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#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(x) & 1 ? -1 : 1); } int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(ll x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u64 x) { return (__builtin_parity(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 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); } #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; }