"Poj1284-primitive Roots" Euler function-the number of primitive roots of odd primes

http://poj.org/problem?id=1284

Test instructions: Given an odd prime p, the number of the original root of P is obtained.

Original root: {(xi mod p) | 1 <= i <= p-1} is equal to {1, ..., p-1}, then x is the original root of P.

Conclusion: The number of the original root of the singular prime P is phi (P-1).

Prove:

for the given prime number p, the first thing to be clear: the Motone of P must be present (this has been proved by Euler, this is not mentioned here), therefore, it may be possible to set one of the meta-Root is A0 (1<=a0<=p-1) according to the definition of the topic, A0^i (1<=i <=p-1) mod p values are different, and then by P is the prime number, the contact Fermat small theorem: q^ (p-1) mod p=1; (1<=q<=p-1) (this is useful below) proves that if B is a different from a motone of p, may make B and A0 ^t about P congruence, then there is bound to be gcd (t,p-1) = 1, that is T and P-1 coprime, and vice versa; Proof: if D=GCD (t,p-1) >1, T=k1*d,p-1=k2*d, then Fermat (a0^ (k1*d)) ^K2 MoD p= ( a0^ (k2*d) ^ (K1) mod p= (a0^ (p-1)) ^ (K1) mod p=1 again by b=a0^t (mod p), combined with the above formula: (a0^ (k1*d)) ^k2 mod n=b^k2 mod p=1; b^0 mod p=1, so B^0=B^K2 (mod p), so b^i mod P's cycle section =k2<p-1, so that B is not meta-root; re-certification, if D=GCD (t,p-1) = 1, that is T and P-1 coprime, then B is necessarily motone; P-1, making b^j=b^i (mod p), i.e. a0^ (j*t) =a0^ (i*t) (mod p), is a meta-root by A0, that is, the length of the loop section of A0 is (p-1), (p-1) | (I*t-j*t), (p-1) | t* (I-J), due to P with T coprime, so (p-1) | (I-J), but according to the hypothesis, 0<i-j<p-1, draw the contradiction, the conclusion is proved; by the above two proofs b=a0^t (mod p), is a meta-root of the necessary and sufficient conditions is T and P-1 coprime, all of these T's total number is Phi (p-1);

Reference from http://poj.org/showmessage?message_id=158630

The proof was very witty, and I didn't even think of splitting p-1 and T. Remember this idea of splitting. Note: Fermat is the Fermat theorem.

Because the data range of the topic is relatively small, so I did not use Euler sieve directly decomposition factorization.

#include <cstdio>#include<cstdlib>#include<cstring>#include<iostream>using namespaceStd;typedefLong LongLL; ll Eular (ll x) {ll ans=x,k=2;  while(x!=1)    {        if(x%k==0) {ans/=K; Ans*= (K-1); }         while(x%k==0) x/=K; K++; }    returnans;}intMain () {//freopen ("a.in", "R", stdin); //freopen ("A.out", "w", stdout);LL x;  while(SCANF ("%i64d", &x)! =EOF) {printf ("%i64d\n", Eular (X-1)); }    return 0;}

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"Poj1284-primitive Roots" Euler function-the number of primitive roots of odd primes