algorithm
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test/yosupo/factorize.cpp
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- Last update: 2022-03-17 11:21:22+09:00
- Link: http://miller-rabin.appspot.com/
- Link: https://joisino.hatenablog.com/entry/2017/08/03/210000
- Link: https://misteer.hatenablog.com/entry/miller-rabin
- Link: https://yukicoder.me/problems/no/3030
Code
//#pragma GCC optimize("Ofast")
//#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using pii = pair<int, int>;
template <class T>
using V = vector<T>;
template <class T>
using VV = V<V<T>>;
template <class T>
V<T> make_vec(size_t a) {
return V<T>(a);
}
template <class T, class... Ts>
auto make_vec(size_t a, Ts... ts) {
return V<decltype(make_vec<T>(ts...))>(a, make_vec<T>(ts...));
}
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define fi first
#define se second
#define rep(i, n) rep2(i, 0, n)
#define rep2(i, m, n) for (int i = m; i < (n); i++)
#define per(i, b) per2(i, 0, b)
#define per2(i, a, b) for (int i = int(b) - 1; i >= int(a); i--)
#define ALL(c) (c).begin(), (c).end()
#define SZ(x) ((int)(x).size())
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T, class U>
void chmin(T& t, const U& u) {
if (t > u) t = u;
}
template <class T, class U>
void chmax(T& t, const U& u) {
if (t < u) t = u;
}
template <class T>
void mkuni(vector<T>& v) {
sort(ALL(v));
v.erase(unique(ALL(v)), end(v));
}
template <class T>
vector<int> sort_by(const vector<T>& v) {
vector<int> res(v.size());
iota(res.begin(), res.end(), 0);
sort(res.begin(), res.end(), [&](int i, int j) { return v[i] < v[j]; });
return res;
}
template <class T, class U>
istream& operator>>(istream& is, pair<T, U>& p) {
is >> p.first >> p.second;
return is;
}
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
os << "(" << p.first << "," << p.second << ")";
return os;
}
template <class T>
istream& operator>>(istream& is, vector<T>& v) {
for (auto& x : v) {
is >> x;
}
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
rep(i, v.size()) {
if (i) os << ",";
os << v[i];
}
os << "}";
return os;
}
#ifdef LOCAL
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << H;
debug_out(T...);
}
#define debug(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif
template <class T>
void scan(vector<T>& v, T offset = T(0)) {
for (auto& x : v) {
cin >> x;
x += offset;
}
}
template <class T>
void print(T x, int suc = 1) {
cout << x;
if (suc == 1)
cout << "\n";
else if (suc == 2)
cout << " ";
}
template <class T>
void print(const vector<T>& v, int suc = 1) {
for (int i = 0; i < v.size(); ++i)
print(v[i], i == int(v.size()) - 1 ? suc : 2);
}
struct prepare_io {
prepare_io() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(10);
}
} prep_io;
template <class T, class U>
T powmod(T x, U k, T m) {
x %= m;
T res = 1 % m;
while (k) {
if (k & 1) {
res = res * x % m;
}
k >>= 1;
x = x * x % m;
}
return res;
}
// see : https://joisino.hatenablog.com/entry/2017/08/03/210000
// see : https://misteer.hatenablog.com/entry/miller-rabin
// test : https://yukicoder.me/problems/no/3030
// ref : http://miller-rabin.appspot.com/
// check if n is prime
// n < 4,759,123,141 -> call miller_rabin
// 2^64 > n >= 4,759,123,141 -> call miller_rabin_big
// O(k log^3 n)
bool miller_rabin(ll n) {
static const vector<ll> v = {2, 7, 61};
if (n < 2) {
return false;
} else if (n == 2) {
return true;
} else if (n % 2 == 0) {
return false;
}
ull d = n - 1;
int s = 0;
while (d % 2 == 0) {
d /= 2;
s++;
}
for (auto a : v) {
if (a % n == 0) continue;
auto x = powmod(a, d, n);
if (x == 1 || x == n - 1) continue;
rep(r, s) {
auto y = x * x % n;
if (y == 1) {
if (x != 1 && x != n - 1) {
return false;
} else {
x = 1;
break;
}
}
x = y;
}
if (x != 1) return false;
}
return true;
}
using R = __int128_t;
using ull = unsigned long long;
bool miller_rabin_big(ull n) {
static const vector<R> v = {2, 325, 9375, 28178,
450775, 9780504, 1795265022};
if (n < 2) {
return false;
} else if (n == 2) {
return true;
} else if (n % 2 == 0) {
return false;
}
ull d = n - 1;
int s = 0;
while (d % 2 == 0) {
d /= 2;
s++;
}
for (auto a : v) {
if (a % n == 0) continue;
auto x = powmod(a, d, R(n));
if (x == 1 || x == n - 1) continue;
rep(r, s) {
auto y = x * x % n;
if (y == 1) {
if (x != 1 && x != n - 1) {
return false;
} else {
x = 1;
break;
}
}
x = y;
}
if (x != 1) return false;
}
return true;
}
// bit op
int popcnt(uint x) { return __builtin_popcount(x); }
int popcnt(ull x) { return __builtin_popcountll(x); }
int bsr(uint x) { return 31 - __builtin_clz(x); }
int bsr(ull x) { return 63 - __builtin_clzll(x); }
int bsf(uint x) { return __builtin_ctz(x); }
int bsf(ull x) { return __builtin_ctzll(x); }
// binary gcd
ll binary_gcd(ll _a, ll _b) {
ull a = abs(_a), b = abs(_b);
if (a == 0) return b;
if (b == 0) return a;
int shift = bsf(a | b);
a >>= bsf(a);
do {
b >>= bsf(b);
if (a > b) swap(a, b);
b -= a;
} while (b);
return (a << shift);
}
using R = __int128;
ll pollrard_single(ll n) {
auto f = [&](ll x) { return ((R)x * x + 1) % n; };
if (miller_rabin_big(n)) return n;
if (n % 2 == 0) return 2;
ll st = 0;
while (true) {
st++;
ll x = st, y = f(x);
while (true) {
ll p = gcd((y - x + n), n);
if (p == 0 || p == n) break;
if (p != 1) return p;
x = f(x);
y = f(f(y));
}
}
}
V<ll> pollrard(ll n) {
if (n == 1) return {};
ll x = pollrard_single(n);
if (x == n) return {x};
V<ll> v1 = pollrard(x), v2 = pollrard(n / x);
v1.insert(v1.end(), ALL(v2));
return v1;
}
void slv() {
ll x;
cin >> x;
auto ans = pollrard(x);
sort(ALL(ans));
debug(x, ans);
print(SZ(ans), 2);
print(ans);
}
int main() {
int cases = 1;
cin >> cases;
rep(i, cases) slv();
return 0;
}#line 1 "test/yosupo/factorize.cpp"
//#pragma GCC optimize("Ofast")
//#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using pii = pair<int, int>;
template <class T>
using V = vector<T>;
template <class T>
using VV = V<V<T>>;
template <class T>
V<T> make_vec(size_t a) {
return V<T>(a);
}
template <class T, class... Ts>
auto make_vec(size_t a, Ts... ts) {
return V<decltype(make_vec<T>(ts...))>(a, make_vec<T>(ts...));
}
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define fi first
#define se second
#define rep(i, n) rep2(i, 0, n)
#define rep2(i, m, n) for (int i = m; i < (n); i++)
#define per(i, b) per2(i, 0, b)
#define per2(i, a, b) for (int i = int(b) - 1; i >= int(a); i--)
#define ALL(c) (c).begin(), (c).end()
#define SZ(x) ((int)(x).size())
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T, class U>
void chmin(T& t, const U& u) {
if (t > u) t = u;
}
template <class T, class U>
void chmax(T& t, const U& u) {
if (t < u) t = u;
}
template <class T>
void mkuni(vector<T>& v) {
sort(ALL(v));
v.erase(unique(ALL(v)), end(v));
}
template <class T>
vector<int> sort_by(const vector<T>& v) {
vector<int> res(v.size());
iota(res.begin(), res.end(), 0);
sort(res.begin(), res.end(), [&](int i, int j) { return v[i] < v[j]; });
return res;
}
template <class T, class U>
istream& operator>>(istream& is, pair<T, U>& p) {
is >> p.first >> p.second;
return is;
}
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
os << "(" << p.first << "," << p.second << ")";
return os;
}
template <class T>
istream& operator>>(istream& is, vector<T>& v) {
for (auto& x : v) {
is >> x;
}
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
rep(i, v.size()) {
if (i) os << ",";
os << v[i];
}
os << "}";
return os;
}
#ifdef LOCAL
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << H;
debug_out(T...);
}
#define debug(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif
template <class T>
void scan(vector<T>& v, T offset = T(0)) {
for (auto& x : v) {
cin >> x;
x += offset;
}
}
template <class T>
void print(T x, int suc = 1) {
cout << x;
if (suc == 1)
cout << "\n";
else if (suc == 2)
cout << " ";
}
template <class T>
void print(const vector<T>& v, int suc = 1) {
for (int i = 0; i < v.size(); ++i)
print(v[i], i == int(v.size()) - 1 ? suc : 2);
}
struct prepare_io {
prepare_io() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(10);
}
} prep_io;
template <class T, class U>
T powmod(T x, U k, T m) {
x %= m;
T res = 1 % m;
while (k) {
if (k & 1) {
res = res * x % m;
}
k >>= 1;
x = x * x % m;
}
return res;
}
// see : https://joisino.hatenablog.com/entry/2017/08/03/210000
// see : https://misteer.hatenablog.com/entry/miller-rabin
// test : https://yukicoder.me/problems/no/3030
// ref : http://miller-rabin.appspot.com/
// check if n is prime
// n < 4,759,123,141 -> call miller_rabin
// 2^64 > n >= 4,759,123,141 -> call miller_rabin_big
// O(k log^3 n)
bool miller_rabin(ll n) {
static const vector<ll> v = {2, 7, 61};
if (n < 2) {
return false;
} else if (n == 2) {
return true;
} else if (n % 2 == 0) {
return false;
}
ull d = n - 1;
int s = 0;
while (d % 2 == 0) {
d /= 2;
s++;
}
for (auto a : v) {
if (a % n == 0) continue;
auto x = powmod(a, d, n);
if (x == 1 || x == n - 1) continue;
rep(r, s) {
auto y = x * x % n;
if (y == 1) {
if (x != 1 && x != n - 1) {
return false;
} else {
x = 1;
break;
}
}
x = y;
}
if (x != 1) return false;
}
return true;
}
using R = __int128_t;
using ull = unsigned long long;
bool miller_rabin_big(ull n) {
static const vector<R> v = {2, 325, 9375, 28178,
450775, 9780504, 1795265022};
if (n < 2) {
return false;
} else if (n == 2) {
return true;
} else if (n % 2 == 0) {
return false;
}
ull d = n - 1;
int s = 0;
while (d % 2 == 0) {
d /= 2;
s++;
}
for (auto a : v) {
if (a % n == 0) continue;
auto x = powmod(a, d, R(n));
if (x == 1 || x == n - 1) continue;
rep(r, s) {
auto y = x * x % n;
if (y == 1) {
if (x != 1 && x != n - 1) {
return false;
} else {
x = 1;
break;
}
}
x = y;
}
if (x != 1) return false;
}
return true;
}
// bit op
int popcnt(uint x) { return __builtin_popcount(x); }
int popcnt(ull x) { return __builtin_popcountll(x); }
int bsr(uint x) { return 31 - __builtin_clz(x); }
int bsr(ull x) { return 63 - __builtin_clzll(x); }
int bsf(uint x) { return __builtin_ctz(x); }
int bsf(ull x) { return __builtin_ctzll(x); }
// binary gcd
ll binary_gcd(ll _a, ll _b) {
ull a = abs(_a), b = abs(_b);
if (a == 0) return b;
if (b == 0) return a;
int shift = bsf(a | b);
a >>= bsf(a);
do {
b >>= bsf(b);
if (a > b) swap(a, b);
b -= a;
} while (b);
return (a << shift);
}
using R = __int128;
ll pollrard_single(ll n) {
auto f = [&](ll x) { return ((R)x * x + 1) % n; };
if (miller_rabin_big(n)) return n;
if (n % 2 == 0) return 2;
ll st = 0;
while (true) {
st++;
ll x = st, y = f(x);
while (true) {
ll p = gcd((y - x + n), n);
if (p == 0 || p == n) break;
if (p != 1) return p;
x = f(x);
y = f(f(y));
}
}
}
V<ll> pollrard(ll n) {
if (n == 1) return {};
ll x = pollrard_single(n);
if (x == n) return {x};
V<ll> v1 = pollrard(x), v2 = pollrard(n / x);
v1.insert(v1.end(), ALL(v2));
return v1;
}
void slv() {
ll x;
cin >> x;
auto ans = pollrard(x);
sort(ALL(ans));
debug(x, ans);
print(SZ(ans), 2);
print(ans);
}
int main() {
int cases = 1;
cin >> cases;
rep(i, cases) slv();
return 0;
}