結果
| 問題 | No.2192 平方数の下１４桁 |
| ユーザー |
 suisen
|
| 提出日時 | 2023-01-13 23:19:38 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 3 ms / 2,000 ms |
| コード長 | 20,541 bytes |
| コンパイル時間 | 2,790 ms |
| コンパイル使用メモリ | 234,128 KB |
| 実行使用メモリ | 5,376 KB |
| 最終ジャッジ日時 | 2024-06-07 00:09:10 |
| 合計ジャッジ時間 | 4,013 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge1 |
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テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 3 ms
5,248 KB |
| testcase_01 | AC | 3 ms
5,248 KB |
| testcase_02 | AC | 3 ms
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| testcase_03 | AC | 3 ms
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| testcase_04 | AC | 3 ms
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| testcase_05 | AC | 3 ms
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| testcase_06 | AC | 3 ms
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| testcase_07 | AC | 3 ms
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| testcase_08 | AC | 3 ms
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| testcase_09 | AC | 3 ms
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| testcase_10 | AC | 2 ms
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| testcase_11 | AC | 3 ms
5,376 KB |
| testcase_12 | AC | 3 ms
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| testcase_13 | AC | 3 ms
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| testcase_14 | AC | 3 ms
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| testcase_15 | AC | 3 ms
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| testcase_16 | AC | 3 ms
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| testcase_17 | AC | 3 ms
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| testcase_18 | AC | 3 ms
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| testcase_19 | AC | 3 ms
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| testcase_20 | AC | 3 ms
5,376 KB |
| testcase_21 | AC | 3 ms
5,376 KB |
| testcase_22 | AC | 3 ms
5,376 KB |
| testcase_23 | AC | 3 ms
5,376 KB |
| testcase_24 | AC | 3 ms
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| testcase_25 | AC | 3 ms
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| testcase_26 | AC | 3 ms
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| testcase_27 | AC | 3 ms
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| testcase_28 | AC | 3 ms
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| testcase_29 | AC | 3 ms
5,376 KB |
| testcase_30 | AC | 3 ms
5,376 KB |
| testcase_31 | AC | 3 ms
5,376 KB |
| testcase_32 | AC | 3 ms
5,376 KB |
| testcase_33 | AC | 3 ms
5,376 KB |
| testcase_34 | AC | 3 ms
5,376 KB |
| testcase_35 | AC | 3 ms
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| testcase_36 | AC | 3 ms
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| testcase_37 | AC | 3 ms
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| testcase_38 | AC | 3 ms
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| testcase_39 | AC | 3 ms
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| testcase_40 | AC | 3 ms
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| testcase_41 | AC | 3 ms
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| testcase_42 | AC | 3 ms
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| testcase_43 | AC | 3 ms
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ソースコード
#include <bits/stdc++.h>
using namespace std;
#include <cmath>
#include <iostream>
#include <random>
#include <numeric>
#include <utility>
#include <cassert>
#include <cstdint>
#include <iterator>
#include <limits>
#include <type_traits>
namespace suisen {
// ! utility
template <typename ...Types>
using constraints_t = std::enable_if_t<std::conjunction_v<Types...>, std::nullptr_t>;
template <bool cond_v, typename Then, typename OrElse>
constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) {
if constexpr (cond_v) {
return std::forward<Then>(then);
} else {
return std::forward<OrElse>(or_else);
}
}
// ! function
template <typename ReturnType, typename Callable, typename ...Args>
using is_same_as_invoke_result = std::is_same<std::invoke_result_t<Callable, Args...>, ReturnType>;
template <typename F, typename T>
using is_uni_op = is_same_as_invoke_result<T, F, T>;
template <typename F, typename T>
using is_bin_op = is_same_as_invoke_result<T, F, T, T>;
template <typename Comparator, typename T>
using is_comparator = std::is_same<std::invoke_result_t<Comparator, T, T>, bool>;
// ! integral
template <typename T, typename = constraints_t<std::is_integral<T>>>
constexpr int bit_num = std::numeric_limits<std::make_unsigned_t<T>>::digits;
template <typename T, unsigned int n>
struct is_nbit { static constexpr bool value = bit_num<T> == n; };
template <typename T, unsigned int n>
static constexpr bool is_nbit_v = is_nbit<T, n>::value;
// ?
template <typename T>
struct safely_multipliable {};
template <>
struct safely_multipliable<int> { using type = long long; };
template <>
struct safely_multipliable<long long> { using type = __int128_t; };
template <>
struct safely_multipliable<unsigned int> { using type = unsigned long long; };
template <>
struct safely_multipliable<unsigned long int> { using type = __uint128_t; };
template <>
struct safely_multipliable<unsigned long long> { using type = __uint128_t; };
template <>
struct safely_multipliable<float> { using type = float; };
template <>
struct safely_multipliable<double> { using type = double; };
template <>
struct safely_multipliable<long double> { using type = long double; };
template <typename T>
using safely_multipliable_t = typename safely_multipliable<T>::type;
template <typename T, typename = void>
struct rec_value_type {
using type = T;
};
template <typename T>
struct rec_value_type<T, std::void_t<typename T::value_type>> {
using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T>
using rec_value_type_t = typename rec_value_type<T>::type;
} // namespace suisen
namespace suisen::miller_rabin {
namespace internal {
constexpr uint32_t THRESHOLD_1 = 341531U;
constexpr uint64_t BASE_1[] { 9345883071009581737ULL };
constexpr uint32_t THRESHOLD_2 = 1050535501U;
constexpr uint64_t BASE_2[] { 336781006125ULL, 9639812373923155ULL };
constexpr uint64_t THRESHOLD_3 = 350269456337ULL;
constexpr uint64_t BASE_3[] { 4230279247111683200ULL, 14694767155120705706ULL, 16641139526367750375ULL };
constexpr uint64_t THRESHOLD_4 = 55245642489451ULL;
constexpr uint64_t BASE_4[] { 2ULL, 141889084524735ULL, 1199124725622454117ULL, 11096072698276303650ULL };
constexpr uint64_t THRESHOLD_5 = 7999252175582851ULL;
constexpr uint64_t BASE_5[] { 2ULL, 4130806001517ULL, 149795463772692060ULL, 186635894390467037ULL, 3967304179347715805ULL };
constexpr uint64_t THRESHOLD_6 = 585226005592931977ULL;
constexpr uint64_t BASE_6[] { 2ULL, 123635709730000ULL, 9233062284813009ULL, 43835965440333360ULL, 761179012939631437ULL, 1263739024124850375ULL };
constexpr uint32_t BASE_7[] { 2U, 325U, 9375U, 28178U, 450775U, 9780504U, 1795265022U };
template <auto BASE, std::size_t SIZE, typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
constexpr bool miller_rabin(T _n) {
using U = std::make_unsigned_t<T>;
using M = safely_multipliable_t<U>;
U n = _n, d = (n - 1) >> __builtin_ctzll(n - 1);
if (n == 2 or n == 3 or n == 5 or n == 7) return true;
if (n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0) return false;
for (std::size_t i = 0; i < SIZE; ++i) {
M y = 1, p = BASE[i] % n;
if (p == 0) continue;
for (U d2 = d; d2; d2 >>= 1) {
if (d2 & 1) y = y * p % n;
p = p * p % n;
}
if (y == 1) continue;
for (U t = d; y != n - 1; t <<= 1) {
y = y * y % n;
if (y == 1 or t == n - 1) return false;
}
}
return true;
}
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
constexpr bool is_prime(T n) {
if (n <= 1) return false;
using U = std::make_unsigned_t<T>;
U n2 = n;
using namespace internal;
if (n2 < THRESHOLD_1) {
return miller_rabin<BASE_1, 1>(n2);
} else if (n2 < THRESHOLD_2) {
return miller_rabin<BASE_2, 2>(n2);
} else if (n2 < THRESHOLD_3) {
return miller_rabin<BASE_3, 3>(n2);
} else if (n2 < THRESHOLD_4) {
return miller_rabin<BASE_4, 4>(n2);
} else if (n2 < THRESHOLD_5) {
return miller_rabin<BASE_5, 5>(n2);
} else if (n2 < THRESHOLD_6) {
return miller_rabin<BASE_6, 6>(n2);
} else {
return miller_rabin<BASE_7, 7>(n2);
}
}
} // namespace suisen::miller_rabin
#include <vector>
namespace suisen::internal::sieve {
constexpr std::uint8_t K = 8;
constexpr std::uint8_t PROD = 2 * 3 * 5;
constexpr std::uint8_t RM[K] = { 1, 7, 11, 13, 17, 19, 23, 29 };
constexpr std::uint8_t DR[K] = { 6, 4, 2, 4, 2, 4, 6, 2 };
constexpr std::uint8_t DF[K][K] = {
{ 0, 0, 0, 0, 0, 0, 0, 1 }, { 1, 1, 1, 0, 1, 1, 1, 1 },
{ 2, 2, 0, 2, 0, 2, 2, 1 }, { 3, 1, 1, 2, 1, 1, 3, 1 },
{ 3, 3, 1, 2, 1, 3, 3, 1 }, { 4, 2, 2, 2, 2, 2, 4, 1 },
{ 5, 3, 1, 4, 1, 3, 5, 1 }, { 6, 4, 2, 4, 2, 4, 6, 1 },
};
constexpr std::uint8_t DRP[K] = { 48, 32, 16, 32, 16, 32, 48, 16 };
constexpr std::uint8_t DFP[K][K] = {
{ 0, 0, 0, 0, 0, 0, 0, 8 }, { 8, 8, 8, 0, 8, 8, 8, 8 },
{ 16, 16, 0, 16, 0, 16, 16, 8 }, { 24, 8, 8, 16, 8, 8, 24, 8 },
{ 24, 24, 8, 16, 8, 24, 24, 8 }, { 32, 16, 16, 16, 16, 16, 32, 8 },
{ 40, 24, 8, 32, 8, 24, 40, 8 }, { 48, 32, 16, 32, 16, 32, 48, 8 },
};
constexpr std::uint8_t MASK[K][K] = {
{ 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80 }, { 0x02, 0x20, 0x10, 0x01, 0x80, 0x08, 0x04, 0x40 },
{ 0x04, 0x10, 0x01, 0x40, 0x02, 0x80, 0x08, 0x20 }, { 0x08, 0x01, 0x40, 0x20, 0x04, 0x02, 0x80, 0x10 },
{ 0x10, 0x80, 0x02, 0x04, 0x20, 0x40, 0x01, 0x08 }, { 0x20, 0x08, 0x80, 0x02, 0x40, 0x01, 0x10, 0x04 },
{ 0x40, 0x04, 0x08, 0x80, 0x01, 0x10, 0x20, 0x02 }, { 0x80, 0x40, 0x20, 0x10, 0x08, 0x04, 0x02, 0x01 },
};
constexpr std::uint8_t OFFSET[K][K] = {
{ 0, 1, 2, 3, 4, 5, 6, 7, },
{ 1, 5, 4, 0, 7, 3, 2, 6, },
{ 2, 4, 0, 6, 1, 7, 3, 5, },
{ 3, 0, 6, 5, 2, 1, 7, 4, },
{ 4, 7, 1, 2, 5, 6, 0, 3, },
{ 5, 3, 7, 1, 6, 0, 4, 2, },
{ 6, 2, 3, 7, 0, 4, 5, 1, },
{ 7, 6, 5, 4, 3, 2, 1, 0, },
};
constexpr std::uint8_t mask_to_index(const std::uint8_t bits) {
switch (bits) {
case 1 << 0: return 0;
case 1 << 1: return 1;
case 1 << 2: return 2;
case 1 << 3: return 3;
case 1 << 4: return 4;
case 1 << 5: return 5;
case 1 << 6: return 6;
case 1 << 7: return 7;
default: assert(false);
}
}
} // namespace suisen::internal::sieve
namespace suisen {
template <unsigned int N>
class SimpleSieve {
private:
static constexpr unsigned int siz = N / internal::sieve::PROD + 1;
static std::uint8_t flag[siz];
public:
SimpleSieve() {
using namespace internal::sieve;
flag[0] |= 1;
unsigned int k_max = (unsigned int) std::sqrt(N + 2) / PROD;
for (unsigned int kp = 0; kp <= k_max; ++kp) {
for (std::uint8_t bits = ~flag[kp]; bits; bits &= bits - 1) {
const std::uint8_t mp = mask_to_index(bits & -bits), m = RM[mp];
unsigned int kr = kp * (PROD * kp + 2 * m) + m * m / PROD;
for (std::uint8_t mq = mp; kr < siz; kr += kp * DR[mq] + DF[mp][mq], ++mq &= 7) {
flag[kr] |= MASK[mp][mq];
}
}
}
}
std::vector<int> prime_list(unsigned int max_val = N) const {
using namespace internal::sieve;
std::vector<int> res { 2, 3, 5 };
res.reserve(max_val / 25);
for (unsigned int i = 0, offset = 0; i < siz and offset < max_val; ++i, offset += PROD) {
for (uint8_t f = ~flag[i]; f;) {
uint8_t g = f & -f;
res.push_back(offset + RM[mask_to_index(g)]);
f ^= g;
}
}
while (res.size() and (unsigned int) res.back() > max_val) res.pop_back();
return res;
}
bool is_prime(const unsigned int p) const {
using namespace internal::sieve;
switch (p) {
case 2: case 3: case 5: return true;
default:
switch (p % PROD) {
case RM[0]: return ((flag[p / PROD] >> 0) & 1) == 0;
case RM[1]: return ((flag[p / PROD] >> 1) & 1) == 0;
case RM[2]: return ((flag[p / PROD] >> 2) & 1) == 0;
case RM[3]: return ((flag[p / PROD] >> 3) & 1) == 0;
case RM[4]: return ((flag[p / PROD] >> 4) & 1) == 0;
case RM[5]: return ((flag[p / PROD] >> 5) & 1) == 0;
case RM[6]: return ((flag[p / PROD] >> 6) & 1) == 0;
case RM[7]: return ((flag[p / PROD] >> 7) & 1) == 0;
default: return false;
}
}
}
};
template <unsigned int N>
std::uint8_t SimpleSieve<N>::flag[SimpleSieve<N>::siz];
template <unsigned int N>
class Sieve {
private:
static constexpr unsigned int base_max = (N + 1) * internal::sieve::K / internal::sieve::PROD;
static unsigned int pf[base_max + internal::sieve::K];
public:
Sieve() {
using namespace internal::sieve;
pf[0] = 1;
unsigned int k_max = ((unsigned int) std::sqrt(N + 1) - 1) / PROD;
for (unsigned int kp = 0; kp <= k_max; ++kp) {
const int base_i = kp * K, base_act_i = kp * PROD;
for (int mp = 0; mp < K; ++mp) {
const int m = RM[mp], i = base_i + mp;
if (pf[i] == 0) {
unsigned int act_i = base_act_i + m;
unsigned int base_k = (kp * (PROD * kp + 2 * m) + m * m / PROD) * K;
for (std::uint8_t mq = mp; base_k <= base_max; base_k += kp * DRP[mq] + DFP[mp][mq], ++mq &= 7) {
pf[base_k + OFFSET[mp][mq]] = act_i;
}
}
}
}
}
bool is_prime(const unsigned int p) const {
using namespace internal::sieve;
switch (p) {
case 2: case 3: case 5: return true;
default:
switch (p % PROD) {
case RM[0]: return pf[p / PROD * K + 0] == 0;
case RM[1]: return pf[p / PROD * K + 1] == 0;
case RM[2]: return pf[p / PROD * K + 2] == 0;
case RM[3]: return pf[p / PROD * K + 3] == 0;
case RM[4]: return pf[p / PROD * K + 4] == 0;
case RM[5]: return pf[p / PROD * K + 5] == 0;
case RM[6]: return pf[p / PROD * K + 6] == 0;
case RM[7]: return pf[p / PROD * K + 7] == 0;
default: return false;
}
}
}
int prime_factor(const unsigned int p) const {
using namespace internal::sieve;
switch (p % PROD) {
case 0: case 2: case 4: case 6: case 8:
case 10: case 12: case 14: case 16: case 18:
case 20: case 22: case 24: case 26: case 28: return 2;
case 3: case 9: case 15: case 21: case 27: return 3;
case 5: case 25: return 5;
case RM[0]: return pf[p / PROD * K + 0] ? pf[p / PROD * K + 0] : p;
case RM[1]: return pf[p / PROD * K + 1] ? pf[p / PROD * K + 1] : p;
case RM[2]: return pf[p / PROD * K + 2] ? pf[p / PROD * K + 2] : p;
case RM[3]: return pf[p / PROD * K + 3] ? pf[p / PROD * K + 3] : p;
case RM[4]: return pf[p / PROD * K + 4] ? pf[p / PROD * K + 4] : p;
case RM[5]: return pf[p / PROD * K + 5] ? pf[p / PROD * K + 5] : p;
case RM[6]: return pf[p / PROD * K + 6] ? pf[p / PROD * K + 6] : p;
case RM[7]: return pf[p / PROD * K + 7] ? pf[p / PROD * K + 7] : p;
default: assert(false);
}
}
/**
* Returns a vector of `{ prime, index }`.
*/
std::vector<std::pair<int, int>> factorize(unsigned int n) const {
assert(0 < n and n <= N);
std::vector<std::pair<int, int>> prime_powers;
while (n > 1) {
int p = prime_factor(n), c = 0;
do { n /= p, ++c; } while (n % p == 0);
prime_powers.emplace_back(p, c);
}
return prime_powers;
}
/**
* Returns the divisors of `n`.
* It is NOT guaranteed that the returned vector is sorted.
*/
std::vector<int> divisors(unsigned int n) const {
assert(0 < n and n <= N);
std::vector<int> divs { 1 };
for (auto [prime, index] : factorize(n)) {
int sz = divs.size();
for (int i = 0; i < sz; ++i) {
int d = divs[i];
for (int j = 0; j < index; ++j) {
divs.push_back(d *= prime);
}
}
}
return divs;
}
};
template <unsigned int N>
unsigned int Sieve<N>::pf[Sieve<N>::base_max + internal::sieve::K];
} // namespace suisen
namespace suisen::fast_factorize {
namespace internal {
template <typename T>
constexpr int floor_log2(T n) {
int i = 0;
while (n) n >>= 1, ++i;
return i - 1;
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T pollard_rho(T n) {
using M = safely_multipliable_t<T>;
const T m = T(1) << (floor_log2(n) / 5);
static std::mt19937_64 rng{std::random_device{}()};
std::uniform_int_distribution<T> dist(0, n - 1);
while (true) {
T c = dist(rng);
auto f = [&](T x) -> T { return (M(x) * x + c) % n; };
T x, y = 2, ys, q = 1, g = 1;
for (T r = 1; g == 1; r <<= 1) {
x = y;
for (T i = 0; i < r; ++i) y = f(y);
for (T k = 0; k < r and g == 1; k += m) {
ys = y;
for (T i = 0; i < std::min(m, r - k); ++i) y = f(y), q = M(q) * (x > y ? x - y : y - x) % n;
g = std::gcd(q, n);
}
}
if (g == n) {
g = 1;
while (g == 1) ys = f(ys), g = std::gcd(x > ys ? x - ys : ys - x, n);
}
if (g < n) {
if (miller_rabin::is_prime(g)) return g;
if (T d = n / g; miller_rabin::is_prime(d)) return d;
return pollard_rho(g);
}
}
}
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<std::pair<T, int>> factorize(T n) {
static constexpr int threshold = 1000000;
static Sieve<threshold> sieve;
std::vector<std::pair<T, int>> res;
if (n <= threshold) {
for (auto [p, q] : sieve.factorize(n)) res.emplace_back(p, q);
return res;
}
if ((n & 1) == 0) {
int q = 0;
do ++q, n >>= 1; while ((n & 1) == 0);
res.emplace_back(2, q);
}
for (T p = 3; p * p <= n; p += 2) {
if (p >= 101 and n >= 1 << 20) {
while (n > 1) {
if (miller_rabin::is_prime(n)) {
res.emplace_back(std::exchange(n, 1), 1);
} else {
p = internal::pollard_rho(n);
int q = 0;
do ++q, n /= p; while (n % p == 0);
res.emplace_back(p, q);
}
}
break;
}
if (n % p == 0) {
int q = 0;
do ++q, n /= p; while (n % p == 0);
res.emplace_back(p, q);
}
}
if (n > 1) res.emplace_back(n, 1);
return res;
}
} // namespace suisen::fast_factorize
constexpr std::pair<__int128_t, __int128_t> inv_gcd(__int128_t a, __int128_t b) {
a %= b;
if (a < 0) a += b;
if (a == 0) return {b, 0};
__int128_t s = b, t = a;
__int128_t m0 = 0, m1 = 1;
while (t) {
__int128_t u = s / t;
s -= t * u;
m0 -= m1 * u;
std::swap(s, t);
std::swap(m0, m1);
}
if (m0 < 0) m0 += b / s;
return {s, m0};
}
__int128_t modpow(__int128_t a, __int128_t b, __int128_t m) {
a %= m;
__int128_t res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res = res * pow_a % m;
pow_a = pow_a * pow_a % m;
}
return res;
}
std::optional<__int128_t> mod_sqrt(__int128_t a, const __int128_t p) {
a %= p;
if (a < 0) a += p;
if (a == 0) return std::make_optional(0);
if (p == 2) return std::make_optional(a);
if (modpow(a, (p - 1) / 2, p) != 1) {
return std::nullopt;
}
__int128_t b = 1;
while (modpow(b, (p - 1) / 2, p) == 1) {
++b;
}
int tlz = __builtin_ctz(p - 1);
__int128_t q = (p - 1) >> tlz;
__int128_t x = modpow(a, (q + 1) / 2, p);
b = modpow(b, q, p);
for (int shift = 2;; ++shift) {
__int128_t x2 = x * x % p;
if (x2 == a) {
return std::make_optional(x2);
}
__int128_t e = inv_gcd(a, p).second * x2 % p;
if (modpow(e, 1 << (tlz - shift), p) != 1) {
x = x * b % p;
}
b = b * b % p;
}
}
int main() {
long long m_, n_;
std::cin >> m_ >> n_;
__int128_t p, q;
for (auto e : suisen::fast_factorize::factorize(m_)) {
std::tie(p, q) = e;
auto ox = mod_sqrt(n_, p);
auto dfs = [&](auto dfs, int i, __int128_t x0, __int128_t pq) -> bool {
if (i == q) return true;
__int128_t f_x0 = (x0 * x0 - n_) / pq % p;
__int128_t df_x0 = 2 * x0 % p;
if (f_x0 < 0) f_x0 += p;
if (df_x0 != 0) {
__int128_t y0 = (-f_x0 * inv_gcd(df_x0, p).second) % p;
if (y0 < 0) y0 += p;
return dfs(dfs, i + 1, x0 + pq * y0, pq * p);
} else if (f_x0 != 0) {
return false;
} else {
for (__int128_t y0 = 0; y0 < p; ++y0) {
if (dfs(dfs, i + 1, x0 + pq * y0, pq * p)) {
return true;
}
}
return false;
}
};
if (not (ox and (dfs(dfs, 1, *ox, p) or dfs(dfs, 1, (p - *ox) % p, p)))) {
std::cout << "NO" << std::endl;
return 0;
}
}
std::cout << "YES" << std::endl;
}