On the largest prime factor of n2 C 1

被引：1

作者：

Merikoski, Jori ^{[1
]}

机构：

[1] Univ Oxford, Math Inst, Radcliffe Observ Quarter, Andrew Wiles Bldg, Woodstock Rd, Oxford OX2 6GG, England

来源：

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY | 2023年 / 25卷 / 04期

关键词：

Prime numbers; sieve methods; Kloosterman sums;

D O I：

10.4171/JEMS/1216

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We show that the largest prime factor of n2 + 1 is infinitely often greater than n1.279. This improves the result of de la Breteche and Drappeau specialIntscript 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 Deshouillers 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.

引用

页码：1253 / 1284

页数：32

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