短区间中整数及其逆的分布  

Distribution of an integer and its inverse in a short interval

作　　者：赵艳[1] 吕星星 ZHAO Yan;LYU Xingxing(School of Information Engineering,Xi'an University,Xi'an 710068,China;School of Mathematics,Northwest University,Xi'an 710127,China)

机构地区：[1]西安文理学院信息工程学院,陕西西安710068 [2]西北大学数学学院,陕西西安710127

出　　处：《浙江大学学报（理学版）》2020年第5期531-534,共4页Journal of Zhejiang University（Science Edition）

基　　金：国家自然科学基金面上项目(11771351);西安市科技计划项目(2019KJWL28);西安市科技计划项目(2020KJWL15).

摘　　要：初等数论中的同余问题,备受学者青睐。利用初等方法、三角和性质及Kloosterman和估计,研究了短区间中整数及其逆的分布问题,从两个不同的角度回答了蔡天新教授提出的猜想:设p是一个奇素数,除p=3,5,7和13外,至少存在1组整数1<i,j<p/2,满足同余式i·j≡1modp。本文不仅证明了同余方程有解,还给出了一个较强的渐近公式,说明解的个数不超过M^2/p+O(1/p^2ln^2p)。Congruence in elementary number theory is favored by many scholars.In this paper,using the elementary method,the properties of trigonometric sums and the estimation of Kloosterman sums,we study the distribution of integers and their inverse on short intervals and solve a number theory conjecture problem proposed by professor CAI Tianxin from two different angles.Assume that p is an odd prime number except for 3,5,7,and 13,there is at least one set of integers 1<i,j<p/2 which satisfy the congruence i·j≡1modp.It not only shows that the congruence equation has a solution,but also gives a strong asymptotic formula,indicating that the number of solutions is less than M^2/p+O(1/p^2ln^2p).

关 键 词：三角和 整数及其逆 渐近公式 

分 类 号：O156.4[理学—数学]