POJ3641:Pseudoprime numbers

最新推荐文章于 2019-08-22 21:09:50 发布

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最新推荐文章于 2019-08-22 21:09:50 发布

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本文介绍了一种利用费马小定理来判断特定整数是否为伪素数的方法，并提供了相应的算法实现。通过给定的输入（一个可能的大素数p和一个整数a），程序能够判断p是否为a的伪素数。

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Description

Fermat's theorem states that for any prime number p and for any integera > 1, a^p = a (mod p). That is, if we raisea to the p^th power and divide by p, the remainder isa. 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 alla.)

Given 2 < p ≤ 1000000000 and 1 < a < p, determine whether or notp 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 containingp 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


#include <iostream>
using namespace std;

int prime(long long a)
{
    int i;
    if(a == 2)
        return 1;
    for(i = 2; i*i<=a; i++)
        if(a%i == 0)
            return 0;
    return 1;
}

long long mod(long long a,long long b,long long m)
{
    long long ans = 1;
    while(b>0)
    {
        if(b&1)
        {
            ans = ans*a%m;
            //b--;
        }
        b>>=1;
        a = a*a%m;
    }
    return ans;
}

int main()
{
    long long a,p;

    while(cin >> p >> a && (p||a))
    {
        long long ans;
        if(prime(p))
        cout << "no" << endl;
        else
        {
            ans = mod(a,p,p);
            if(ans == a)
            cout << "yes" << endl;
            else
            cout << "no" << endl;
        }
    }

    return 0;
}