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.
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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