CP-Algorithms Library
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tests/two_squares.cpp
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- Last update: 2026-10-16 14:16:08+02:00
Depends on
cp-algo/math/common.hpp- cp-algo/number_theory/euler.hpp
- cp-algo/number_theory/factorize.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/number_theory/primality.hpp
- cp-algo/number_theory/two_squares.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/complex.hpp
- cp-algo/util/generator.hpp
- cp-algo/util/simd.hpp
Code
#include "cp-algo/number_theory/two_squares.hpp"
#include <set>
using namespace cp_algo::math;
void check(int64_t n) {
std::set<std::pair<int64_t, int64_t>> want, got;
for(int64_t x = 0, y = int64_t(std::sqrt(n)); x * x <= n; x++) {
while(x * x + y * y > n) {y--;}
if(x * x + y * y == n) {want.emplace(x, y);}
}
for(auto z: two_squares_all(n)) {
assert(z.real() >= 0 && z.imag() >= 0 && z.norm() == n);
got.emplace(z.real(), z.imag());
}
assert(got == want);
}
int main() {
cp_algo::random::gen.seed(71283);
for(int64_t n = 0; n <= 2500; n++) {check(n);}
for(int64_t n: {3125, 28125, 400000, 422500, 1267500}) {check(n);}
std::cout << "Sums of two squares passed exhaustive enumeration checks\n";
}
#line 1 "cp-algo/number_theory/two_squares.hpp"
#line 1 "cp-algo/number_theory/euler.hpp"
#line 1 "cp-algo/number_theory/factorize.hpp"
#line 1 "cp-algo/number_theory/primality.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#include <bit>
#include <vector>
#include <algorithm>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
// Nonnegative 64-bit exponents, with an associative operation and its identity.
// Windows >1 precompute odd powers only when that saves operations.
template<int window = 1>
auto bpow(auto const& x, auto n, auto const& one, auto op) {
static_assert(window >= 1 && window <= 6);
if constexpr(window > 1) {
if(n == 0) {return one;}
int bits = std::bit_width(uint64_t(n));
auto low_bit = [&](int high) {
int low = std::max(0, high - window + 1);
while(!((n >> low) & 1)) {low++;}
return low;
};
int first = low_bit(bits - 1);
int cost = (1 << (window - 1)) + first;
for(int j = first - 1; j >= 0;) {
if(!((n >> j) & 1)) {j--;}
else {cost++; j = low_bit(j) - 1;}
}
// Do not pay for the table when binary powering uses fewer operations.
if(cost >= bits + std::popcount(uint64_t(n)) - 2) {return bpow<1>(x, n, one, op);}
using T = std::decay_t<decltype(x)>;
std::vector<T> odd;
odd.reserve(1 << (window - 1));
odd.push_back(x);
auto square = op(x, x);
while(odd.size() < size_t(1 << (window - 1))) {odd.push_back(op(odd.back(), square));}
auto ans = odd[(n >> first) / 2];
for(int j = first - 1; j >= 0;) {
if(!((n >> j) & 1)) {ans = op(ans, ans); j--;}
else {
int low = low_bit(j), length = j - low + 1;
auto digit = (n >> low) & ((1u << length) - 1);
for(int i = 0; i < length; i++) {ans = op(ans, ans);}
ans = op(ans, odd[digit / 2]);
j = low - 1;
}
}
return ans;
} else {
if (n == 0) {
return one;
}
auto ans = x;
for(int j = std::bit_width<uint64_t>(n) - 2; ~j; j--) {
ans = op(ans, ans);
if((n >> j) & 1) {
ans = op(ans, x);
}
}
return ans;
}
}
template<int window = 1>
auto bpow(auto x, auto n, auto ans) {
return bpow<window>(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
}
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
constexpr static Int mod() {
return modint::mod();
}
constexpr static UInt remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
constexpr modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator-() const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator/=(const modint& t) {
return to_modint() *= t.inv();
}
modint& operator*=(const modint& t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator+=(const modint& t) {
r += t.r;
r = std::min(r, r - remod());
return to_modint();
}
modint& operator-=(const modint& t) {
r -= t.r;
r = std::min(r, r + remod());
return to_modint();
}
modint operator+(const modint& t) const { return modint(to_modint()) += t; }
modint operator-(const modint& t) const { return modint(to_modint()) -= t; }
modint operator*(const modint& t) const { return modint(to_modint()) *= t; }
modint operator/(const modint& t) const { return modint(to_modint()) /= t; }
// Why <=> doesn't work?..
auto operator==(const modint& t) const { return to_modint().getr() == t.getr(); }
auto operator!=(const modint& t) const { return to_modint().getr() != t.getr(); }
auto operator<=(const modint& t) const { return to_modint().getr() <= t.getr(); }
auto operator>=(const modint& t) const { return to_modint().getr() >= t.getr(); }
auto operator<(const modint& t) const { return to_modint().getr() < t.getr(); }
auto operator>(const modint& t) const { return to_modint().getr() > t.getr(); }
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static uint64_t modmod8() { return uint64_t(8 * modmod()); }
void add_unsafe(UInt t) { r += t; }
void pseudonormalize() { r = std::min(r, r - modmod8()); }
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) { r = rr; }
UInt getr_direct() const { return r; }
protected:
UInt r;
private:
constexpr modint& to_modint() { return static_cast<modint&>(*this); }
constexpr modint const& to_modint() const { return static_cast<modint const&>(*this); }
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator>>(decltype(std::cin)& in, modint& x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator<<(decltype(std::cout)& out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() { return m; }
static constexpr Base::UInt remod() { return m; }
auto getr() const { return Base::r; }
};
// Odd moduli up to a quarter of the unsigned word keep Montgomery residues lazily in
// [0, 2 mod): remod() = 2 mod, and both 4 mod and ab + q * mod fit. Any other modulus keeps
// fully reduced residues, remod() = mod, which is what the sums of modint_base need then:
// an even one without the Montgomery form, a wide odd one with a reduction that subtracts
// high words instead of adding double words that would overflow.
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
// Out of line, so that the hot path stays as small as it was.
[[gnu::noinline, gnu::cold]] static Base::UInt m_reduce_reduced(Base::UInt2 ab) {
if(mod() % 2 == 0) {
return typename Base::UInt(ab % mod());
}
// q * mod has the low word of ab, so the difference of the high words is exact.
typename Base::UInt q = -(typename Base::UInt(ab) * inverse);
auto high = typename Base::UInt(ab >> Base::bits);
auto low = typename Base::UInt(typename Base::UInt2(q) * typename Base::UInt(mod()) >> Base::bits);
return high >= low ? high - low : high - low + mod();
}
static Base::UInt m_reduce(Base::UInt2 ab) {
if(imod() == 0) [[unlikely]] {
return m_reduce_reduced(ab);
}
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator*=(const dynamic_modint& t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() { return m; }
static Base::UInt remod() { return rm; }
static Base::UInt imod() { return im; }
static Base::UInt2 pw128() { return r2; }
static void switch_mod(Int nm) {
m = nm;
bool lazy = m % 2 && typename Base::UInt(m) <= typename Base::UInt(-1) / 4;
rm = typename Base::UInt(m) * (lazy ? 2 : 1);
inverse = m % 2 ? inv2(-m) : 0;
im = lazy ? inverse : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() { switch_mod(prev); }
} _;
switch_mod(tmp);
return callback();
}
private:
static inline thread_local Int m = 1;
// im: -1 / mod modulo 2^bits for lazy residues and 0 for reduced ones; inverse: the same
// for every odd mod; rm: the value of remod().
static inline thread_local Base::UInt im = -1, r2 = 0, inverse = -1, rm = 2;
};
}
#line 6 "cp-algo/number_theory/primality.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
template<typename _Int>
bool is_prime(_Int m) {
using Int = std::make_signed_t<_Int>;
using UInt = std::make_unsigned_t<Int>;
if(m == 1 || m % 2 == 0) {
return m == 2;
}
// m - 1 = 2^s * d
int s = std::countr_zero(UInt(m - 1));
auto d = (m - 1) >> s;
using base = dynamic_modint<Int>;
auto test = [&](base x) {
x = bpow(x, d);
if(std::abs(x.rem()) <= 1) {
return true;
}
for(int i = 1; i < s && x != -1; i++) {
x *= x;
}
return x == -1;
};
return base::with_mod(m, [&]() {
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
uint16_t base2 = 7, base3 = 61;
if (m != uint32_t(m)) {
base2 = base_table1[uint32_t(m * 0xAD625B89) >> 18];
base3 = base_table2[base2 >> 13];
}
return test(2) && test(base2) && test(base3);
#else
return std::ranges::all_of(std::array{2, 325, 9375, 28178, 450775, 9780504, 1795265022}, test);
#endif
});
}
}
#line 1 "cp-algo/util/generator.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#line 10 "cp-algo/util/big_alloc.hpp"
#include <string>
#include <cstddef>
#line 14 "cp-algo/util/big_alloc.hpp"
#include <forward_list>
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#define CP_ALGO_USE_MMAP 1
#include <sys/mman.h>
#else
#define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template<typename T, size_t Align = 32>
class big_alloc {
static_assert(Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::has_single_bit(Align), "Align must be a power of two");
public:
using value_type = T;
template<class U>
struct rebind {
using other = big_alloc<U, Align>;
};
constexpr bool operator==(const big_alloc&) const = default;
big_alloc() noexcept = default;
template<typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded, PROT_READ | PROT_WRITE, MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if(!p) {
return;
}
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
munmap(p, padded);
return;
}
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
template<typename T>
using big_vector = std::vector<T, big_alloc<T>>;
template<typename T>
using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
template<typename T>
using big_deque = std::deque<T, big_alloc<T>>;
template<typename T>
using big_stack = std::stack<T, big_deque<T>>;
template<typename T>
using big_queue = std::queue<T, big_deque<T>>;
template<typename T>
using big_priority_queue = std::priority_queue<T, big_vector<T>>;
template<typename T>
using big_forward_list = std::forward_list<T, big_alloc<T>>;
using big_string = big_basic_string<char>;
template<typename Key, typename Value, typename Compare = std::less<Key>>
using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
template<typename T, typename Compare = std::less<T>>
using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
template<typename T, typename Compare = std::less<T>>
using big_set = std::set<T, Compare, big_alloc<T>>;
}
#line 5 "cp-algo/util/generator.hpp"
#include <generator>
namespace cp_algo {
template<typename Ref, typename V = void>
using big_generator = std::generator<Ref, V, big_alloc<std::byte>>;
}
namespace std::ranges {
template<typename Ref, typename V>
elements_of(cp_algo::big_generator<Ref, V>&&) -> elements_of<cp_algo::big_generator<Ref, V>&&, cp_algo::big_alloc<std::byte>>;
}
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random {
std::mt19937_64 gen(
std::chrono::steady_clock::now().time_since_epoch().count()
);
uint64_t rng() {
return gen();
}
}
#line 6 "cp-algo/number_theory/factorize.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
template<typename _Int>
auto proper_divisor(_Int m) {
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return m % 2 == 0 ? 2 : base::with_mod(m, [&]() {
base t = random::rng();
auto f = [&](auto x) {
return x * x + t;
};
const base zero = 0, one = 1;
base x = zero, y = zero;
base g = one;
while(g == one) {
for(int i = 0; i < 64; i++) {
x = f(x);
y = f(f(y));
if(x == y) [[unlikely]] {
t = random::rng();
x = y = zero;
} else {
base t = g * (x - y);
g = t == zero ? g : t;
}
}
g = std::gcd(g.getr(), m);
}
return g.getr();
});
}
template<typename Int>
big_generator<Int> factorize(Int m) {
if(is_prime(m)) {
co_yield m;
} else if(m > 1) {
auto g = proper_divisor(m);
co_yield std::ranges::elements_of(factorize(g));
co_yield std::ranges::elements_of(factorize(m / g));
}
}
template<typename Int>
big_generator<Int> divisors_sqrt(Int m) {
for(Int i = 1; i * i <= m; i++) {
if(m % i == 0) {
co_yield i;
if(i * i != m) {
co_yield m / i;
}
}
}
}
}
#line 5 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math {
auto euler_phi(auto m) {
using T = std::decay_t<decltype(m)>;
auto primes = to<big_vector<T>>(factorize(m));
std::ranges::sort(primes);
auto [from, to] = std::ranges::unique(primes);
primes.erase(from, to);
auto ans = m;
for(auto it: primes) {
ans -= ans / it;
}
return ans;
}
template<modint_type base>
auto period(base x) {
auto ans = euler_phi(base::mod());
base x0 = bpow(x, ans);
for(auto t: factorize(ans)) {
while(ans % t == 0 && x0 * bpow(x, ans / t) == x0) {
ans /= t;
}
}
return ans;
}
template<typename _Int>
_Int primitive_root(_Int p) {
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return base::with_mod(p, [p](){
base t = 1;
while(period(t) != p - 1) {
t = random::rng();
}
return t.getr();
});
}
}
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2,fma\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using u64x8 = simd<uint64_t, 8>;
using u32x16 = simd<uint32_t, 16>;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using u16x16 = simd<uint16_t, 16>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using u16x8 = simd<uint16_t, 8>;
using u16x4 = simd<uint16_t, 4>;
using i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using u8x16 = simd<uint8_t, 16>;
using u8x8 = simd<uint8_t, 8>;
using u8x4 = simd<uint8_t, 4>;
using dx4 = simd<double, 4>;
inline dx4 abs(dx4 a) {
return dx4{
std::abs(a[0]),
std::abs(a[1]),
std::abs(a[2]),
std::abs(a[3])
};
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
auto x_ninv = u64x4(u32x8(low32(x)) * imod);
x += x_ninv * uint64_t(mod);
#endif
return swap_bytes(x);
}
// x - mod where x >= mod, for x in [0, 2 mod). Unsigned, so it holds up to mod < 2^31;
// zero upper halves of 64-bit lanes stay zero.
inline u32x8 reduce_once(u32x8 x, uint32_t mod) {
auto y = x - mod;
return x < y ? x : y;
}
inline u64x4 reduce_once(u64x4 x, uint32_t mod) {
return u64x4(reduce_once(u32x8(x), mod));
}
inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
return montgomery_reduce(x * y, mod, imod);
#endif
}
inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
}
inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
template<std::size_t Align = 32>
inline bool is_aligned(const auto *p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#pragma GCC pop_options
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
// Custom implementation, since std::complex is UB on non-floating types
template<typename T>
struct complex {
using value_type = T;
T x, y;
inline constexpr complex(): x(), y() {}
inline constexpr complex(T const& x): x(x), y() {}
inline constexpr complex(T const& x, T const& y): x(x), y(y) {}
inline complex& operator *= (T const& t) {x *= t; y *= t; return *this;}
inline complex& operator /= (T const& t) {x /= t; y /= t; return *this;}
inline complex operator * (T const& t) const {return complex(*this) *= t;}
inline complex operator / (T const& t) const {return complex(*this) /= t;}
inline complex& operator += (complex const& t) {x += t.x; y += t.y; return *this;}
inline complex& operator -= (complex const& t) {x -= t.x; y -= t.y; return *this;}
inline complex operator * (complex const& t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
inline complex operator / (complex const& t) const {return *this * t.conj() / t.norm();}
inline complex operator + (complex const& t) const {return complex(*this) += t;}
inline complex operator - (complex const& t) const {return complex(*this) -= t;}
inline complex& operator *= (complex const& t) {return *this = *this * t;}
inline complex& operator /= (complex const& t) {return *this = *this / t;}
inline complex operator - () const {return {-x, -y};}
inline complex conj() const {return {x, -y};}
inline T norm() const {return x * x + y * y;}
inline T abs() const {return std::sqrt(norm());}
inline T const real() const {return x;}
inline T const imag() const {return y;}
inline T& real() {return x;}
inline T& imag() {return y;}
inline static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
inline auto operator <=> (complex const& t) const = default;
};
template<typename T> inline complex<T> conj(complex<T> const& x) {return x.conj();}
template<typename T> inline T norm(complex<T> const& x) {return x.norm();}
template<typename T> inline T abs(complex<T> const& x) {return x.abs();}
template<typename T> inline T& real(complex<T> &x) {return x.real();}
template<typename T> inline T& imag(complex<T> &x) {return x.imag();}
template<typename T> inline T const real(complex<T> const& x) {return x.real();}
template<typename T> inline T const imag(complex<T> const& x) {return x.imag();}
template<typename T>
inline constexpr complex<T> polar(T r, T theta) {
return complex<T>::polar(r, theta);
}
template<typename T>
inline std::ostream& operator << (std::ostream &out, complex<T> const& x) {
return out << x.real() << ' ' << x.imag();
}
}
#pragma GCC pop_options
#line 7 "cp-algo/number_theory/two_squares.hpp"
#include <utility>
#line 10 "cp-algo/number_theory/two_squares.hpp"
namespace cp_algo::math {
template<typename T>
using gaussint = complex<T>;
template<typename _Int>
auto two_squares_prime_any(_Int p) {
if(p == 2) {
return gaussint<_Int>{1, 1};
}
assert(p % 4 == 1);
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return base::with_mod(p, [&](){
base g = primitive_root(p);
int64_t i = bpow(g, (p - 1) / 4).getr();
int64_t q0 = 1, q1 = 0;
int64_t r = i, m = p;
// TODO: Use library contfrac?
do {
int64_t d = r / m;
q0 = std::exchange(q1, q0 + d * q1);
r = std::exchange(m, r % m);
} while(q1 < p / q1);
return gaussint<_Int>{q0, (base(i) * base(q0)).rem()};
});
}
template<typename Int>
big_vector<gaussint<Int>> two_squares_all(Int n) {
if(n == 0) {
return {0};
}
auto primes = factorize(n);
big_map<Int, int> cnt;
for(auto p: primes) {
cnt[p]++;
}
big_vector<gaussint<Int>> res = {1};
for(auto [p, c]: cnt) {
if(p % 4 == 3 && c % 2) {
return {};
}
if(p % 4 != 1) {
auto mul = p == 2 ? bpow(gaussint<Int>(1, 1), c) : bpow(gaussint<Int>(p), c / 2);
for(auto &value: res) {
value *= mul;
}
} else {
big_vector<gaussint<Int>> nres;
auto base = two_squares_prime_any(p);
for(int i = 0; i <= c; i++) {
auto mul = bpow(base, i) * bpow(conj(base), c - i);
for(auto value: res) {
nres.push_back(value * mul);
}
}
res = std::move(nres);
}
}
big_vector<gaussint<Int>> nres;
for(auto p: res) {
while(p.real() < 0 || p.imag() < 0) {
p *= gaussint<Int>(0, 1);
}
nres.push_back(p);
if(!p.real() || !p.imag()) {
nres.emplace_back(p.imag(), p.real());
}
}
return nres;
}
}
#line 3 "tests/two_squares.cpp"
using namespace cp_algo::math;
void check(int64_t n) {
std::set<std::pair<int64_t, int64_t>> want, got;
for(int64_t x = 0, y = int64_t(std::sqrt(n)); x * x <= n; x++) {
while(x * x + y * y > n) {y--;}
if(x * x + y * y == n) {want.emplace(x, y);}
}
for(auto z: two_squares_all(n)) {
assert(z.real() >= 0 && z.imag() >= 0 && z.norm() == n);
got.emplace(z.real(), z.imag());
}
assert(got == want);
}
int main() {
cp_algo::random::gen.seed(71283);
for(int64_t n = 0; n <= 2500; n++) {check(n);}
for(int64_t n: {3125, 28125, 400000, 422500, 1267500}) {check(n);}
std::cout << "Sums of two squares passed exhaustive enumeration checks\n";
}
#line 1 "cp-algo/number_theory/two_squares.hpp"
#line 1 "cp-algo/number_theory/euler.hpp"
#line 1 "cp-algo/number_theory/factorize.hpp"
#line 1 "cp-algo/number_theory/primality.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#include <bit>
#include <vector>
#include <algorithm>
namespace cp_algo::math{
#ifdef CP_ALGO_MAXN
const int maxn=CP_ALGO_MAXN;
#else
const int maxn=1<<19;
#endif
const int magic=64; template<int window=1> auto bpow(auto const&x,auto n,auto const&one,auto op){ static_assert(window>=1&&window<=6); if constexpr(window>1){ if(n==0){return one;} int bits=std::bit_width(uint64_t(n)); auto low_bit= [&](int high){ int low=std::max(0,high-window+1); while(!((n>>low)&1)){low++;} return low; }; int first=low_bit(bits-1); int cost=(1<<(window-1))+first; for(int j=first-1;j>=0;){ if(!((n>>j)&1)){j--;} else{cost++;j=low_bit(j)-1;} } if(cost>=bits+std::popcount(uint64_t(n))-2){return bpow<1>(x,n,one,op);} using T=std::decay_t<decltype(x)>; std::vector<T>odd; odd.reserve(1<<(window-1)); odd.push_back(x); auto square=op(x,x); while(odd.size()<size_t(1<<(window-1))){odd.push_back(op(odd.back(),square));} auto ans=odd[(n>>first)/2]; for(int j=first-1;j>=0;){ if(!((n>>j)&1)){ans=op(ans,ans);j--;} else{ int low=low_bit(j),length=j-low+1; auto digit=(n>>low)&((1u<<length)-1); for(int i=0;i<length;i++){ans=op(ans,ans);} ans=op(ans,odd[digit/2]); j=low-1; } } return ans; }else{ if(n==0){ return one; } auto ans=x; for(int j=std::bit_width<uint64_t>(n)-2;~j;j--){ ans=op(ans,ans); if((n>>j)&1){ ans=op(ans,x); } } return ans; } } template<int window=1> auto bpow(auto x,auto n,auto ans){ return bpow<window>(x,n,ans,std::multiplies{}); } template<typename T> T bpow(T const&x,auto n){ return bpow(x,n,T(1)); } inline constexpr auto inv2(auto x){ assert(x%2); std::make_unsigned_t<decltype(x)>y=1; while(y*x!=1){ y*=2-x*y; } return y; } }
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math{ template<typename modint,typename _Int> struct modint_base{ using Int=_Int; using UInt=std::make_unsigned_t<Int>; static constexpr size_t bits=sizeof(Int)*8; using Int2=std::conditional_t<bits<=32,int64_t,__int128_t>; using UInt2=std::conditional_t<bits<=32,uint64_t,__uint128_t>; constexpr static Int mod(){ return modint::mod(); } constexpr static UInt remod(){ return modint::remod(); } constexpr static UInt2 modmod(){ return UInt2(mod())*mod(); } constexpr modint_base()=default; constexpr modint_base(Int2 rr){ to_modint().setr(UInt((rr+modmod())%mod())); } constexpr modint inv()const{ return bpow(to_modint(),mod()-2); } modint operator-()const{ modint neg; neg.r=std::min(-r,remod()-r); return neg; } modint&operator/=(const modint&t){ return to_modint()*=t.inv(); } modint&operator*=(const modint&t){ r=UInt(UInt2(r)*t.r%mod()); return to_modint(); } modint&operator+=(const modint&t){ r+=t.r; r=std::min(r,r-remod()); return to_modint(); } modint&operator-=(const modint&t){ r-=t.r; r=std::min(r,r+remod()); return to_modint(); } modint operator+(const modint&t)const{return modint(to_modint())+=t;} modint operator-(const modint&t)const{return modint(to_modint())-=t;} modint operator*(const modint&t)const{return modint(to_modint())*=t;} modint operator/(const modint&t)const{return modint(to_modint())/=t;} auto operator==(const modint&t)const{return to_modint().getr()==t.getr();} auto operator!=(const modint&t)const{return to_modint().getr()!=t.getr();} auto operator<=(const modint&t)const{return to_modint().getr()<=t.getr();} auto operator>=(const modint&t)const{return to_modint().getr()>=t.getr();} auto operator<(const modint&t)const{return to_modint().getr()<t.getr();} auto operator>(const modint&t)const{return to_modint().getr()>t.getr();} Int rem()const{ UInt R=to_modint().getr(); return R-(R>(UInt)mod()/2)*mod(); } constexpr void setr(UInt rr){ r=rr; } constexpr UInt getr()const{ return r; } static uint64_t modmod8(){return uint64_t(8*modmod());} void add_unsafe(UInt t){r+=t;} void pseudonormalize(){r=std::min(r,r-modmod8());} modint const&normalize(){ if(r>=(UInt)mod()){ r%=mod(); } return to_modint(); } void setr_direct(UInt rr){r=rr;} UInt getr_direct()const{return r;} protected: UInt r; private: constexpr modint&to_modint(){return static_cast<modint&>(*this);} constexpr modint const&to_modint()const{return static_cast<modint const&>(*this);} }; template<typename modint> concept modint_type=std::is_base_of_v<modint_base<modint,typename modint::Int>,modint>; template<modint_type modint> decltype(std::cin)&operator>>(decltype(std::cin)&in,modint&x){ typename modint::UInt r; auto&res=in>>r; x.setr(r); return res; } template<modint_type modint> decltype(std::cout)&operator<<(decltype(std::cout)&out,modint const&x){ return out<<x.getr(); } template<auto m> struct modint:modint_base<modint<m>,decltype(m)>{ using Base=modint_base<modint<m>,decltype(m)>; using Base::Base; static constexpr Base::Int mod(){return m;} static constexpr Base::UInt remod(){return m;} auto getr()const{return Base::r;} }; template<typename Int=int> struct dynamic_modint:modint_base<dynamic_modint<Int>,Int>{ using Base=modint_base<dynamic_modint<Int>,Int>; using Base::Base; [[gnu::noinline,gnu::cold]]static Base::UInt m_reduce_reduced(Base::UInt2 ab){ if(mod()%2==0){ return typename Base::UInt(ab%mod()); } typename Base::UInt q= -(typename Base::UInt(ab)*inverse); auto high=typename Base::UInt(ab>>Base::bits); auto low=typename Base::UInt(typename Base::UInt2(q)*typename Base::UInt(mod())>>Base::bits); return high>=low?high-low:high-low+mod(); } static Base::UInt m_reduce(Base::UInt2 ab){ if(imod()==0)[[unlikely]]{ return m_reduce_reduced(ab); } typename Base::UInt2 m=typename Base::UInt(ab)*imod(); return typename Base::UInt((ab+m*mod())>>Base::bits); } static Base::UInt m_transform(Base::UInt a){ if(mod()%2==0)[[unlikely]]{ return a; }else{ return m_reduce(a*pw128()); } } dynamic_modint&operator*=(const dynamic_modint&t){ Base::r=m_reduce(typename Base::UInt2(Base::r)*t.r); return*this; } void setr(Base::UInt rr){ Base::r=m_transform(rr); } Base::UInt getr()const{ typename Base::UInt res=m_reduce(Base::r); return std::min(res,res-mod()); } static Int mod(){return m;} static Base::UInt remod(){return rm;} static Base::UInt imod(){return im;} static Base::UInt2 pw128(){return r2;} static void switch_mod(Int nm){ m=nm; bool lazy=m%2&&typename Base::UInt(m)<=typename Base::UInt(-1)/4; rm=typename Base::UInt(m)*(lazy?2:1); inverse=m%2?inv2(-m):0; im=lazy?inverse:0; r2=static_cast<Base::UInt>(static_cast<Base::UInt2>(-1)%m+1); } auto static with_mod(Int tmp,auto callback){ struct scoped{ Int prev=mod(); ~scoped(){switch_mod(prev);} }_; switch_mod(tmp); return callback(); } private: static inline thread_local Int m=1; static inline thread_local Base::UInt im= -1,r2=0,inverse= -1,rm=2; }; }
#line 6 "cp-algo/number_theory/primality.hpp"
namespace cp_algo::math{ template<typename _Int> bool is_prime(_Int m){ using Int=std::make_signed_t<_Int>; using UInt=std::make_unsigned_t<Int>; if(m==1||m%2==0){ return m==2; } int s=std::countr_zero(UInt(m-1)); auto d=(m-1)>>s; using base=dynamic_modint<Int>; auto test= [&](base x){ x=bpow(x,d); if(std::abs(x.rem())<=1){ return true; } for(int i=1;i<s&&x!= -1;i++){ x*=x; } return x== -1; }; return base::with_mod(m,[&](){
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
uint16_t base2=7,base3=61; if(m!=uint32_t(m)){ base2=base_table1[uint32_t(m*0xAD625B89)>>18]; base3=base_table2[base2>>13]; } return test(2)&&test(base2)&&test(base3);
#else
return std::ranges::all_of(std::array{2,325,9375,28178,450775,9780504,1795265022},test);
#endif
}); } }
#line 1 "cp-algo/util/generator.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#line 10 "cp-algo/util/big_alloc.hpp"
#include <string>
#include <cstddef>
#line 14 "cp-algo/util/big_alloc.hpp"
#include <forward_list>
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#define CP_ALGO_USE_MMAP 1
#include <sys/mman.h>
#else
#define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo{ template<typename T,size_t Align=32> class big_alloc{ static_assert(Align>=alignof(void*),"Align must be at least pointer-size"); static_assert(std::has_single_bit(Align),"Align must be a power of two"); public: using value_type=T; template<class U> struct rebind{ using other=big_alloc<U,Align>; }; constexpr bool operator==(const big_alloc&)const=default; big_alloc()noexcept=default; template<typename U,std::size_t A> big_alloc(const big_alloc<U,A>&)noexcept{} [[nodiscard]]T*allocate(std::size_t n){ std::size_t padded=round_up(n*sizeof(T)); std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){ void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0); madvise(raw,padded,MADV_HUGEPAGE); return static_cast<T*>(raw); }
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align))); } void deallocate(T*p,std::size_t n)noexcept{ if(!p){ return; } std::size_t padded=round_up(n*sizeof(T)); std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){ munmap(p,padded); return; }
#endif
::operator delete(p,padded,std::align_val_t(align)); } private: static constexpr std::size_t MEGABYTE=1<<20; static constexpr std::size_t round_up(std::size_t x)noexcept{ return(x+Align-1)/Align*Align; } }; template<typename T> using big_vector=std::vector<T,big_alloc<T>>; template<typename T> using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>; template<typename T> using big_deque=std::deque<T,big_alloc<T>>; template<typename T> using big_stack=std::stack<T,big_deque<T>>; template<typename T> using big_queue=std::queue<T,big_deque<T>>; template<typename T> using big_priority_queue=std::priority_queue<T,big_vector<T>>; template<typename T> using big_forward_list=std::forward_list<T,big_alloc<T>>; using big_string=big_basic_string<char>; template<typename Key,typename Value,typename Compare=std::less<Key>> using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>; template<typename T,typename Compare=std::less<T>> using big_multiset=std::multiset<T,Compare,big_alloc<T>>; template<typename T,typename Compare=std::less<T>> using big_set=std::set<T,Compare,big_alloc<T>>; }
#line 5 "cp-algo/util/generator.hpp"
#include <generator>
namespace cp_algo{ template<typename Ref,typename V=void> using big_generator=std::generator<Ref,V,big_alloc<std::byte>>; } namespace std::ranges{ template<typename Ref,typename V> elements_of(cp_algo::big_generator<Ref,V>&&)->elements_of<cp_algo::big_generator<Ref,V>&&,cp_algo::big_alloc<std::byte>>; }
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random{ std::mt19937_64 gen( std::chrono::steady_clock::now().time_since_epoch().count() ); uint64_t rng(){ return gen(); } }
#line 6 "cp-algo/number_theory/factorize.hpp"
namespace cp_algo::math{ template<typename _Int> auto proper_divisor(_Int m){ using Int=std::make_signed_t<_Int>; using base=dynamic_modint<Int>; return m%2==0?2:base::with_mod(m,[&](){ base t=random::rng(); auto f= [&](auto x){ return x*x+t; }; const base zero=0,one=1; base x=zero,y=zero; base g=one; while(g==one){ for(int i=0;i<64;i++){ x=f(x); y=f(f(y)); if(x==y)[[unlikely]]{ t=random::rng(); x=y=zero; }else{ base t=g*(x-y); g=t==zero?g:t; } } g=std::gcd(g.getr(),m); } return g.getr(); }); } template<typename Int> big_generator<Int>factorize(Int m){ if(is_prime(m)){ co_yield m; }else if(m>1){ auto g=proper_divisor(m); co_yield std::ranges::elements_of(factorize(g)); co_yield std::ranges::elements_of(factorize(m/g)); } } template<typename Int> big_generator<Int>divisors_sqrt(Int m){ for(Int i=1;i*i<=m;i++){ if(m%i==0){ co_yield i; if(i*i!=m){ co_yield m/i; } } } } }
#line 5 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math{ auto euler_phi(auto m){ using T=std::decay_t<decltype(m)>; auto primes=to<big_vector<T>>(factorize(m)); std::ranges::sort(primes); auto[from,to] =std::ranges::unique(primes); primes.erase(from,to); auto ans=m; for(auto it:primes){ ans-=ans/it; } return ans; } template<modint_type base> auto period(base x){ auto ans=euler_phi(base::mod()); base x0=bpow(x,ans); for(auto t:factorize(ans)){ while(ans%t==0&&x0*bpow(x,ans/t)==x0){ ans/=t; } } return ans; } template<typename _Int> _Int primitive_root(_Int p){ using Int=std::make_signed_t<_Int>; using base=dynamic_modint<Int>; return base::with_mod(p,[p](){ base t=1; while(period(t)!=p-1){ t=random::rng(); } return t.getr(); }); } }
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2,fma\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH _Pragma("GCC push_options") CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo{ template<typename T,size_t len> using simd[[gnu::vector_size(len*sizeof(T))]] =T; using u64x8=simd<uint64_t,8>; using u32x16=simd<uint32_t,16>; using i64x4=simd<int64_t,4>; using u64x4=simd<uint64_t,4>; using u32x8=simd<uint32_t,8>; using u16x16=simd<uint16_t,16>; using i32x4=simd<int32_t,4>; using u32x4=simd<uint32_t,4>; using u16x8=simd<uint16_t,8>; using u16x4=simd<uint16_t,4>; using i16x4=simd<int16_t,4>; using u8x32=simd<uint8_t,32>; using u8x16=simd<uint8_t,16>; using u8x8=simd<uint8_t,8>; using u8x4=simd<uint8_t,4>; using dx4=simd<double,4>; inline dx4 abs(dx4 a){ return dx4{ std::abs(a[0]), std::abs(a[1]), std::abs(a[2]), std::abs(a[3]) }; } static constexpr dx4 magic=dx4()+(3ULL<<51); inline i64x4 lround(dx4 x){ return i64x4(x+magic)-i64x4(magic); } inline dx4 to_double(i64x4 x){ return dx4(x+i64x4(magic))-magic; } inline dx4 round(dx4 a){ return dx4{ std::nearbyint(a[0]), std::nearbyint(a[1]), std::nearbyint(a[2]), std::nearbyint(a[3]) }; } inline u64x4 low32(u64x4 x){ return x&uint32_t(-1); } inline auto swap_bytes(auto x){ return decltype(x)(__builtin_shufflevector(u32x8(x),u32x8(x),1,0,3,2,5,4,7,6)); } inline u64x4 montgomery_reduce(u64x4 x,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
auto x_ninv=u64x4(_mm256_mul_epu32(__m256i(x),__m256i()+imod)); x+=u64x4(_mm256_mul_epu32(__m256i(x_ninv),__m256i()+mod));
#else
auto x_ninv=u64x4(u32x8(low32(x))*imod); x+=x_ninv*uint64_t(mod);
#endif
return swap_bytes(x); } inline u32x8 reduce_once(u32x8 x,uint32_t mod){ auto y=x-mod; return x<y?x:y; } inline u64x4 reduce_once(u64x4 x,uint32_t mod){ return u64x4(reduce_once(u32x8(x),mod)); } inline u64x4 montgomery_mul(u64x4 x,u64x4 y,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x),__m256i(y))),mod,imod);
#else
return montgomery_reduce(x*y,mod,imod);
#endif
} inline u32x8 montgomery_mul(u32x8 x,u32x8 y,uint32_t mod,uint32_t imod){ return u32x8(montgomery_mul(u64x4(x),u64x4(y),mod,imod))| u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)),u64x4(swap_bytes(y)),mod,imod))); } inline dx4 rotate_right(dx4 x){ static constexpr u64x4 shuffler={3,0,1,2}; return __builtin_shuffle(x,shuffler); } template<std::size_t Align=32> inline bool is_aligned(const auto*p)noexcept{ return(reinterpret_cast<std::uintptr_t>(p)%Align)==0; } template<class Target> inline Target&vector_cast(auto&&p){ return*reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p)); } }
#pragma GCC pop_options
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo{ template<typename T> struct complex{ using value_type=T; T x,y; inline constexpr complex():x(),y(){} inline constexpr complex(T const&x):x(x),y(){} inline constexpr complex(T const&x,T const&y):x(x),y(y){} inline complex&operator*=(T const&t){x*=t;y*=t;return*this;} inline complex&operator/=(T const&t){x/=t;y/=t;return*this;} inline complex operator*(T const&t)const{return complex(*this)*=t;} inline complex operator/(T const&t)const{return complex(*this)/=t;} inline complex&operator+=(complex const&t){x+=t.x;y+=t.y;return*this;} inline complex&operator-=(complex const&t){x-=t.x;y-=t.y;return*this;} inline complex operator*(complex const&t)const{return{x*t.x-y*t.y,x*t.y+y*t.x};} inline complex operator/(complex const&t)const{return*this*t.conj()/t.norm();} inline complex operator+(complex const&t)const{return complex(*this)+=t;} inline complex operator-(complex const&t)const{return complex(*this)-=t;} inline complex&operator*=(complex const&t){return*this= *this*t;} inline complex&operator/=(complex const&t){return*this= *this/t;} inline complex operator-()const{return{-x,-y};} inline complex conj()const{return{x,-y};} inline T norm()const{return x*x+y*y;} inline T abs()const{return std::sqrt(norm());} inline T const real()const{return x;} inline T const imag()const{return y;} inline T&real(){return x;} inline T&imag(){return y;} inline static constexpr complex polar(T r,T theta){return{T(r*cos(theta)),T(r*sin(theta))};} inline auto operator<=>(complex const&t)const=default; }; template<typename T>inline complex<T>conj(complex<T>const&x){return x.conj();} template<typename T>inline T norm(complex<T>const&x){return x.norm();} template<typename T>inline T abs(complex<T>const&x){return x.abs();} template<typename T>inline T&real(complex<T> &x){return x.real();} template<typename T>inline T&imag(complex<T> &x){return x.imag();} template<typename T>inline T const real(complex<T>const&x){return x.real();} template<typename T>inline T const imag(complex<T>const&x){return x.imag();} template<typename T> inline constexpr complex<T>polar(T r,T theta){ return complex<T>::polar(r,theta); } template<typename T> inline std::ostream&operator<<(std::ostream&out,complex<T>const&x){ return out<<x.real()<<' '<<x.imag(); } }
#pragma GCC pop_options
#line 7 "cp-algo/number_theory/two_squares.hpp"
#include <utility>
#line 10 "cp-algo/number_theory/two_squares.hpp"
namespace cp_algo::math{ template<typename T> using gaussint=complex<T>; template<typename _Int> auto two_squares_prime_any(_Int p){ if(p==2){ return gaussint<_Int>{1,1}; } assert(p%4==1); using Int=std::make_signed_t<_Int>; using base=dynamic_modint<Int>; return base::with_mod(p,[&](){ base g=primitive_root(p); int64_t i=bpow(g,(p-1)/4).getr(); int64_t q0=1,q1=0; int64_t r=i,m=p; do{ int64_t d=r/m; q0=std::exchange(q1,q0+d*q1); r=std::exchange(m,r%m); }while(q1<p/q1); return gaussint<_Int>{q0,(base(i)*base(q0)).rem()}; }); } template<typename Int> big_vector<gaussint<Int>>two_squares_all(Int n){ if(n==0){ return{0}; } auto primes=factorize(n); big_map<Int,int>cnt; for(auto p:primes){ cnt[p]++; } big_vector<gaussint<Int>>res={1}; for(auto[p,c]:cnt){ if(p%4==3&&c%2){ return{}; } if(p%4!=1){ auto mul=p==2?bpow(gaussint<Int>(1,1),c):bpow(gaussint<Int>(p),c/2); for(auto&value:res){ value*=mul; } }else{ big_vector<gaussint<Int>>nres; auto base=two_squares_prime_any(p); for(int i=0;i<=c;i++){ auto mul=bpow(base,i)*bpow(conj(base),c-i); for(auto value:res){ nres.push_back(value*mul); } } res=std::move(nres); } } big_vector<gaussint<Int>>nres; for(auto p:res){ while(p.real()<0||p.imag()<0){ p*=gaussint<Int>(0,1); } nres.push_back(p); if(!p.real()|| !p.imag()){ nres.emplace_back(p.imag(),p.real()); } } return nres; } }
#line 3 "tests/two_squares.cpp"
using namespace cp_algo::math; void check(int64_t n){ std::set<std::pair<int64_t,int64_t>>want,got; for(int64_t x=0,y=int64_t(std::sqrt(n));x*x<=n;x++){ while(x*x+y*y>n){y--;} if(x*x+y*y==n){want.emplace(x,y);} } for(auto z:two_squares_all(n)){ assert(z.real()>=0&&z.imag()>=0&&z.norm()==n); got.emplace(z.real(),z.imag()); } assert(got==want); } int main(){ cp_algo::random::gen.seed(71283); for(int64_t n=0;n<=2500;n++){check(n);} for(int64_t n:{3125,28125,400000,422500,1267500}){check(n);} std::cout<<"Sums of two squares passed exhaustive enumeration checks\n"; }