数学简单专题

#include <map>
#include <set>
#include <stack>
#include <queue>
#include <cmath>
#include <ctime>
#include <vector>
#include <cstdio>
#include <cctype>
#include <cstring>
#include <cstdlib>
#include <iostream>
//#include <algorithm>
using namespace std;
#define INF 0x3f3f3f3f
#define inf ((LL)1<<40)
#define lson k<<1, L, mid
#define rson k<<1|1, mid+1, R
#define mem0(a) memset(a,0,sizeof(a))
#define mem1(a) memset(a,-1,sizeof(a))
#define mem(a, b) memset(a, b, sizeof(a))
#define FOPENIN(IN) freopen(IN, "r", stdin)
#define FOPENOUT(OUT) freopen(OUT, "w", stdout)
template<class T> T ABS ( T a) { return a >= 0 ? a : -a;   }
template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }

typedef __int64 LL;
//typedef long long LL;
const int MAXN = 10005;
const int MAXM = 110000;
const double eps = 1e-14;
const double PI = 4.0 * atan(1.0);

LL n, ans;
int t;

#define Times 10

//产生[0, n-1]的一个随机数
LL random(LL n)
{
return ((double)rand() / RAND_MAX * n + 0.5);
}
//乘法，采用加法模拟，避免中间结果超出LL
LL multi(LL a,LL b,LL mod)
{
LL ans=0;
while(b)
{
if(b & 1)
{
b--;
ans=(ans+a) % mod;
}
else
{
b/=2;
a=(2*a) % mod;
}
}
return ans;
}

//快速幂，同样避免超出LL的做法
LL Pow(LL a, LL b, LL mod)
{
LL ans=1;
while(b)
{
if(b&1)
{
b--;
ans=multi(ans,a,mod);
}
else
{
b/=2;
a=multi(a,a,mod);
}
}
return ans;
}

//miller_rabin的一遍探测，返回false表示是合数
bool witness(LL a,LL n)
{
LL d=n-1;
while( !(d&1) )
d >>= 1;
LL t=Pow(a, d, n);
while(d!=n-1 && t!=1 && t!=n-1)
{
t=multi(t,t,n);
d<<=1;
}
return t==n-1 || d&1;
}

//miller_rabin算法，返回false表示是合数，否则是素数
//返回素数出错的概率(最高)为 1 / (4 ^ times)
bool miller_rabin(LL n)
{
if(n == 2)
return true;
if( n<2 || !(n&1) )
return false;
for(int i = 1; i <= Times ; i++ )
{
LL a = random(n-2) + 1;
if( !witness(a, n) )
return false;
}
return true;
}

//************************************************
//pollard_rho 算法进行质因数分解
//************************************************
LL factor[100];//质因数分解结果（刚返回时是无序的）
int tol;//质因数的个数。数组小标从0开始

LL gcd(LL a,LL b)
{
if(a==0)return 1;//???????
if(a<0) return gcd(-a,b);
while(b)
{
LL t=a%b;
a=b;
b=t;
}
return a;
}

LL Pollard_rho(LL x, LL c)
{
LL i=1,k=2;
LL x0=rand()%x;
LL y=x0;
while(1)
{
i++;
x0=(multi(x0, x0, x) + c) % x;
LL d=gcd(y-x0,x);
if(d!=1&&d!=x) return d;
if(y==x0) return x;
if(i==k){y=x0;k+=k;}
}
}

//对n进行素因子分解
void findfac(LL n)
{
if(miller_rabin(n))//素数
{
factor[tol++]=n;
ans = MIN(ans, n);
return;
}
LL p=n;
while(p>=n)
p=Pollard_rho(p, rand()%(n-1)+1);
findfac(p);
findfac(n/p);
}

int main()
{
//FOPENIN("in.txt");
while(~scanf("%d", &t))while(t--)
{
ans = inf;
scanf("%I64d", &n);
findfac(n);
if(miller_rabin(n))
printf("Prime\n");
else printf("%I64d\n", ans);
}
return 0;
}