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浙江大学学报（理学版）
浙江大学学报（理学版）  2020, Vol. 47 Issue (5): 531-534    DOI: 10.3785/j.issn.1008-9497.2020.05.002
数学与计算机科学     
短区间中整数及其逆的分布
赵艳1, 吕星星2
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
Distribution of an integer and its inverse in a short interval
ZHAO Yan1, LYU Xingxing2
1.School of Information Engineering, Xi'an University, Xi'an 710068, China
2.School of Mathematics, Northwest University, Xi'an 710127, China);.
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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<p2，满足同余式i?j≡1?mod?p。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M2p+Op12ln2p。
关键词： 整数及其逆;  渐近公式;  三角和    
Abstract: 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<p2 which satisfy the congruence i?j≡1?mod?p. 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 M2p+Op12ln2p.
Key words: asymptotic formula    trigonometric sums    integer and its inverse
收稿日期: 2019-11-02 出版日期: 2020-09-25
CLC:  O156.4  
基金资助: 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).
通讯作者: 吕星星（1994—），ORCID：http：//orcid.org/0000-0001-7785-5638，E-mail：lvxingxing@stumail.nwu.edu.cn.     E-mail: lvxingxing@stumail.nwu.edu.cn
作者简介: 赵艳（1978—），ORCID：http：//orcid.org/0000-0003-0466-1765,女，硕士，讲师，主要从事数论、P-adic数域研究，E-mail：1838156757@qq.com.。
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引用本文:

赵艳, 吕星星. 短区间中整数及其逆的分布[J]. 浙江大学学报（理学版）, 2020, 47(5): 531-534.

ZHAO Yan, LYU Xingxing. Distribution of an integer and its inverse in a short interval. Journal of Zhejiang University (Science Edition), 2020, 47(5): 531-534.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2020.05.002    或    https://www.zjujournals.com/sci/CN/Y2020/V47/I5/531

1 APOSTOL T M. Introduction to Analytic Number Theory[M]. New York: Springer-Verlag, 1976.
2 VAN HAMME L. Some conjectures concerning partial sums of generalized hyper geometric series in p-adic functional analysis[J]. Lecture Notes in Pure and Applied Mathematics, 1997, 192: 223-236.
3 PAN H, SUN Z W. A combinatorial identity with application to Catalan numbers[J]. Discrete Math ematics, 2006, 306(16): 1921-1940. DOI: 10.1016/j.disc.2006.03.050
4 ZHAO L L, PAN H, SUN Z W. Congruences for the second-order Catalan numbers[J]. Proceedings of the American Mathematical Society, 2010, 138(1): 37-46. DOI: 10.1090/S0002-9939-09-10067-9
5 DUAN, R, SHEN S M. Bernoulli polynomials and their some new congruence properties[J]. Symmetry, 2019, 11(3): 365. DOI: 10.3390/sym11030365
6 HOU Y W, SHEN S M. The Euler numbers and recursive properties of Dirichlet L-functions[J]. Advances in Difference Equations, 2018: 397. DOI: 10.1186/s13662-018-1853-y
7 ZHANG W P. Some identities involving the Euler and the central factorial numbers[J]. The Fibonacci Quarterly, 1998, 36(2): 154-157.
8 ZHAO J H, CHEN Z Y. Some symmetric identities involving Fubini polynomials and Euler numbers[J]. Symmetry, 2018, 10(8): 303. DOI: 10.3390/sym10080303
9 ZHANG W P, XU Z F. On a conjecture of the Euler numbers[J]. Journal of Number Theory, 2007, 127 (2): 283-291. DOI: 10.1016/j.jnt.2007.04.004
10 CHO B, PARK H. Evaluating binomial convolution sums of divisor functions in terms of Euler and Bernoulli polynomials[J]. International Journal of Number Theory, 2018, 14(2): 509-525. DOI: 10.1142/S1793042118500318
11 WANG L Q, CAI, T X. A curious congruence modulo prime powers[J]. Journal of Number Theory, 2014, 144(14): 15-24. DOI: 10.1016/j.jnt.2014.04.004
12 SUN Z H. Congruences concerning Bernoulli numbers and Bernoulli polynomials[J]. Discrete Applied Mathematics, 2000, 105(1): 193-223. DOI: 10.1016/s0166-218x(00)00184-0
13 ZHOU X, CAI T X. A generalization of a curious congruence on harmonic sums[J]. Proceedings of the American Mathematical Society, 2007, 135(5): 1329-1334. DOI: 10.1090/s0002-9939-06-08777-6
14 CHOWLA S. On Kloosterman's sum[J]. Norkse Vid Selbsk Fak Frondheim, 1967, 40(4): 70-72.
15 IWANIEC H. Topics in Classical Automorphic Forms[M]. Providence: American Mathematical Society Publications, 1997: 61-63. DOI: http://dx.doi.org/10.1090/gsm/017
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