同余式2^(n-4)≡1(mod n)的解

Some solutions of the congruence 2^(n-4)≡1(mod n)

设a,b,c,k为给定的正整数.同余式acn-k≡b(mod n)的求解问题是数论中一个基本而重要的课题,Rotkiewicz,Shen Mok-Kong,Kiss和Phong,袁平之,张明志等学者均作过许多工作.本文研究了同余式2n-4≡1(mod n)的解,获得了同余式解的一些充要条件.借助计算机,作者求出了当n≤1010时该同余式的所有解,并得到了当n>1010时的许多解,包括含有k个因子的解,其中k=3,4,…,8.最后,提出了关于同余式的一些问题与猜想.

Let a, b, c, d are given positive integers. Solving the congruence acn-k≡b（mod n） is a basic and important issue in number theory. Rotkiewicz, Shen Mok-Kong, Kiss and Phong, Yuan, Zhang and other scholars have made some works. In this paper, we study the congruence 2n-4≡1（mod n）, and find all solutions of the congruence for n≤10^10. By using the same method, we obtain some new solutions n of the congruence when n〉10^10. Finally, we set some questions and conjectures about congruence 2n-4 ≡1（mod n） and 2n≡5（mod n）.