Pseudoprime numbers

最新推荐文章于 2020-10-22 21:33:13 发布

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最新推荐文章于 2020-10-22 21:33:13 发布

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分类专栏： gcd 快速幂素筛

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本文介绍了一个利用费马小定理进行素数判定的方法，并通过快速幂运算来提高效率。针对输入的整数对 (p,a)，判断 p 是否为 a 的伪素数。文中提供了一段 C++ 代码实现，包括素数判定和快速幂函数。

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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-apseudoprimes, 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-apseudoprime.

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

3 2
10 3
341 2
341 3
1105 2
1105 3
0 0

素数判定+快速幂 （素数判定的另一种方式
#include<cstdio>
#include<cmath>
bool set_prime(long long a){
	if(a==3||a==2)
	return true;
	if (a%2==0||a%3==0||a==1)
	return false;
	int k = sqrt(a)+1;
	for(int i=5;i<k;i+=6)
	   if(a%i==0||a%(i+2)==0)
		return false;
	return true;
}
long long set_pow(long long a,long long b,long long c){
        long long temp = 1;
		a = a%c;
        while (b){
        	if (b&1){
        		temp = temp*a%c;
			}
			a = a*a%c;
			b>>=1;
		}
		return temp;
}
int main(){
	long long an,p;
	while (scanf("%lld %lld",&p,&an),an+p){
		if (!set_prime(p) && set_pow(an,p,p) == an){
				printf("yes");
		}
		else
		printf("no");
		printf("\n");
	}
	return 0;
}