Zoj -- 1489--2 ^ x mod n = 1

This question requires the smallest positive integer x, N> 0 that satisfies 2 ^ x limit 1 (mod N.

First consider the Euler's Theorem 2 ^ Eular (n) limit 1 (mod N), which requires n> 1. So when n = 1, in fact, all K numbers have k limit 0 (mod N), which is a special decision.

In the Euler's theorem, Eular (n) must be cyclic, but not necessarily the smallest cyclic section: When eg: n = 7:

2 ^ 0 mod 7 = 1;

2 ^ 1 mod 7 = 2;

2 ^ 2 mod 7 = 4;

2 ^ 3 mod 7 = 1;

2 ^ 4 mod 7 = 2;

The shortest cycle is 3, which is a factor of the Euler's function Eular (7) = 6.

There are two methods for finding the shortest cycle: 1. brute-force enumeration, with a maximum of Eular (n). The complexity is O (n. 2. enumerate Eular (n) factors from small to large, and then quickly determine the power.