数学--数论---P4718 Pollard-Rho算法 大数分解
P4718 【模板】Pollard-Rho算法
题目描述
输入格式
第一行,TT代表数据组数(不大于350350)
以下TT行,每行一个整数nn,保证1\le n\le 10^{18}1≤n≤10
18
。
输出格式
输出TT行。
对于每组测试数据输出结果。
输入输出样例
输入 #1复制
6
2
13
134
8897
1234567654321
1000000000000
输出 #1复制
Prime
Prime
67
41
4649
5
这个题目之考察了计算,没考察分解,我博客里有带输出元素的代码。
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
ll pr;
ll pmod(ll a, ll b, ll p) { return (a * b - (ll)((long double)a / p * b) * p + p) % p; } //普通的快速乘会T
ll gmod(ll a, ll b, ll p)
{
ll res = 1;
while (b)
{
if (b & 1) res = pmod(res, a, p);
a = pmod(a, a, p);
b >>= 1;
}
return res;
}
inline ll gcd(ll a, ll b)
{ //听说二进制算法特快
if (!a) return b;
if (!b)return a;
int t = __builtin_ctzll(a | b);
a >>= __builtin_ctzll(a);
do
{
b >>= __builtin_ctzll(b);
if (a > b)
{
ll t = b;
b = a, a = t;
}
b -= a;
} while (b);
return a << t;
}
bool Miller_Rabin(ll n)
{
if (n == 46856248255981ll || n < 2)
return false; //强伪素数
if (n == 2 || n == 3 || n == 7 || n == 61 || n == 24251)
return true;
if (!(n & 1) || !(n % 3) || !(n % 61) || !(n % 24251))
return false;
ll m = n - 1, k = 0;
while (!(m & 1))
k++, m >>= 1;
for (int i = 1; i <= 20; ++i) // 20为Miller-Rabin测试的迭代次数
{
ll a = rand() % (n - 1) + 1, x = gmod(a, m, n), y;
for (int j = 1; j <= k; ++j)
{
y = pmod(x, x, n);
if (y == 1 && x != 1 && x != n - 1)
return 0;
x = y;
}
if (y != 1)
return 0;
}
return 1;
}
ll Pollard_Rho(ll x)
{
ll n = 0, m = 0, t = 1, q = 1, c = rand() % (x - 1) + 1;
for (ll k = 2;; k <<= 1, m = n, q = 1)
{
for (ll i = 1; i <= k; ++i)
{
n = (pmod(n, n, x) + c) % x;
q = pmod(q, abs(m - n), x);
}
t = gcd(x, q);
if (t > 1)
return t;
}
}
void fid(ll n)
{
if (n == 1)
return;
if (Miller_Rabin(n))
{
pr = max(pr, n);
return;
}
ll p = n;
while (p >= n)
p = Pollard_Rho(p);
fid(p);
fid(n / p);
}
int main()
{
int T;
ll n;
scanf("%d", &T);
while (T--)
{
scanf("%lld", &n);
pr = 0;
fid(n);
if (pr == n)
puts("Prime");
else
printf("%lld\n", pr);
}
return 0;
}