【poj1284】Primitive Roots

2014年12月29日1,6520

Description

We say that integer x, 0 < x < p, is a primitive root modulo odd prime p if and only if the set { (xi mod p) | 1 <= i <= p-1 } is equal to { 1, …, p-1 }. For example, the consecutive powers of 3 modulo 7 are 3, 2, 6, 4, 5, 1, and thus 3 is a primitive root modulo 7.
Write a program which given any odd prime 3 <= p < 65536 outputs the number of primitive roots modulo p.

Input

Each line of the input contains an odd prime numbers p. Input is terminated by the end-of-file seperator.

Output

For each p, print a single number that gives the number of primitive roots in a single line.

Sample Input

Sample Output

Source

题意是求奇素数的原根个数

所有的奇素数有原根

若o有原根,则其有φ(φ(n))个模p不同余的原根

若p是素数,则其有φ(p-1)个模p不同余的原根(由2式显然可得)

证明2式好麻烦,先略

结论:p是素数,b = a0^t(mod p),是一个原根的充要条件是t与p-1互质,所有的这些t的总个数就是φ(p-1)