【poj 3641】 Pseudoprime numbers 【Waterloo Local Contest, 2007.9.23】

最新推荐文章于 2019-10-05 20:07:50 发布

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最新推荐文章于 2019-10-05 20:07:50 发布

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分类专栏： poj 数学/数论

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本文介绍如何使用费马小定理来判断一个数是否为特定基数的伪素数，并提供了通过Miller测试实现这一过程的C++代码示例。

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Description

Fermat's theorem states that for any prime number p and for any integer a > 1, a^p = a (mod p). That is, if we raise a to the p^th power and divide by p, the remainder is a. Some (but not very many) non-prime values of p, known as base-a pseudoprimes, have this property for some a. (And some, known as Carmichael Numbers, are base-a pseudoprimes for all a.)

Given 2 < p ≤ 1000000000 and 1 < a < p, determine whether or not p is a base-a pseudoprime.

Input

Input contains several test cases followed by a line containing "0 0". Each test case consists of a line containing p and a.

Output

For each test case, output "yes" if p is a base-a pseudoprime; otherwise output "no".

Sample Input

Sample Output

no
no
yes
no
yes
yes

这道题是判断p是否为一个a-伪素数的题目，可以用Miller测试解决，下面是代码：

#include<stdio.h>
#include<time.h>
#include<stdlib.h>
#include<iostream>
#define ll long long
using namespace std;
ll random(ll n){
	return ((ll)(rand())*rand()*rand())%n+1;
}
ll read(){
	ll s=0;
	char c=getchar();
	while(c<'0'||c>'9'){
		c=getchar();
	}
	while(c>='0'&&c<='9'){
		s*=10;
		s+=c-'0';
		c=getchar();
	}
	return s;
}
ll q(ll a,ll b,ll k){
	a%=k;
	ll r=a,s=1;
	while(b){
		if(b&1){
			s*=r;
			s%=k;
		}
		r*=r;
		r%=k;
		b>>=1;
	}
	return s;
}
bool p(ll a,ll n){
	ll m=n-1;
	int k=0;
	while(!(m&1)){
		k++;
		m>>=1;
	}
	ll x=q(a,m,n);
	if(x==1||x==n-1){
		return 0;
	}
	while(k--){
		x=(x*x)%n;
		if(x==n-1){
			return 0;
		}
	}
	return 1;
}
bool check(ll n){
	int i;
	if(n==1||(!(n&1))){
		return 0;
	}
	if(n==2){
		return 1;
	}
	for(i=0;i<10;i++){
		ll x=random(n-2)+1;
		if(p(x,n)){
			return 0;
		}
	}
	return 1;
}
int main(){
	srand((unsigned)(time(NULL)));
	ll p,a;
	p=read();
	a=read();
	while(p||a){
		if(check(p)){
			printf("no\n");
		}
		else{
			if(q(a,p,p)==a){
				printf("yes\n");
			}
			else{
				printf("no\n");
			}
		}
		p=read();
		a=read();
	}
	return 0;
}

注：随机次数越多，错误几率越小。