质因数分解

前置知识

大数判断质数Miller Rabin

因数分解 Pollard rho

质因数分解

利用Miller Rabin判断$n$是不是质数，如果不是就进行分解，直到$n$是质数。

每次利用Pollard rho分解出$n$的一个因数，不妨记为$d$。

同样利用Miller Rabin判断$d$是不是质数，如果$d$不是，则继续用Pollard rho分解和用Miller Rabin判断，直到$d$为质数。

此时，就已经找到了$n$的一个质因子$d$，然后将$n$中所有的$d$给除去。

再重复上述步骤，直到$n$的质数。

Code

//#pragma GCC optimize (2)
//#pragma G++ optimize (2)
#include <cstdio>
#include <algorithm>
#include <cstring>
#include <cmath>
#include <iostream>
#include <vector>
#include <queue>
#define G getchar
#define ll long long
using namespace std;

ll read()
{
    char ch;
    for(ch = G();(ch < '0' || ch > '9') && ch != '-';ch = G());
    ll n = 0 , w;
    if (ch == '-')
    {
        w = -1;
        ch = G();
    } else w = 1;
    for(;'0' <= ch && ch <= '9';ch = G())n = (n<<1)+(n<<3)+ch-48;
    return n * w;
}

const int N = 100003;
ll n , ans;
ll pri[N] , k[N] , m;
int ss[16] = {2 , 3 , 5 , 7 , 11 ,13 , 17 , 19 , 23 , 29 , 325 , 9375 , 28187 , 450775 , 9870504 , 1795265022};

ll mul (ll a , ll b , ll p) //a*b%p
{
    /*ll s = 0;
    for ( ; b ; )
    {
        if (b & 1) s = (s + a) % p;
        a = (a + a) % p;
        b = b >> 1;
    }
    return s;*/
    ll s = a * b - (ll)((long double) a * b / p +0.5) * p;
    return s < 0 ? s + p : s;
}

ll ksm (ll x , ll y , ll p)
{
    ll s = 1;
    for ( ; y ; )
    {
        if (y & 1) s = mul(s , x , p);
        x = mul (x , x , p);
        y = y >> 1;
    }
    return s;
}

bool MR_detect (ll n , ll a)
{
    if (n == a) return 1;
    if (a % n == 0) return 1;
    if (ksm(a , n - 1 , n) != 1) return 0;
    ll p = n - 1 , lst = 1;
    for ( ; ((p & 1) ^ 1) && lst == 1 ; )
    {
        p = p >> 1;
        lst = ksm(a , p , n);
        if (lst != 1 && lst != n - 1) return 0;
    }
    return 1;
}

bool MR(ll  n)
{
    if (n < 2) return 0;
    for (int i = 0 ; i < 16 ; i ++)
        if (! MR_detect(n , ss[i])) return 0;
    return 1;
}

ll F(ll x , ll C , ll p)
{
    return (mul(x , x , p) + C) % p;
}

ll gcd(ll a , ll b)
{
    return a % b ? gcd(b , a % b) : b;
}

ll Rand()
{
    return (ll)rand() + ((ll)rand() << 15) + ((ll)rand() << 30) + ((ll)rand() << 45);
}

ll PR(ll n)
{
    if (MR(n)) return n;
    if (n == 4) return 2;
    for ( ; ; )
    {
        ll C = Rand() % (n - 1) + 1;
        ll p1 = 0 , p2 = 0;
        ll s = 1 , tmp;
        for ( ; ; )
        {
            for (int i = 0 ; i < 128 ; i++)
            {
                p2 = F(F(p2 , C , n) , C , n);
                p1 = F(p1 , C , n);
                tmp = mul(s , abs(p1 - p2) , n);
                if (tmp == 0 || p1 == p2) break;
                s = tmp;
            }
            tmp = gcd(n , s);
            if (tmp > 1) return tmp;
            if (p1 == p2) break;
        }
    }
}

int main()
{
    //freopen("1.txt","r",stdin);
    //freopen("2.txt","w",stdout);

    srand((unsigned)0);
    //printf("350\n");
    //for (int i = 1 ; i <= 350 ; i++)printf("%lld\n", Rand());
    for (int T = read() ; T ;T--)
    {
        n = read();
        if (MR(n)) printf("Prime\n"); else
        {
            m = 0;
            for ( ; n != 1 ; )
            {
                ll tmp = PR(n);
                for ( ; !MR(tmp) ; )tmp = PR(tmp);
                m++;
                pri[m] = tmp;
                k[m] = 0;
                for ( ; n % tmp == 0 ; )
                {
                    k[m]++;
                    n = n / tmp;
                }
            }
            //for (int i = 1 ; i <= m ; i++)
            //    printf("%lld^%lld\n", pri[i] , k[i]);
            ans = 0;
            for (int i = 1 ; i <= m ; i++)
                ans = max(ans , pri[i]);
            printf("%lld\n", ans);
        }
     }   

    return 0;
}