米勒罗宾大素数测定

 

/*
判断素数和分解合数
Miller-Rabin测试素数，Pollard_Rho分解质因子
由于算法本身基于概率，所以存在TLE、WA的可能，多次提交即可
*/
//////////////模板开始//////////////
#include <ctime>
#include <iostream>
#include <algorithm>
#include <cmath>
using namespace std;
#define M 1510
#define N 100010
#define inf 200000000
long long factor[100], fac_top = -1;
//计算两个数的gcd
long long gcd(long long a,long long b) {
    if(a==0) return b;
    long long c;
    while(b!=0) {
        c=b;
        b=a%b;
        a=c;
    }
    return a;
}
//ret=(a*b)%n (n<2^62)
long long muti_mod(long long a,long long b,long long n) {
    long long exp=a%n,res=0;
    while(b) {
        if(b&1) {
            res+=exp;
            if(res>n) res-=n;
        }
        exp<<=1;
        if(exp>n)
            exp-=n;
        b>>=1;
    }
    return res;
}
//ret=(a^b)%n
long long mod_exp(long long a,long long p,long long m) {
    long long exp=a%m,res=1; //
    while(p>1) {
        if(p&1) res=muti_mod(res,exp,m);
        exp=muti_mod(exp,exp,m);
        p>>=1;
    }
    return muti_mod(res,exp,m);
}
//miller-rabin法测试素数,time 测试次数
bool miller_rabin(long long n,int times) {
    if(n==2) return 1;
    if(n==1||(n!=2&&!(n%2))||(n!=3&&!(n%3))||(n!=5&&!(n%5))||(n!=7&&!(n%7)))
        return 0;
    long long a,u=n-1,x,y;
    int t=0;
    while(u%2==0) {
        ++t;
        u/=2;
    }
    srand(time(0));
    for(int i=0; i<times; i++) {
        a=rand()%(n-1)+1;
        x=mod_exp(a,u,n);
        for(int j=0; j<t; j++) {
            y=muti_mod(x,x,n);
            if(y==1&&x!=1&&x!=n-1) return false;
            x=y;
        }
        if(y!=1) return false;
    }
    return true;
}

long long pollard_rho(long long n,int c) {
    //找出一个因子
    long long x,y,d,i=1,k=2;
    srand(time(0));
    x=rand()%(n-1)+1;
    y=x;
    while(true) {
        ++i;
        x=(muti_mod(x,x,n)+c)%n;
        d=gcd(y-x,n);
        if(1<d&&d<n) return d;
        if(y==x) return n;
        if(i==k) {
            y=x;
            k<<=1;
        }
    }
}

void findFactor(long long n,int k) {
    //二分找出所有质因子，存入factor
    if(n==1) return;
    if(miller_rabin(n,6)) {
        factor[++fac_top]=n;
        return;
    }
    long long p=n;
    while(p>=n)
        p=pollard_rho(p,k--);//k值变化，防止死循环
    findFactor(p,k);
    findFactor(n/p,k);
}
//////////////模板完毕//////////////

const int MAXN = 100010;
int it, prime_num[MAXN], plist[MAXN] = {0};
void getPrime() {
    for(it = 2; it <= MAXN; it++) {
        if(!plist[it])
            for(int j = it * 2; j <= MAXN; j += it)
                plist[j] = 1;
    }
    it = 0;
    for(int i = 2; i <= MAXN; i++)
        if(!plist[i])
            prime_num[it++] = i;
}

int main() {
    int t;
    getPrime();
    scanf("%d", &t);
    while (t--) {
        int num = 0;
        long long n, ans = 0, now, pos;
        scanf("%I64d", &n);
        if (n <= 1000000000) {
            now = n;
            long long limit = sqrt((double)n);
            for (int i = 0; i < it && prime_num[i] <= limit; i++) {
                if (n % prime_num[i] == 0) {
                    pos = prime_num[i];
                    num++;
                    long long tmp = prime_num[i];
                    n /= prime_num[i];
                    while (n % prime_num[i] == 0) {
                        tmp *= prime_num[i];
                        n /= prime_num[i];
                    }
                    ans += tmp;
                }
            }
            if (n != 1)
                num++, ans += n;
            if (num == 1) {
                printf("%d %I64d\n", num, now/pos);
                continue;
            }
            printf("%d %I64d\n", num, ans);
            continue;
        }
        fac_top = -1;
        if (miller_rabin(n, 10)) { //随机测试10次
            printf("1 %I64d\n", n);
        } else {
            findFactor(n, 30);
            sort(factor, factor+fac_top+1);
            long long tmp = factor[0];
            for (int i = 1; i <= fac_top; i++) {
                if (factor[i] != factor[i-1]) {
                    ans += tmp;
                    tmp = factor[i];
                    num++;
                } else {
                    tmp *= factor[i-1];
                }
            }
            num++;
            ans += tmp;
            if (num == 1) {
                printf("%d %I64d\n", num, n/factor[0]);
                continue;
            }
            printf("%d %I64d\n", num, ans);
        }
    }
    return 0;
}