[poj 1811]质数分解

题目链接，给定一个数，判断它是否是素数，如果是合数，就输出最小的质因子。

matrix67 大牛的文章，讲得很详细，好评！

Pallord−rhoPallord-rho 算法本质上是随机一个数 xx 判断 gcd(x,n)gcd(x,n) 是否是 nn 的一个约数

期望时间复杂度: O(n14∗logn)O(n^{\frac 1 4} * \log n)

Miller−rabinMiller-rabin 算法是通过费马小定理和二次探测来判断这个数是否是质数，可以处理强伪素数。

这个算法正确率为 1−(14)k1-(\frac 1 4)^k(kk 是选取的强伪证据的组数)。

时间复杂度: O(k∗logn)O(k*\log n)

注意一定要手写快速乘，不然会爆 long long。。。

#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <ctime>
#include <vector>
#include <utility>
#include <stack>
#include <queue>
#include <iostream>
#include <algorithm>

template<class Num>void read(Num &x)
{
    char c; int flag = 1;
    while((c = getchar()) < '0' || c > '9')
        if(c == '-') flag *= -1;
    x = c - '0';
    while((c = getchar()) >= '0' && c <= '9')
        x = (x<<3) + (x<<1) + (c-'0');
    x *= flag;
    return;
}
template<class Num>void write(Num x)
{
    if(x < 0) putchar('-'), x = -x;
    static char s[20];int sl = 0;
    while(x) s[sl++] = x%10 + '0',x /= 10;
    if(!sl) {putchar('0');return;}
    while(sl) putchar(s[--sl]);
}

const int Case = 3, prime[] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 61, 24251}, tot = 16;
const long long LINF = 1LL << 60;

long long N, ans;

long long Mult_Mod(long long a,long long k,long long p)
{
    long long r = 0;

    a %= p;

    while(k)
    {
        if(k&1) r += a, r %= p;
        a <<= 1, a %= p, k >>= 1; 
    }

    return r;
}
long long Power_Mod(long long a,long long k,long long p)
{
    long long r = 1;

    a %= p;

    while(k)
    {
        if(k&1) r = Mult_Mod(r, a, p);
        a = Mult_Mod(a, a, p), k >>= 1; 
    }

    return r;
}
long long gcd(long long a,long long b)
{
    return b ? gcd(b, a % b) : a;
}
long long Pollard_Rho(long long n)
{
    long long f = rand()%n, g = f, t;

    while(true)
    {
        for(int i = 0; i <= 1; i++)
        {
            g = (Mult_Mod(g, g, n) + 1) % n;

            t = gcd(llabs(g - f), n); 
            if(t != 1 && t != n) return t;

            if(f == g) return n;
        }

        f = (Mult_Mod(f, f, n) + 1) % n;
    }
}
bool Miller_Rabin(long long n)
{
    if(n == 2) return true;
    if(!(n&1)) return false;

    long long y = n - 1;
    int t = 0;

    while(!(y&1)) y >>= 1, t++;

    for (int i = 0; i < tot; i++)
    {  
        if(n == prime[i]) return true;  

        long long f = Power_Mod(prime[i], y, n), g = f;

        for(int j = 1; j <= t; j++)
        {
            f = Mult_Mod(f, f, n);
            if(f == 1 && g != 1 && g != n - 1) return false;
            g = f;
        }
        if(f != 1) return false;
    }
    return true;  
}
void solve(long long n)
{
    if(Miller_Rabin(n))
    {
        ans = std::min(n, ans);
        return;
    }

    for(int i = 1; i <= Case; i++)
    {
        long long p = Pollard_Rho(n);

        if(p != n)
        {
            solve(p), solve(n/p);
            return;
        }
    }
}

int main()
{
    int T;

#ifndef ONLINE_JUDGE
    freopen("1811.in","r",stdin);
    freopen("1811.out","w",stdout);
#endif

    srand(23333);

    read(T);

    while(T--)
    {
        read(N);
        ans = N;

        solve(N); 

        if(ans < N)
            write(ans), puts("");
        else
            puts("Prime");  
    }

#ifndef ONLINE_JUDGE
    fclose(stdin);
    fclose(stdout);
#endif
    return 0;
}