[POJ 1811]Prime Test

Description

Given a big integer number, you are required to find out whether it's a prime number.

Input

The first line contains the number of test cases T (1 <= T <= 20 ), then the following T lines each contains an integer number N (2 <= N < 2⁵⁴).

Output

For each test case, if N is a prime number, output a line containing the word "Prime", otherwise, output a line containing the smallest prime factor of N.

Sample Input

2
5
10

Sample Output

Prime
2

题解

Miller-Rabin 素数测试

主要利用费马小定理 $a^{p-1} equiv 1 pmod{p}$ ，实现的过程如下：

1. 1. 计算奇数 $M$ ，使得 $N = 2^r imes M+1$
   2. 选择随机数 $A < N$
   3. 对于任意 $i < r$ ，若 $A^{2^i imes M}~mod~N = N-1$ ，则 $N$ 通过随机数 $A$ 的测试
   4. 或者，若 $A^M~mod~N = 1$ ，则 $N$ 通过随机数 $A$ 的测试
   5. 让 $A$ 取不同的值对 $N$ 进行 $5$ 次测试，若全部通过则判定 $N$ 为素数

而在实际运用中，我们可以直接取 $r = 0$ ，从而省去步骤 $3$ 的测试，提高速度；另外的可以首先用几个小素数对 $N$ 进行测试。除此之外也可以引入“二次探测”的思想，防止被 Carmichael数 卡。

二次探测：对于一个奇素数 $p$ ，对于关于 $x$ 的同余方程 $x^2 equiv 1 pmod{p}$ ，在 $[1, p)$ 上的解仅有 $x = 1$ 及 $x = p-1$ 。

Pollard-Rho 分解大数因子

//It is made by Awson on 2018.1.15
#include <set>
#include <map>
#include <cmath>
#include <ctime>
#include <queue>
#include <stack>
#include <cstdio>
#include <string>
#include <vector>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <algorithm>
#define LL long long
#define Abs(a) ((a) < 0 ? (-(a)) : (a))
#define Max(a, b) ((a) > (b) ? (a) : (b))
#define Min(a, b) ((a) < (b) ? (a) : (b))
#define Swap(a, b) ((a) ^= (b), (b) ^= (a), (a) ^= (b))
using namespace std;
const LL INF = 1e18;
void read(LL &x) {
    char ch; bool flag = 0;
    for (ch = getchar(); !isdigit(ch) && ((flag |= (ch == '-')) || 1); ch = getchar());
    for (x = 0; isdigit(ch); x = (x<<1)+(x<<3)+ch-48, ch = getchar());
    x *= 1-2*flag;
}
void write(LL x) {
    if (x > 9) write(x/10);
    putchar(x%10+48);
}

LL n, ans;
const int prime[10] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29};

LL quick_multi(LL a, LL b, LL p) {
    LL ans = 0;
    while (b) {
    if (b&1) ans = (ans+a)%p;
    a = (a+a)%p, b >>= 1;
    }
    return ans;
}
LL quick_pow(LL a, LL b, LL p) {
    LL ans = 1;
    while (b) {
    if (b&1) ans = quick_multi(ans, a, p);
    a = quick_multi(a, a, p), b >>= 1;
    }
    return ans;
}
bool Miller_Rabin(LL x) {
    if (x == 1) return false;
    for (int i = 0; i < 10; i++) {
    if (x == prime[i]) return true;
    if (!(x%prime[i])) return false;
    }
    LL m = x-1, k = 0;
    while (!(m&1)) m >>= 1, ++k;
    for (int i = 0; i < 10; i++) {
    LL a = rand()%(x-2)+2, pre = quick_pow(a, m, x), y;
    for (int j = 0; j < k; j++) {
        y = quick_multi(pre, pre, x);
        if (y == 1 && pre != 1 && pre != x-1) return false;
        pre = y;
    }
    if (pre != 1) return false;
    }
    return true;
}
LL gcd(LL a, LL b) {return b ? gcd(b, a%b) : a; }
LL Pollard_Rho(LL x, int c) {
    LL i = 1, k = 2, a1, a2;
    a1 = a2 = rand()%(x-1)+1;
    while (true) {
    ++i, a1 = (quick_multi(a1, a1, x)+c)%x;
    LL d = gcd(Abs(a1-a2), x);
    if (1 < d && d < x) return d;
    if (a1 == a2) return x;
    if (i == k) a2 = a1, k <<= 1;
    }
}
void find(LL x, int c) {
    if (x == 1) return;
    if (Miller_Rabin(x)) {ans = Min(ans, x); return; }
    LL p = x, k = c;
    while (p == x) p = Pollard_Rho(x, c--);
    find(p, k), find(x/p, k);
}
void work() {
    srand(time(0));
    LL t; read(t);
    while (t--) {
    read(n); ans = INF;
    if (Miller_Rabin(n)) printf("Prime
");
    else find(n, 120), write(ans), putchar('
');
    }
}
int main() {
    work();
    return 0;
}