstanding quality and interest, according to the highest international standards.
Occasionally, substantial survey papers on topics of exceptional interest will also be published. Starting in 1999, the Journal was published by Springer-Verlag until the end of 2003. Since 2004 it is published by the EMS Publishing House. The first Editor-in-Chief of the Journal was J. Jost, succeeded by H. Brezis in 2004.
The Journal of the European Mathematical Society is covered in:
Mathematical Reviews (MR), Current Mathematical Publications (CMP), MathSciNet, Zentralblatt für Mathematik, Zentralblatt MATH Database, Science Citation Index (SCI), Science Citation Index Expanded (SCIE), CompuMath Citation Index (CMCI), Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), ISI Alerting Services, Journal Citation Reports/Science Edition, Web of Science.
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