Poj1284_primitive roots [Euler's function]

Primitive rootstime limit: 1000 ms memory limit: 10000 ktotal submissions: 3032 accepted: 1734 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


23
31
79
Sample output


10
8
24
Source

Jia Yi @ PKU

P is an odd prime number. If {x ^ I % p | 1 <= I <= p-1} = {1, 2 ,..., P is the original root of P.
Give a P and ask how many original roots there are.

Ideas:

{X ^ I % p | 1 <= I <= p-1} = {,..., P-1} is equivalent
{X ^ I % (p-1) | 1 <= I <= p-1} = {, 2,..., P-2 },

{X ^ 1, x ^ 2, x ^ 3 ,..., X ^ (p-1)} is equivalent to the full remainder of P.

If the interconnectivity between x and P-1 (gcd (x, P-1) = 1), then {x ^ 0, x ^ 1, x ^ 2 ,..., X ^ (P-2)} is the full residual system of (p-1 ).

Proof below:

If x ^ I! = X ^ J (mod P-1), then x * x ^ I! = X * x ^ J (mod P-1, then gcd (x, PM )! = 0, contrary to the above.

Then x ^ I = x ^ J (mod P-1 ).

According to the ferma theorem and Euler's theorem, we know that I = PHI (p-1 ).

For more information about Euler's functions, ferma's theorem, and Euler's theorem, see the question of Euler's function:

Http://blog.csdn.net/lianai911/article/details/40114675

Reference blog: http://www.cnblogs.com/lnever/p/3969117.html

By the way, worship the reasoning gods.

# Include <stdio. h> # include <math. h> int Euler (int n) {int I, ret = N; for (I = 2; I <= SQRT (1.0 * n); I ++) {If (N % I = 0) {n/= I; ret = ret-RET/I;} while (N % I = 0) N/= I ;} if (n> 1) {ret = ret-RET/N;} return ret;} int main () {int P; while (~ Scanf ("% d", & P) {printf ("% d \ n", Euler (p-1); // If P is a prime number, euler (p) = Euler (p-1)} return 0 ;}

Poj1284_primitive roots [Euler's function]