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test/2_library_checker/number_theory/two_square.test.cpp
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- Last update: 2024-12-13 13:55:16+09:00
- Problem: https://judge.yosupo.jp/problem/two_square_sum
Depends on
mod/barrett.hpp
- mod/mod_pow.hpp
- mod/mongomery_modint.hpp
- my_template.hpp
- nt/factor.hpp
- nt/gaussian_integers.hpp
- nt/primetest.hpp
- nt/two_square.hpp
- other/io.hpp
- random/base.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/two_square_sum"
#include "my_template.hpp"
#include "other/io.hpp"
#include "nt/two_square.hpp"
void solve() {
LL(N);
vc<pi> ANS = two_square(N, true);
print(len(ANS));
for (auto& x: ANS) print(x);
}
signed main() {
INT(T);
FOR(T) solve();
return 0;
}#line 1 "test/2_library_checker/number_theory/two_square.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/two_square_sum"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
// https://codeforces.com/blog/entry/96344
#pragma GCC optimize("Ofast,unroll-loops")
// いまの CF だとこれ入れると動かない？
// #pragma GCC target("avx2,popcnt")
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;
template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'010'000'000;
template <>
constexpr ll infty<ll> = 2'020'000'000'000'000'000;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;
using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;
#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...) \
vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))
// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)
#define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if
#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second
#define stoi stoll
int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_sgn(int x) { return (__builtin_parity(unsigned(x)) & 1 ? -1 : 1); }
int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(ll x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
int popcnt_sgn(u64 x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
template <typename T>
T kth_bit(int k) {
return T(1) << k;
}
template <typename T>
bool has_kth_bit(T x, int k) {
return x >> k & 1;
}
template <typename T>
T floor(T a, T b) {
return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
T q = floor(x, y);
return {q, x - q * y};
}
template <typename T, typename U>
T SUM(const vector<U> &A) {
T sm = 0;
for (auto &&a: A) sm += a;
return sm;
}
#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()
template <typename T>
T POP(deque<T> &que) {
T a = que.front();
que.pop_front();
return a;
}
template <typename T>
T POP(pq<T> &que) {
T a = que.top();
que.pop();
return a;
}
template <typename T>
T POP(pqg<T> &que) {
T a = que.top();
que.pop();
return a;
}
template <typename T>
T POP(vc<T> &que) {
T a = que.back();
que.pop_back();
return a;
}
template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
if (check_ok) assert(check(ok));
while (abs(ok - ng) > 1) {
auto x = (ng + ok) / 2;
(check(x) ? ok : ng) = x;
}
return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
FOR(iter) {
double x = (ok + ng) / 2;
(check(x) ? ok : ng) = x;
}
return (ok + ng) / 2;
}
template <class T, class S>
inline bool chmax(T &a, const S &b) {
return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
return (a > b ? a = b, 1 : 0);
}
// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
vc<int> A(S.size());
FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
return A;
}
template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
int N = A.size();
vector<T> B(N + 1);
FOR(i, N) { B[i + 1] = B[i] + A[i]; }
if (off == 0) B.erase(B.begin());
return B;
}
// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
vector<int> ids(len(A));
iota(all(ids), 0);
sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
return ids;
}
// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
vc<T> B(len(I));
FOR(i, len(I)) B[i] = A[I[i]];
return B;
}
template <typename T, typename... Vectors>
void concat(vc<T> &first, const Vectors &... others) {
vc<T> &res = first;
(res.insert(res.end(), others.begin(), others.end()), ...);
}
#endif
#line 1 "other/io.hpp"
#define FASTIO
#include <unistd.h>
// https://judge.yosupo.jp/submission/21623
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
char num[10000][4];
constexpr Pre() : num() {
for (int i = 0; i < 10000; i++) {
int n = i;
for (int j = 3; j >= 0; j--) {
num[i][j] = n % 10 | '0';
n /= 10;
}
}
}
} constexpr pre;
inline void load() {
memcpy(ibuf, ibuf + pil, pir - pil);
pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
pil = 0;
if (pir < SZ) ibuf[pir++] = '\n';
}
inline void flush() {
fwrite(obuf, 1, por, stdout);
por = 0;
}
void rd(char &c) {
do {
if (pil + 1 > pir) load();
c = ibuf[pil++];
} while (isspace(c));
}
void rd(string &x) {
x.clear();
char c;
do {
if (pil + 1 > pir) load();
c = ibuf[pil++];
} while (isspace(c));
do {
x += c;
if (pil == pir) load();
c = ibuf[pil++];
} while (!isspace(c));
}
template <typename T>
void rd_real(T &x) {
string s;
rd(s);
x = stod(s);
}
template <typename T>
void rd_integer(T &x) {
if (pil + 100 > pir) load();
char c;
do
c = ibuf[pil++];
while (c < '-');
bool minus = 0;
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (c == '-') { minus = 1, c = ibuf[pil++]; }
}
x = 0;
while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (minus) x = -x;
}
}
void rd(int &x) { rd_integer(x); }
void rd(ll &x) { rd_integer(x); }
void rd(i128 &x) { rd_integer(x); }
void rd(u32 &x) { rd_integer(x); }
void rd(u64 &x) { rd_integer(x); }
void rd(u128 &x) { rd_integer(x); }
void rd(double &x) { rd_real(x); }
void rd(long double &x) { rd_real(x); }
void rd(f128 &x) { rd_real(x); }
template <class T, class U>
void rd(pair<T, U> &p) {
return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T>
void rd_tuple(T &t) {
if constexpr (N < std::tuple_size<T>::value) {
auto &x = std::get<N>(t);
rd(x);
rd_tuple<N + 1>(t);
}
}
template <class... T>
void rd(tuple<T...> &tpl) {
rd_tuple(tpl);
}
template <size_t N = 0, typename T>
void rd(array<T, N> &x) {
for (auto &d: x) rd(d);
}
template <class T>
void rd(vc<T> &x) {
for (auto &d: x) rd(d);
}
void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
rd(h), read(t...);
}
void wt(const char c) {
if (por == SZ) flush();
obuf[por++] = c;
}
void wt(const string s) {
for (char c: s) wt(c);
}
void wt(const char *s) {
size_t len = strlen(s);
for (size_t i = 0; i < len; i++) wt(s[i]);
}
template <typename T>
void wt_integer(T x) {
if (por > SZ - 100) flush();
if (x < 0) { obuf[por++] = '-', x = -x; }
int outi;
for (outi = 96; x >= 10000; outi -= 4) {
memcpy(out + outi, pre.num[x % 10000], 4);
x /= 10000;
}
if (x >= 1000) {
memcpy(obuf + por, pre.num[x], 4);
por += 4;
} else if (x >= 100) {
memcpy(obuf + por, pre.num[x] + 1, 3);
por += 3;
} else if (x >= 10) {
int q = (x * 103) >> 10;
obuf[por] = q | '0';
obuf[por + 1] = (x - q * 10) | '0';
por += 2;
} else
obuf[por++] = x | '0';
memcpy(obuf + por, out + outi + 4, 96 - outi);
por += 96 - outi;
}
template <typename T>
void wt_real(T x) {
ostringstream oss;
oss << fixed << setprecision(15) << double(x);
string s = oss.str();
wt(s);
}
void wt(int x) { wt_integer(x); }
void wt(ll x) { wt_integer(x); }
void wt(i128 x) { wt_integer(x); }
void wt(u32 x) { wt_integer(x); }
void wt(u64 x) { wt_integer(x); }
void wt(u128 x) { wt_integer(x); }
void wt(double x) { wt_real(x); }
void wt(long double x) { wt_real(x); }
void wt(f128 x) { wt_real(x); }
template <class T, class U>
void wt(const pair<T, U> val) {
wt(val.first);
wt(' ');
wt(val.second);
}
template <size_t N = 0, typename T>
void wt_tuple(const T t) {
if constexpr (N < std::tuple_size<T>::value) {
if constexpr (N > 0) { wt(' '); }
const auto x = std::get<N>(t);
wt(x);
wt_tuple<N + 1>(t);
}
}
template <class... T>
void wt(tuple<T...> tpl) {
wt_tuple(tpl);
}
template <class T, size_t S>
void wt(const array<T, S> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i) wt(' ');
wt(val[i]);
}
}
template <class T>
void wt(const vector<T> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i) wt(' ');
wt(val[i]);
}
}
void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
wt(head);
if (sizeof...(Tail)) wt(' ');
print(forward<Tail>(tail)...);
}
// gcc expansion. called automaticall after main.
void __attribute__((destructor)) _d() { flush(); }
} // namespace fastio
using fastio::read;
using fastio::print;
using fastio::flush;
#if defined(LOCAL)
#define SHOW(...) SHOW_IMPL(__VA_ARGS__, SHOW6, SHOW5, SHOW4, SHOW3, SHOW2, SHOW1)(__VA_ARGS__)
#define SHOW_IMPL(_1, _2, _3, _4, _5, _6, NAME, ...) NAME
#define SHOW1(x) print(#x, "=", (x)), flush()
#define SHOW2(x, y) print(#x, "=", (x), #y, "=", (y)), flush()
#define SHOW3(x, y, z) print(#x, "=", (x), #y, "=", (y), #z, "=", (z)), flush()
#define SHOW4(x, y, z, w) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w)), flush()
#define SHOW5(x, y, z, w, v) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v)), flush()
#define SHOW6(x, y, z, w, v, u) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v), #u, "=", (u)), flush()
#else
#define SHOW(...)
#endif
#define INT(...) \
int __VA_ARGS__; \
read(__VA_ARGS__)
#define LL(...) \
ll __VA_ARGS__; \
read(__VA_ARGS__)
#define U32(...) \
u32 __VA_ARGS__; \
read(__VA_ARGS__)
#define U64(...) \
u64 __VA_ARGS__; \
read(__VA_ARGS__)
#define STR(...) \
string __VA_ARGS__; \
read(__VA_ARGS__)
#define CHAR(...) \
char __VA_ARGS__; \
read(__VA_ARGS__)
#define DBL(...) \
double __VA_ARGS__; \
read(__VA_ARGS__)
#define VEC(type, name, size) \
vector<type> name(size); \
read(name)
#define VV(type, name, h, w) \
vector<vector<type>> name(h, vector<type>(w)); \
read(name)
void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
void YA(bool t = 1) { print(t ? "YA" : "TIDAK"); }
void TIDAK(bool t = 1) { YES(!t); }
#line 4 "test/2_library_checker/number_theory/two_square.test.cpp"
#line 2 "nt/factor.hpp"
#line 2 "random/base.hpp"
u64 RNG_64() {
static u64 x_ = u64(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
x_ ^= x_ << 7;
return x_ ^= x_ >> 9;
}
u64 RNG(u64 lim) { return RNG_64() % lim; }
ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "mod/mongomery_modint.hpp"
// odd mod.
// x の代わりに rx を持つ
template <int id, typename U1, typename U2>
struct Mongomery_modint {
using mint = Mongomery_modint;
inline static U1 m, r, n2;
static constexpr int W = numeric_limits<U1>::digits;
static void set_mod(U1 mod) {
assert(mod & 1 && mod <= U1(1) << (W - 2));
m = mod, n2 = -U2(m) % m, r = m;
FOR(5) r *= 2 - m * r;
r = -r;
assert(r * m == U1(-1));
}
static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; }
U1 x;
Mongomery_modint() : x(0) {}
Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){};
U1 val() const {
U1 y = reduce(x);
return y >= m ? y - m : y;
}
mint &operator+=(mint y) {
x = ((x += y.x) >= m ? x - m : x);
return *this;
}
mint &operator-=(mint y) {
x -= (x >= y.x ? y.x : y.x - m);
return *this;
}
mint &operator*=(mint y) {
x = reduce(U2(x) * y.x);
return *this;
}
mint operator+(mint y) const { return mint(*this) += y; }
mint operator-(mint y) const { return mint(*this) -= y; }
mint operator*(mint y) const { return mint(*this) *= y; }
bool operator==(mint y) const {
return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
}
bool operator!=(mint y) const { return not operator==(y); }
mint pow(ll n) const {
assert(n >= 0);
mint y = 1, z = *this;
for (; n; n >>= 1, z *= z)
if (n & 1) y *= z;
return y;
}
};
template <int id>
using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>;
template <int id>
using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>;
#line 3 "nt/primetest.hpp"
bool primetest(const u64 x) {
assert(x < u64(1) << 62);
if (x == 2 or x == 3 or x == 5 or x == 7) return true;
if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false;
if (x < 121) return x > 1;
const u64 d = (x - 1) >> lowbit(x - 1);
using mint = Mongomery_modint_64<202311020>;
mint::set_mod(x);
const mint one(u64(1)), minus_one(x - 1);
auto ok = [&](u64 a) -> bool {
auto y = mint(a).pow(d);
u64 t = d;
while (y != one && y != minus_one && t != x - 1) y *= y, t <<= 1;
if (y != minus_one && t % 2 == 0) return false;
return true;
};
if (x < (u64(1) << 32)) {
for (u64 a: {2, 7, 61})
if (!ok(a)) return false;
} else {
for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
if (!ok(a)) return false;
}
}
return true;
}
#line 5 "nt/factor.hpp"
template <typename mint>
ll rho(ll n, ll c) {
assert(n > 1);
const mint cc(c);
auto f = [&](mint x) { return x * x + cc; };
mint x = 1, y = 2, z = 1, q = 1;
ll g = 1;
const ll m = 1LL << (__lg(n) / 5);
for (ll r = 1; g == 1; r <<= 1) {
x = y;
FOR(r) y = f(y);
for (ll k = 0; k < r && g == 1; k += m) {
z = y;
FOR(min(m, r - k)) y = f(y), q *= x - y;
g = gcd(q.val(), n);
}
}
if (g == n) do {
z = f(z);
g = gcd((x - z).val(), n);
} while (g == 1);
return g;
}
ll find_prime_factor(ll n) {
assert(n > 1);
if (primetest(n)) return n;
FOR(100) {
ll m = 0;
if (n < (1 << 30)) {
using mint = Mongomery_modint_32<20231025>;
mint::set_mod(n);
m = rho<mint>(n, RNG(0, n));
} else {
using mint = Mongomery_modint_64<20231025>;
mint::set_mod(n);
m = rho<mint>(n, RNG(0, n));
}
if (primetest(m)) return m;
n = m;
}
assert(0);
return -1;
}
// ソートしてくれる
vc<pair<ll, int>> factor(ll n) {
assert(n >= 1);
vc<pair<ll, int>> pf;
FOR(p, 2, 100) {
if (p * p > n) break;
if (n % p == 0) {
ll e = 0;
do { n /= p, e += 1; } while (n % p == 0);
pf.eb(p, e);
}
}
while (n > 1) {
ll p = find_prime_factor(n);
ll e = 0;
do { n /= p, e += 1; } while (n % p == 0);
pf.eb(p, e);
}
sort(all(pf));
return pf;
}
vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) {
vc<pair<ll, int>> res;
while (n > 1) {
int p = lpf[n];
int e = 0;
while (n % p == 0) {
n /= p;
++e;
}
res.eb(p, e);
}
return res;
}
#line 2 "mod/mod_pow.hpp"
#line 2 "mod/barrett.hpp"
// https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp
struct Barrett {
u32 m;
u64 im;
explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {}
u32 umod() const { return m; }
u32 modulo(u64 z) {
if (m == 1) return 0;
u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
u64 y = x * m;
return (z - y + (z < y ? m : 0));
}
u64 floor(u64 z) {
if (m == 1) return z;
u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
u64 y = x * m;
return (z < y ? x - 1 : x);
}
pair<u64, u32> divmod(u64 z) {
if (m == 1) return {z, 0};
u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
u64 y = x * m;
if (z < y) return {x - 1, z - y + m};
return {x, z - y};
}
u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); }
};
struct Barrett_64 {
u128 mod, mh, ml;
explicit Barrett_64(u64 mod = 1) : mod(mod) {
u128 m = u128(-1) / mod;
if (m * mod + mod == u128(0)) ++m;
mh = m >> 64;
ml = m & u64(-1);
}
u64 umod() const { return mod; }
u64 modulo(u128 x) {
u128 z = (x & u64(-1)) * ml;
z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64);
z = (x >> 64) * mh + (z >> 64);
x -= z * mod;
return x < mod ? x : x - mod;
}
u64 mul(u64 a, u64 b) { return modulo(u128(a) * b); }
};
#line 5 "mod/mod_pow.hpp"
u32 mod_pow(int a, ll n, int mod) {
assert(n >= 0);
a = ((a %= mod) < 0 ? a + mod : a);
if ((mod & 1) && (mod < (1 << 30))) {
using mint = Mongomery_modint_32<202311021>;
mint::set_mod(mod);
return mint(a).pow(n).val();
}
Barrett bt(mod);
int r = 1;
while (n) {
if (n & 1) r = bt.mul(r, a);
a = bt.mul(a, a), n >>= 1;
}
return r;
}
u64 mod_pow_64(ll a, ll n, u64 mod) {
assert(n >= 0);
a = ((a %= mod) < 0 ? a + mod : a);
if ((mod & 1) && (mod < (u64(1) << 62))) {
using mint = Mongomery_modint_64<202311021>;
mint::set_mod(mod);
return mint(a).pow(n).val();
}
Barrett_64 bt(mod);
ll r = 1;
while (n) {
if (n & 1) r = bt.mul(r, a);
a = bt.mul(a, a), n >>= 1;
}
return r;
}
#line 3 "nt/gaussian_integers.hpp"
template <typename T>
struct Gaussian_Integer {
T x, y;
using G = Gaussian_Integer;
Gaussian_Integer(T x = 0, T y = 0) : x(x), y(y) {}
Gaussian_Integer(pair<T, T> p) : x(p.fi), y(p.se) {}
T norm() const { return x * x + y * y; }
G conjugate() const { return G(x, -y); }
G &operator+=(const G &g) {
x += g.x, y += g.y;
return *this;
}
G &operator-=(const G &g) {
x -= g.x, y -= g.y;
return *this;
}
G &operator*=(const G &g) {
tie(x, y) = mp(x * g.x - y * g.y, x * g.y + y * g.x);
return *this;
}
G &operator/=(const G &g) {
*this *= g.conjugate();
T n = g.norm();
x = floor(x + n / 2, n);
y = floor(y + n / 2, n);
return *this;
}
G &operator%=(const G &g) {
auto q = G(*this) / g;
q *= g;
(*this) -= q;
return *this;
}
G operator-() { return G(-x, -y); }
G operator+(const G &g) { return G(*this) += g; }
G operator-(const G &g) { return G(*this) -= g; }
G operator*(const G &g) { return G(*this) *= g; }
G operator/(const G &g) { return G(*this) /= g; }
G operator%(const G &g) { return G(*this) %= g; }
bool operator==(const G &g) { return (x == g.x && y == g.y); }
static G gcd(G a, G b) {
while (b.x != 0 || b.y != 0) {
a %= b;
swap(a, b);
}
return a;
}
G pow(ll n) const {
assert(n >= 0);
G ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
// (g,x,y) s.t ax+by=g
static tuple<G, G, G> extgcd(G a, G b) {
if (b.x != 0 || b.y != 0) {
G q = a / b;
auto [g, x, y] = extgcd(b, a - q * b);
return {g, y, x - q * y};
}
return {a, G{1, 0}, G{0, 0}};
}
};
pair<ll, ll> solve_norm_equation_prime(ll p) {
using G = Gaussian_Integer<i128>;
assert(p == 2 || p % 4 == 1);
if (p == 2) return {1, 1};
ll x = [&]() -> ll {
ll x = 1;
while (1) {
++x;
ll pow_x = 1;
if (p < (1 << 30)) {
pow_x = mod_pow(x, (p - 1) / 4, p);
if (pow_x * pow_x % p == p - 1) return pow_x;
} else {
pow_x = mod_pow_64(x, (p - 1) / 4, p);
if (i128(pow_x) * pow_x % p == p - 1) return pow_x;
}
}
return -1;
}();
G a(p, 0), b(x, 1);
a = G::gcd(a, b);
assert(a.norm() == p);
return {a.x, a.y};
}
template <typename T>
vc<Gaussian_Integer<T>> solve_norm_equation_factor(vc<pair<ll, int>> pfs) {
using G = Gaussian_Integer<T>;
vc<G> res;
for (auto &&[p, e]: pfs) {
if (p % 4 == 3 && e % 2 == 1) return {};
}
res.eb(G(1, 0));
for (auto &&[p, e]: pfs) {
if (p % 4 == 3) {
T pp = 1;
FOR(e / 2) pp *= p;
for (auto &&g: res) {
g.x *= pp;
g.y *= pp;
}
continue;
}
auto [pix, piy] = solve_norm_equation_prime(p);
G pi(pix, piy);
vc<G> pows(e + 1);
pows[0] = G(1, 0);
FOR(i, e) pows[i + 1] = pows[i] * pi;
if (p == 2) {
for (auto &&g: res) g *= pows[e];
continue;
}
vc<G> pis(e + 1);
FOR(j, e + 1) { pis[j] = pows[j] * (pows[e - j].conjugate()); }
vc<G> new_res;
new_res.reserve(len(res) * (e + 1));
for (auto &&g: res) {
for (auto &&a: pis) { new_res.eb(g * a); }
}
swap(res, new_res);
}
for (auto &&g: res) {
while (g.x <= 0 || g.y < 0) { g = G(-g.y, g.x); }
}
return res;
}
// i128 を使うと N <= 10^{18} もできる
// ノルムがとれるように、2 乗してもオーバーフローしない型を使おう
// 0 <= arg < 90 となるもののみ返す。
// 単数倍は作らないので、使うときに気を付ける。
template <typename T>
vc<Gaussian_Integer<T>> solve_norm_equation(T N) {
using G = Gaussian_Integer<T>;
vc<G> res;
if (N < 0) return {};
if (N == 0) {
res.eb(G(0, 0));
return res;
}
auto pfs = factor(N);
return solve_norm_equation_factor<T>(pfs);
}
#line 2 "nt/two_square.hpp"
vc<pair<ll, ll>> two_square(ll N, bool only_nonnegative) {
vc<pi> ANS;
if (N == 0) {
ANS.eb(0, 0);
return ANS;
}
if (only_nonnegative) {
for (auto& g: solve_norm_equation<i128>(N)) { ANS.eb(g.x, g.y); }
ll sqN = sqrtl(N);
if (sqN * sqN == N) ANS.eb(0, sqN);
return ANS;
}
for (auto& g: solve_norm_equation<i128>(N)) {
FOR(4) {
ANS.eb(g.x, g.y);
tie(g.x, g.y) = mp(g.y, g.x);
}
}
return ANS;
}
#line 6 "test/2_library_checker/number_theory/two_square.test.cpp"
void solve() {
LL(N);
vc<pi> ANS = two_square(N, true);
print(len(ANS));
for (auto& x: ANS) print(x);
}
signed main() {
INT(T);
FOR(T) solve();
return 0;
}