sum_two_squares.hpp¶
判断并构造 \(n=a^2+b^2\)，同时按二平方和定理统计带符号有序表示数。
\[
\displaystyle n=a^2+b^2
\]
Complexity: Time: Expected integer-factorization time plus O(r log n) to enumerate r representations. Space: O(r + log n).
AC 记录：two_square_sum。
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: Expected integer-factorization time plus O(r log n) to
/// enumerate r representations. Space: O(r + log n).
#include "noya/factorize.hpp"
#include "noya/mod_sqrt.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <optional>
#include <utility>
#include <vector>
namespace noya {
/// @brief Decide and construct n = a^2 + b^2 for an unsigned 64-bit integer;
/// also count ordered signed representations using the two-square theorem.
namespace sum_two_squares_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
using i128 = __int128;
inline u64 integer_sqrt(u64 val) {
u64 rt = u64(std::sqrt(static_cast<long double>(val)));
while (u128(rt + 1) * (rt + 1) <= val) {
rt++;
}
while (u128(rt) * rt > val) {
rt--;
}
return rt;
}
inline std::pair<u64, u64> multiply(std::pair<u64, u64> lhs,
std::pair<u64, u64> rhs) {
i128 re = i128(lhs.first) * rhs.first - i128(lhs.second) * rhs.second;
i128 im = i128(lhs.first) * rhs.second + i128(lhs.second) * rhs.first;
u64 a = u64(re < 0 ? -re : re);
u64 b = u64(im < 0 ? -im : im);
if (a > b) {
std::swap(a, b);
}
return {a, b};
}
inline std::optional<std::pair<u64, u64>> prime_representation(u64 p) {
assert(p % 4 == 1 && is_prime(p));
auto sqr = mod_sqrt(p - 1, p);
assert(sqr.has_value());
for (u64 rt : {*sqr, p - *sqr}) {
u64 pre = p;
u64 cur = rt;
while (u128(cur) * cur > p) {
u64 nxt = pre % cur;
pre = cur;
cur = nxt;
}
u64 sq = p - cur * cur;
u64 oth = integer_sqrt(sq);
if (oth * oth == sq) {
return std::pair<u64, u64>{std::min(cur, oth), std::max(cur, oth)};
}
}
return std::nullopt;
}
inline std::pair<u64, u64> power(std::pair<u64, u64> bas, int exp) {
std::pair<u64, u64> res{1, 0};
while (exp > 0) {
if (exp & 1) {
res = multiply(res, bas);
}
exp >>= 1;
if (exp > 0) {
bas = multiply(bas, bas);
}
}
return res;
}
using gaussian = std::pair<i128, i128>;
inline gaussian signed_multiply(gaussian lhs, gaussian rhs) {
return {lhs.first * rhs.first - lhs.second * rhs.second,
lhs.first * rhs.second + lhs.second * rhs.first};
}
inline gaussian signed_power(gaussian bas, int exp) {
gaussian res{1, 0};
while (exp > 0) {
if (exp & 1) {
res = signed_multiply(res, bas);
}
exp >>= 1;
if (exp > 0) {
bas = signed_multiply(bas, bas);
}
}
return res;
}
} // namespace sum_two_squares_internal
inline bool is_sum_two_squares(std::uint64_t n) {
if (n == 0) {
return true;
}
for (auto [p, exp] : factorize(n)) {
if (p % 4 == 3 && exp % 2 == 1) {
return false;
}
}
return true;
}
/// @brief Return one pair 0 <= a <= b with a^2 + b^2 = n, or nullopt.
inline std::optional<std::pair<std::uint64_t, std::uint64_t>>
sum_two_squares(std::uint64_t n) {
using namespace sum_two_squares_internal;
if (n == 0) {
return std::pair<u64, u64>{0, 0};
}
std::pair<u64, u64> res{1, 0};
for (auto [p, exp] : factorize(n)) {
if (p == 2) {
res = multiply(res, power({1, 1}, exp));
} else if (p % 4 == 1) {
auto rep = prime_representation(p);
assert(rep.has_value());
res = multiply(res, power(*rep, exp));
} else {
if (exp % 2 == 1) {
return std::nullopt;
}
u64 scl = 1;
for (int i = 0; i < exp / 2; i++) {
scl *= p;
}
res = multiply(res, {scl, 0});
}
}
if (res.first > res.second) {
std::swap(res.first, res.second);
}
return res;
}
/// @brief Count integer pairs (a,b), including signs and order, satisfying
/// a^2 + b^2 = n.
inline unsigned __int128 sum_two_squares_representation_count(std::uint64_t n) {
if (n == 0) {
return 1;
}
unsigned __int128 res = 4;
for (auto [p, exp] : factorize(n)) {
if (p % 4 == 3 && exp % 2 == 1) {
return 0;
}
if (p % 4 == 1) {
res *= exp + 1;
}
}
return res;
}
/// @brief Return every ordered non-negative pair (a,b) with a^2 + b^2 = n.
/// The result is sorted and contains no duplicates. Requires n <= 1e18.
inline std::vector<std::pair<std::uint64_t, std::uint64_t>>
all_sum_two_squares(std::uint64_t n) {
using namespace sum_two_squares_internal;
assert(n <= 1000000000000000000ULL);
if (n == 0) {
return {{0, 0}};
}
auto fs = factorize(n);
for (auto [p, exp] : fs) {
if (p % 4 == 3 && exp % 2 == 1) {
return {};
}
}
std::vector<gaussian> rs{{1, 0}};
for (auto [p, exp] : fs) {
if (p % 4 == 3) {
i128 scl = 1;
for (int i = 0; i < exp / 2; i++) {
scl *= p;
}
for (auto &[re, im] : rs) {
re *= scl;
im *= scl;
}
continue;
}
gaussian pf;
if (p == 2) {
pf = {1, 1};
gaussian mul = signed_power(pf, exp);
for (auto &val : rs) {
val = signed_multiply(val, mul);
}
continue;
}
auto rep = prime_representation(p);
assert(rep.has_value());
pf = {rep->first, rep->second};
std::vector<gaussian> pw(exp + 1, {1, 0});
for (int i = 0; i < exp; i++) {
pw[i + 1] = signed_multiply(pw[i], pf);
}
std::vector<gaussian> nxt;
nxt.reserve(rs.size() * (exp + 1));
for (gaussian cur : rs) {
for (int cho = 0; cho <= exp; cho++) {
gaussian cjg = pw[exp - cho];
cjg.second = -cjg.second;
gaussian fct = signed_multiply(pw[cho], cjg);
nxt.push_back(signed_multiply(cur, fct));
}
}
rs.swap(nxt);
}
std::vector<std::pair<u64, u64>> res;
for (auto [re, im] : rs) {
while (re <= 0 || im < 0) {
i128 ore = re;
re = -im;
im = ore;
}
res.emplace_back(u64(re), u64(im));
if (im == 0) {
res.emplace_back(0, u64(re));
}
}
std::sort(res.begin(), res.end());
res.erase(std::unique(res.begin(), res.end()), res.end());
return res;
}
} // namespace noya
#ifndef NOYA_SUM_TWO_SQUARES_HPP
#define NOYA_SUM_TWO_SQUARES_HPP 1
/// @complexity Time: Expected integer-factorization time plus O(r log n) to
/// enumerate r representations. Space: O(r + log n).
#include "noya/factorize.hpp"
#include "noya/mod_sqrt.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <optional>
#include <utility>
#include <vector>
namespace noya {
/// @brief Decide and construct n = a^2 + b^2 for an unsigned 64-bit integer;
/// also count ordered signed representations using the two-square theorem.
namespace sum_two_squares_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
using i128 = __int128;
inline u64 integer_sqrt(u64 val) {
u64 rt = u64(std::sqrt(static_cast<long double>(val)));
while (u128(rt + 1) * (rt + 1) <= val) {
rt++;
}
while (u128(rt) * rt > val) {
rt--;
}
return rt;
}
inline std::pair<u64, u64> multiply(std::pair<u64, u64> lhs,
std::pair<u64, u64> rhs) {
i128 re = i128(lhs.first) * rhs.first - i128(lhs.second) * rhs.second;
i128 im = i128(lhs.first) * rhs.second + i128(lhs.second) * rhs.first;
u64 a = u64(re < 0 ? -re : re);
u64 b = u64(im < 0 ? -im : im);
if (a > b) {
std::swap(a, b);
}
return {a, b};
}
inline std::optional<std::pair<u64, u64>> prime_representation(u64 p) {
assert(p % 4 == 1 && is_prime(p));
auto sqr = mod_sqrt(p - 1, p);
assert(sqr.has_value());
for (u64 rt : {*sqr, p - *sqr}) {
u64 pre = p;
u64 cur = rt;
while (u128(cur) * cur > p) {
u64 nxt = pre % cur;
pre = cur;
cur = nxt;
}
u64 sq = p - cur * cur;
u64 oth = integer_sqrt(sq);
if (oth * oth == sq) {
return std::pair<u64, u64>{std::min(cur, oth), std::max(cur, oth)};
}
}
return std::nullopt;
}
inline std::pair<u64, u64> power(std::pair<u64, u64> bas, int exp) {
std::pair<u64, u64> res{1, 0};
while (exp > 0) {
if (exp & 1) {
res = multiply(res, bas);
}
exp >>= 1;
if (exp > 0) {
bas = multiply(bas, bas);
}
}
return res;
}
using gaussian = std::pair<i128, i128>;
inline gaussian signed_multiply(gaussian lhs, gaussian rhs) {
return {lhs.first * rhs.first - lhs.second * rhs.second,
lhs.first * rhs.second + lhs.second * rhs.first};
}
inline gaussian signed_power(gaussian bas, int exp) {
gaussian res{1, 0};
while (exp > 0) {
if (exp & 1) {
res = signed_multiply(res, bas);
}
exp >>= 1;
if (exp > 0) {
bas = signed_multiply(bas, bas);
}
}
return res;
}
} // namespace sum_two_squares_internal
inline bool is_sum_two_squares(std::uint64_t n) {
if (n == 0) {
return true;
}
for (auto [p, exp] : factorize(n)) {
if (p % 4 == 3 && exp % 2 == 1) {
return false;
}
}
return true;
}
/// @brief Return one pair 0 <= a <= b with a^2 + b^2 = n, or nullopt.
inline std::optional<std::pair<std::uint64_t, std::uint64_t>>
sum_two_squares(std::uint64_t n) {
using namespace sum_two_squares_internal;
if (n == 0) {
return std::pair<u64, u64>{0, 0};
}
std::pair<u64, u64> res{1, 0};
for (auto [p, exp] : factorize(n)) {
if (p == 2) {
res = multiply(res, power({1, 1}, exp));
} else if (p % 4 == 1) {
auto rep = prime_representation(p);
assert(rep.has_value());
res = multiply(res, power(*rep, exp));
} else {
if (exp % 2 == 1) {
return std::nullopt;
}
u64 scl = 1;
for (int i = 0; i < exp / 2; i++) {
scl *= p;
}
res = multiply(res, {scl, 0});
}
}
if (res.first > res.second) {
std::swap(res.first, res.second);
}
return res;
}
/// @brief Count integer pairs (a,b), including signs and order, satisfying
/// a^2 + b^2 = n.
inline unsigned __int128 sum_two_squares_representation_count(std::uint64_t n) {
if (n == 0) {
return 1;
}
unsigned __int128 res = 4;
for (auto [p, exp] : factorize(n)) {
if (p % 4 == 3 && exp % 2 == 1) {
return 0;
}
if (p % 4 == 1) {
res *= exp + 1;
}
}
return res;
}
/// @brief Return every ordered non-negative pair (a,b) with a^2 + b^2 = n.
/// The result is sorted and contains no duplicates. Requires n <= 1e18.
inline std::vector<std::pair<std::uint64_t, std::uint64_t>>
all_sum_two_squares(std::uint64_t n) {
using namespace sum_two_squares_internal;
assert(n <= 1000000000000000000ULL);
if (n == 0) {
return {{0, 0}};
}
auto fs = factorize(n);
for (auto [p, exp] : fs) {
if (p % 4 == 3 && exp % 2 == 1) {
return {};
}
}
std::vector<gaussian> rs{{1, 0}};
for (auto [p, exp] : fs) {
if (p % 4 == 3) {
i128 scl = 1;
for (int i = 0; i < exp / 2; i++) {
scl *= p;
}
for (auto &[re, im] : rs) {
re *= scl;
im *= scl;
}
continue;
}
gaussian pf;
if (p == 2) {
pf = {1, 1};
gaussian mul = signed_power(pf, exp);
for (auto &val : rs) {
val = signed_multiply(val, mul);
}
continue;
}
auto rep = prime_representation(p);
assert(rep.has_value());
pf = {rep->first, rep->second};
std::vector<gaussian> pw(exp + 1, {1, 0});
for (int i = 0; i < exp; i++) {
pw[i + 1] = signed_multiply(pw[i], pf);
}
std::vector<gaussian> nxt;
nxt.reserve(rs.size() * (exp + 1));
for (gaussian cur : rs) {
for (int cho = 0; cho <= exp; cho++) {
gaussian cjg = pw[exp - cho];
cjg.second = -cjg.second;
gaussian fct = signed_multiply(pw[cho], cjg);
nxt.push_back(signed_multiply(cur, fct));
}
}
rs.swap(nxt);
}
std::vector<std::pair<u64, u64>> res;
for (auto [re, im] : rs) {
while (re <= 0 || im < 0) {
i128 ore = re;
re = -im;
im = ore;
}
res.emplace_back(u64(re), u64(im));
if (im == 0) {
res.emplace_back(0, u64(re));
}
}
std::sort(res.begin(), res.end());
res.erase(std::unique(res.begin(), res.end()), res.end());
return res;
}
} // namespace noya
#endif // NOYA_SUM_TWO_SQUARES_HPP
#include <algorithm>
#include <array>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <numeric>
#include <optional>
#include <utility>
#include <vector>
/// @complexity Time: Expected integer-factorization time plus O(r log n) to
/// enumerate r representations. Space: O(r + log n).
/// @complexity Time: O(log^3 n) primality testing; Pollard-rho factorization is expected about O(n^(1/4)).
/// Space: O(log n) recursion and factors.
namespace noya {
namespace factorize_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
inline u64 multiply_mod(u64 a, u64 b, u64 mod) {
return u64(u128(a) * b % mod);
}
inline u64 power_mod(u64 a, u64 exp, u64 mod) {
u64 res = 1;
while (exp > 0) {
if (exp & 1) {
res = multiply_mod(res, a, mod);
}
a = multiply_mod(a, a, mod);
exp >>= 1;
}
return res;
}
inline bool miller_rabin(u64 n) {
if (n < 2) {
return false;
}
for (u64 p :
std::array<u64, 12>{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}) {
if (n % p == 0) {
return n == p;
}
}
int shf = __builtin_ctzll(n - 1);
u64 odd = (n - 1) >> shf;
for (u64 bas :
std::array<u64, 7>{2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
if (bas % n == 0) {
continue;
}
u64 val = power_mod(bas % n, odd, n);
if (val == 1 || val == n - 1) {
continue;
}
bool cmp = true;
for (int i = 1; i < shf; i++) {
val = multiply_mod(val, val, n);
if (val == n - 1) {
cmp = false;
break;
}
}
if (cmp) {
return false;
}
}
return true;
}
inline u64 splitmix64(u64 &st) {
u64 z = (st += 0x9e3779b97f4a7c15ULL);
z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9ULL;
z = (z ^ (z >> 27)) * 0x94d049bb133111ebULL;
return z ^ (z >> 31);
}
inline u64 pollard_rho(u64 n) {
if (n % 2 == 0) {
return 2;
}
if (n % 3 == 0) {
return 3;
}
static u64 st = 0x123456789abcdef0ULL;
while (true) {
u64 y = splitmix64(st) % (n - 1) + 1;
u64 c = splitmix64(st) % (n - 1) + 1;
constexpr u64 blk = 128;
u64 g = 1;
u64 r = 1;
u64 q = 1;
u64 x = 0;
u64 sy = 0;
auto nxt = [&](u64 val) {
return u64((u128(multiply_mod(val, val, n)) + c) % n);
};
while (g == 1) {
x = y;
for (u64 i = 0; i < r; i++) {
y = nxt(y);
}
for (u64 off = 0; off < r && g == 1; off += blk) {
sy = y;
for (u64 i = 0; i < std::min(blk, r - off); i++) {
y = nxt(y);
u64 dif = x > y ? x - y : y - x;
q = multiply_mod(q, dif, n);
}
g = std::gcd(q, n);
}
r <<= 1;
}
if (g == n) {
do {
sy = nxt(sy);
u64 dif = x > sy ? x - sy : sy - x;
g = std::gcd(dif, n);
} while (g == 1);
}
if (g != n) {
return g;
}
}
}
inline void collect_factors(u64 n, std::vector<u64> &res) {
if (n == 1) {
return;
}
if (miller_rabin(n)) {
res.push_back(n);
return;
}
u64 fct = pollard_rho(n);
collect_factors(fct, res);
collect_factors(n / fct, res);
}
} // namespace factorize_internal
/// @brief Deterministic Miller-Rabin primality test for unsigned 64-bit
/// integers.
inline bool is_prime(std::uint64_t n) {
return factorize_internal::miller_rabin(n);
}
/// @brief Return the prime factors of n with multiplicity in increasing order.
inline std::vector<std::uint64_t> prime_factors(std::uint64_t n) {
assert(n >= 1);
std::vector<std::uint64_t> res;
factorize_internal::collect_factors(n, res);
std::sort(res.begin(), res.end());
return res;
}
/// @brief Return the prime factorization of n as (prime, exponent) pairs.
inline std::vector<std::pair<std::uint64_t, int>> factorize(std::uint64_t n) {
std::vector<std::pair<std::uint64_t, int>> res;
for (std::uint64_t p : prime_factors(n)) {
if (res.empty() || res.back().first != p) {
res.emplace_back(p, 1);
} else {
res.back().second++;
}
}
return res;
}
} // namespace noya
/// @complexity Time: O(log^2 p).
/// Space: O(1).
namespace noya {
/// @brief Compute the smaller square root modulo a prime, or nullopt if no
/// square root exists.
inline std::optional<std::uint64_t> mod_sqrt(std::uint64_t val,
std::uint64_t mod) {
assert(mod >= 2 && is_prime(mod));
val %= mod;
if (mod == 2 || val == 0) {
return val;
}
using factorize_internal::multiply_mod;
using factorize_internal::power_mod;
if (power_mod(val, (mod - 1) / 2, mod) != 1) {
return std::nullopt;
}
if (mod % 4 == 3) {
std::uint64_t rt = power_mod(val, (mod + 1) / 4, mod);
return std::min(rt, mod - rt);
}
std::uint64_t odd = mod - 1;
int exp = 0;
while ((odd & 1) == 0) {
odd >>= 1;
exp++;
}
std::uint64_t nqr = 2;
while (power_mod(nqr, (mod - 1) / 2, mod) != mod - 1) {
nqr++;
}
std::uint64_t rt = power_mod(val, (odd + 1) / 2, mod);
std::uint64_t rem = power_mod(val, odd, mod);
std::uint64_t ste = power_mod(nqr, odd, mod);
int rmn = exp;
while (rem != 1) {
std::uint64_t squ = rem;
int shf = 0;
while (squ != 1 && shf < rmn) {
squ = multiply_mod(squ, squ, mod);
shf++;
}
assert(shf < rmn);
std::uint64_t mul =
power_mod(ste, std::uint64_t(1) << (rmn - shf - 1), mod);
rt = multiply_mod(rt, mul, mod);
ste = multiply_mod(mul, mul, mod);
rem = multiply_mod(rem, ste, mod);
rmn = shf;
}
return std::min(rt, mod - rt);
}
} // namespace noya
namespace noya {
/// @brief Decide and construct n = a^2 + b^2 for an unsigned 64-bit integer;
/// also count ordered signed representations using the two-square theorem.
namespace sum_two_squares_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
using i128 = __int128;
inline u64 integer_sqrt(u64 val) {
u64 rt = u64(std::sqrt(static_cast<long double>(val)));
while (u128(rt + 1) * (rt + 1) <= val) {
rt++;
}
while (u128(rt) * rt > val) {
rt--;
}
return rt;
}
inline std::pair<u64, u64> multiply(std::pair<u64, u64> lhs,
std::pair<u64, u64> rhs) {
i128 re = i128(lhs.first) * rhs.first - i128(lhs.second) * rhs.second;
i128 im = i128(lhs.first) * rhs.second + i128(lhs.second) * rhs.first;
u64 a = u64(re < 0 ? -re : re);
u64 b = u64(im < 0 ? -im : im);
if (a > b) {
std::swap(a, b);
}
return {a, b};
}
inline std::optional<std::pair<u64, u64>> prime_representation(u64 p) {
assert(p % 4 == 1 && is_prime(p));
auto sqr = mod_sqrt(p - 1, p);
assert(sqr.has_value());
for (u64 rt : {*sqr, p - *sqr}) {
u64 pre = p;
u64 cur = rt;
while (u128(cur) * cur > p) {
u64 nxt = pre % cur;
pre = cur;
cur = nxt;
}
u64 sq = p - cur * cur;
u64 oth = integer_sqrt(sq);
if (oth * oth == sq) {
return std::pair<u64, u64>{std::min(cur, oth), std::max(cur, oth)};
}
}
return std::nullopt;
}
inline std::pair<u64, u64> power(std::pair<u64, u64> bas, int exp) {
std::pair<u64, u64> res{1, 0};
while (exp > 0) {
if (exp & 1) {
res = multiply(res, bas);
}
exp >>= 1;
if (exp > 0) {
bas = multiply(bas, bas);
}
}
return res;
}
using gaussian = std::pair<i128, i128>;
inline gaussian signed_multiply(gaussian lhs, gaussian rhs) {
return {lhs.first * rhs.first - lhs.second * rhs.second,
lhs.first * rhs.second + lhs.second * rhs.first};
}
inline gaussian signed_power(gaussian bas, int exp) {
gaussian res{1, 0};
while (exp > 0) {
if (exp & 1) {
res = signed_multiply(res, bas);
}
exp >>= 1;
if (exp > 0) {
bas = signed_multiply(bas, bas);
}
}
return res;
}
} // namespace sum_two_squares_internal
inline bool is_sum_two_squares(std::uint64_t n) {
if (n == 0) {
return true;
}
for (auto [p, exp] : factorize(n)) {
if (p % 4 == 3 && exp % 2 == 1) {
return false;
}
}
return true;
}
/// @brief Return one pair 0 <= a <= b with a^2 + b^2 = n, or nullopt.
inline std::optional<std::pair<std::uint64_t, std::uint64_t>>
sum_two_squares(std::uint64_t n) {
using namespace sum_two_squares_internal;
if (n == 0) {
return std::pair<u64, u64>{0, 0};
}
std::pair<u64, u64> res{1, 0};
for (auto [p, exp] : factorize(n)) {
if (p == 2) {
res = multiply(res, power({1, 1}, exp));
} else if (p % 4 == 1) {
auto rep = prime_representation(p);
assert(rep.has_value());
res = multiply(res, power(*rep, exp));
} else {
if (exp % 2 == 1) {
return std::nullopt;
}
u64 scl = 1;
for (int i = 0; i < exp / 2; i++) {
scl *= p;
}
res = multiply(res, {scl, 0});
}
}
if (res.first > res.second) {
std::swap(res.first, res.second);
}
return res;
}
/// @brief Count integer pairs (a,b), including signs and order, satisfying
/// a^2 + b^2 = n.
inline unsigned __int128 sum_two_squares_representation_count(std::uint64_t n) {
if (n == 0) {
return 1;
}
unsigned __int128 res = 4;
for (auto [p, exp] : factorize(n)) {
if (p % 4 == 3 && exp % 2 == 1) {
return 0;
}
if (p % 4 == 1) {
res *= exp + 1;
}
}
return res;
}
/// @brief Return every ordered non-negative pair (a,b) with a^2 + b^2 = n.
/// The result is sorted and contains no duplicates. Requires n <= 1e18.
inline std::vector<std::pair<std::uint64_t, std::uint64_t>>
all_sum_two_squares(std::uint64_t n) {
using namespace sum_two_squares_internal;
assert(n <= 1000000000000000000ULL);
if (n == 0) {
return {{0, 0}};
}
auto fs = factorize(n);
for (auto [p, exp] : fs) {
if (p % 4 == 3 && exp % 2 == 1) {
return {};
}
}
std::vector<gaussian> rs{{1, 0}};
for (auto [p, exp] : fs) {
if (p % 4 == 3) {
i128 scl = 1;
for (int i = 0; i < exp / 2; i++) {
scl *= p;
}
for (auto &[re, im] : rs) {
re *= scl;
im *= scl;
}
continue;
}
gaussian pf;
if (p == 2) {
pf = {1, 1};
gaussian mul = signed_power(pf, exp);
for (auto &val : rs) {
val = signed_multiply(val, mul);
}
continue;
}
auto rep = prime_representation(p);
assert(rep.has_value());
pf = {rep->first, rep->second};
std::vector<gaussian> pw(exp + 1, {1, 0});
for (int i = 0; i < exp; i++) {
pw[i + 1] = signed_multiply(pw[i], pf);
}
std::vector<gaussian> nxt;
nxt.reserve(rs.size() * (exp + 1));
for (gaussian cur : rs) {
for (int cho = 0; cho <= exp; cho++) {
gaussian cjg = pw[exp - cho];
cjg.second = -cjg.second;
gaussian fct = signed_multiply(pw[cho], cjg);
nxt.push_back(signed_multiply(cur, fct));
}
}
rs.swap(nxt);
}
std::vector<std::pair<u64, u64>> res;
for (auto [re, im] : rs) {
while (re <= 0 || im < 0) {
i128 ore = re;
re = -im;
im = ore;
}
res.emplace_back(u64(re), u64(im));
if (im == 0) {
res.emplace_back(0, u64(re));
}
}
std::sort(res.begin(), res.end());
res.erase(std::unique(res.begin(), res.end()), res.end());
return res;
}
} // namespace noya