質數快速判斷演算法Miller-Rabin

不想詳細寫了，因為也不會證。
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bool Miller_Rabin(ll n){
    if (n==1) return 0;
    int s=5,t=0,i;
    ll a,p,k=n-1;
    while (k%2==0) k/=2,t++;
    while (s--){
        a=random(n-1)+1;
        p=a=qsm(a,k,n);
        fo(i,1,t){
            a=mul(a,a,n);
            if (a==1&&p!=1
 
&&p!=n-1) return 0;
            p=a;
        }
        if (a!=1) return 0;
    }
    return 1;
}

因數快速分解演算法Pollard-Rho

不想詳細寫了，因為也不會證。
貼板子！

ll Pollard_Rho(ll n){
    ll k,x,y,c,d,i=1;
    while (1){
        c=random(n-1);
        k=2;y=x=random(n);
        i=1;
        while (1){
            y=(mul(y,y,n)+c)%n;
            d=gcd(abs
 
(x-y),n);
            if (i==k) x=y,k<<=1;
            i++;
            if (d!=1) break;
        }
        if (d!=n) return d;
    }
}

BZOJ3667

裸題

#include<cstdio>
#include<algorithm>
#include<cmath>
#define fo(i,a,b) for(i=a;i<=b;i++)
using namespace std;
typedef long
 
 long ll;
int i,j,k,l,t,m,ca;
ll n,ans;
ll random(ll x){
    ll t=rand()%10000;
    t=t*10000+rand()%10000;
    t=t*10000+rand()%10000;
    t=t*10000+rand()%10000;
    return t%x;
}
ll mul(ll a,ll b,ll p){
    ll tmp=(a*b-(ll)((long double)a/p*b+1e-8)*p);
    return tmp<0?tmp+p:(tmp>=mo?tmp-mo:tmp);
}
ll qsm(ll x,ll y,ll mo){
    if (!y) return 1;
    ll t=qsm(x,y/2,mo);
    t=mul(t,t,mo);
    if (y%2) t=mul(t,x,mo);
    return t;
}
bool Miller_Rabin(ll n){
    if (n==1) return 0;
    int s=5,t=0,i;
    ll a,p,k=n-1;
    while (k%2==0) k/=2,t++;
    while (s--){
        a=random(n-1)+1;
        p=a=qsm(a,k,n);
        fo(i,1,t){
            a=mul(a,a,n);
            if (a==1&&p!=1&&p!=n-1) return 0;
            p=a;
        }
        if (a!=1) return 0;
    }
    return 1;
}
ll gcd(ll a,ll b){
    return b?gcd(b,a%b):a;
}
ll Pollard_Rho(ll n){
    ll k,x,y,c,d,i=1;
    while (1){
        c=random(n-1);
        k=2;y=x=random(n);
        i=1;
        while (1){
            y=(mul(y,y,n)+c)%n;
            d=gcd(abs(x-y),n);
            if (i==k) x=y,k<<=1;
            i++;
            if (d!=1) break;
        }
        if (d!=n) return d;
    }
}
void work(ll n){
    if (Miller_Rabin(n)){
        ans=max(ans,n);
        return;
    }
    ll p=Pollard_Rho(n);
    ll q=n/p;
    work(p);work(q);
}
int main(){
    srand(19890604);
    scanf("%d",&ca);
    while (ca--){
        scanf("%lld",&n);
        ans=0;
        work(n);
        if (ans==n) printf("Prime\n");else printf("%lld\n",ans);
    }
}