关于短区间模逆的分布

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-09-27 DOI:10.1112/mtk.12224

Moubariz Z. Garaev, Igor E. Shparlinski

{"title":"关于短区间模逆的分布","authors":"Moubariz Z. Garaev, Igor E. Shparlinski","doi":"10.1112/mtk.12224","DOIUrl":null,"url":null,"abstract":"For a prime number p and integer x with <math>\n <semantics>\n <mrow>\n <mo>gcd</mo>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>p</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$\\gcd (x,p)=1$</annotation>\n </semantics></math>, let <math>\n <semantics>\n <mover>\n <mi>x</mi>\n <mo>¯</mo>\n </mover>\n <annotation>$\\overline{x}$</annotation>\n </semantics></math> denote the multiplicative inverse of x modulo p. In this paper, we are interested in the problem of distribution modulo p of the sequence\n\n For any fixed <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$A > 1$</annotation>\n </semantics></math> and for any sufficiently large integer N, there exists a prime number p with\n\n For any fixed positive <math>\n <semantics>\n <mrow>\n <mi>γ</mi>\n <mo><</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$\\gamma < 1$</annotation>\n </semantics></math>, there exists a positive constant c such that the following holds: for any sufficiently large integer N there is a prime number <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>></mo>\n <mi>N</mi>\n </mrow>\n <annotation>$p > N$</annotation>\n </semantics></math> such that\n\n ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12224","citationCount":"0","resultStr":"{\"title\":\"On the distribution of modular inverses from short intervals\",\"authors\":\"Moubariz Z. Garaev, Igor E. Shparlinski\",\"doi\":\"10.1112/mtk.12224\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a prime number p and integer x with <math>\\n <semantics>\\n <mrow>\\n <mo>gcd</mo>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>p</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\gcd (x,p)=1$</annotation>\\n </semantics></math>, let <math>\\n <semantics>\\n <mover>\\n <mi>x</mi>\\n <mo>¯</mo>\\n </mover>\\n <annotation>$\\\\overline{x}$</annotation>\\n </semantics></math> denote the multiplicative inverse of x modulo p. In this paper, we are interested in the problem of distribution modulo p of the sequence\\n\\n For any fixed <math>\\n <semantics>\\n <mrow>\\n <mi>A</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$A > 1$</annotation>\\n </semantics></math> and for any sufficiently large integer N, there exists a prime number p with\\n\\n For any fixed positive <math>\\n <semantics>\\n <mrow>\\n <mi>γ</mi>\\n <mo><</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\gamma < 1$</annotation>\\n </semantics></math>, there exists a positive constant c such that the following holds: for any sufficiently large integer N there is a prime number <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>></mo>\\n <mi>N</mi>\\n </mrow>\\n <annotation>$p > N$</annotation>\\n </semantics></math> such that\\n\\n \",\"PeriodicalId\":18463,\"journal\":{\"name\":\"Mathematika\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-09-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12224\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12224\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12224","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

对于素数p和整数x，其中gcd（x，p）=1$\gcd（x，p）=1$，设x$\overline{x}$表示x模p的乘法逆。在本文中，我们感兴趣的是对于任何固定的A>；1$A>；1$，并且对于任何足够大的整数N，存在一个素数p，对于任何固定的正γ<；1$\gamma<；1$，存在一个正常数c，使得以下成立：对于任何足够大的整数N，都有一个素数p>；N$p>；N$

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On the distribution of modular inverses from short intervals

For a prime number p and integer x with $gcd (x, p) = 1$ , let $\bar{x}$ denote the multiplicative inverse of x modulo p. In this paper, we are interested in the problem of distribution modulo p of the sequence

For any fixed $A > 1$ and for any sufficiently large integer N, there exists a prime number p with

For any fixed positive $γ < 1$ , there exists a positive constant c such that the following holds: for any sufficiently large integer N there is a prime number $p > N$ such that

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.