#include using namespace std; #include #include #include #include #include #include #include #include #include #include namespace suisen { // ! utility template using constraints_t = std::enable_if_t, std::nullptr_t>; template constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) { if constexpr (cond_v) { return std::forward(then); } else { return std::forward(or_else); } } // ! function template using is_same_as_invoke_result = std::is_same, ReturnType>; template using is_uni_op = is_same_as_invoke_result; template using is_bin_op = is_same_as_invoke_result; template using is_comparator = std::is_same, bool>; // ! integral template >> constexpr int bit_num = std::numeric_limits>::digits; template struct is_nbit { static constexpr bool value = bit_num == n; }; template static constexpr bool is_nbit_v = is_nbit::value; // ? template struct safely_multipliable {}; template <> struct safely_multipliable { using type = long long; }; template <> struct safely_multipliable { using type = __int128_t; }; template <> struct safely_multipliable { using type = unsigned long long; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = float; }; template <> struct safely_multipliable { using type = double; }; template <> struct safely_multipliable { using type = long double; }; template using safely_multipliable_t = typename safely_multipliable::type; template struct rec_value_type { using type = T; }; template struct rec_value_type> { using type = typename rec_value_type::type; }; template using rec_value_type_t = typename rec_value_type::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 , std::nullptr_t> = nullptr> constexpr bool miller_rabin(T _n) { using U = std::make_unsigned_t; using M = safely_multipliable_t; 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 , std::nullptr_t> = nullptr> constexpr bool is_prime(T n) { if (n <= 1) return false; using U = std::make_unsigned_t; U n2 = n; using namespace internal; if (n2 < THRESHOLD_1) { return miller_rabin(n2); } else if (n2 < THRESHOLD_2) { return miller_rabin(n2); } else if (n2 < THRESHOLD_3) { return miller_rabin(n2); } else if (n2 < THRESHOLD_4) { return miller_rabin(n2); } else if (n2 < THRESHOLD_5) { return miller_rabin(n2); } else if (n2 < THRESHOLD_6) { return miller_rabin(n2); } else { return miller_rabin(n2); } } } // namespace suisen::miller_rabin #include 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 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 prime_list(unsigned int max_val = N) const { using namespace internal::sieve; std::vector 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 std::uint8_t SimpleSieve::flag[SimpleSieve::siz]; template 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> factorize(unsigned int n) const { assert(0 < n and n <= N); std::vector> 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 divisors(unsigned int n) const { assert(0 < n and n <= N); std::vector 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 Sieve::pf[Sieve::base_max + internal::sieve::K]; } // namespace suisen namespace suisen::fast_factorize { namespace internal { template constexpr int floor_log2(T n) { int i = 0; while (n) n >>= 1, ++i; return i - 1; } template , std::nullptr_t> = nullptr> T pollard_rho(T n) { using M = safely_multipliable_t; const T m = T(1) << (floor_log2(n) / 5); static std::mt19937_64 rng{std::random_device{}()}; std::uniform_int_distribution 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 , std::nullptr_t> = nullptr> std::vector> factorize(T n) { static constexpr int threshold = 1000000; static Sieve sieve; std::vector> 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 #include #include namespace suisen { namespace internal { long long inv_mod64(long long a, long long m) { return atcoder::inv_mod(a, m); } long long pow_mod64(long long a, long long b, long long m) { if ((a %= m) < 0) a += m; long long res = 1, pow_a = a; for (; b; b >>= 1) { if (b & 1) { res = __int128_t(res) * pow_a % m; } pow_a = __int128_t(pow_a) * pow_a % m; } return res; } long long mul_mod64(long long a, long long b, long long m) { return __int128_t(a) * b % m; } } std::optional prime_mod_sqrt(long long a, const long long p) { using namespace internal; if ((a %= p) < 0) a += p; if (a == 0) return 0; if (p == 2) return a; if (pow_mod64(a, (p - 1) / 2, p) != 1) { return std::nullopt; } long long b = 1; while (pow_mod64(b, (p - 1) / 2, p) == 1) { ++b; } int tlz = __builtin_ctz(p - 1); long long q = (p - 1) >> tlz; long long ia = inv_mod64(a, p); long long x = pow_mod64(a, (q + 1) / 2, p); b = pow_mod64(b, q, p); for (int shift = 2;; ++shift) { long long x2 = mul_mod64(x, x, p); if (x2 == a) { return x; } long long e = mul_mod64(ia, x2, p); if (pow_mod64(e, 1 << (tlz - shift), p) != 1) { x = mul_mod64(x, b, p); } b = mul_mod64(b, b, p); } } namespace internal { std::optional prime_power_mod_sqrt(long long a, long long p, int q) { std::vector pq(q + 1); pq[0] = 1; for (int i = 1; i <= q; ++i) { pq[i] = pq[i - 1] * p; } a %= pq[q]; if (a == 0) return 0; int b = 0; for (; a % p == 0; a /= p) { ++b; } if (b % 2) { return std::nullopt; } const long long c = pq[b / 2]; q -= b; auto ox0 = prime_mod_sqrt(a, p); if (not ox0) { return std::nullopt; } long long x0 = *ox0; // f(x) = x^2 - a // f(x_i) = 0 mod p^i => f(x_{i+1}) = 0 mod p^{i+1} if (p != 2) { // reference: http://aozoragakuen.sakura.ne.jp/suuron/node24.html const long long inv_df_x0 = inv_mod64(2 * x0, p); for (int i = 1; i < q; ++i) { __int128_t f_x0 = __int128_t(x0) * x0 - a; long long y0 = mul_mod64(-f_x0 / pq[i] % p, inv_df_x0, p); if (y0 < 0) y0 += p; x0 += pq[i] * y0; } return c * x0; } else { // reference: https://twitter.com/maspy_stars/status/1613931151718244352?s=20&t=lAf7ztW2fb_IZa544lo2xw if (a % 8 != 1) return std::nullopt; if (q <= 2) return c * 1; long long x0 = 1; for (int i = 3; i < q; ++i) { __int128_t f_x0 = __int128_t(x0) * x0 - a; x0 |= ((f_x0 >> i) & 1) << (i - 1); } return c * x0; } } } template std::optional composite_mod_sqrt(long long a, const PrimePowers& factorized) { std::vector rs, ms; for (auto [p, q] : factorized) { auto x = internal::prime_power_mod_sqrt(a, p, q); if (not x) { return std::nullopt; } rs.push_back(*x); long long& pq = ms.emplace_back(1); for (int i = 0; i < q; ++i) pq *= p; } return atcoder::crt(rs, ms).first; } } // namespace suisen int main() { long long m, n; std::cin >> m >> n; auto x = suisen::composite_mod_sqrt(n, suisen::fast_factorize::factorize(m)); if (x) { std::cout << "YES" << std::endl; assert(suisen::internal::pow_mod64(*x, 2, m) == n); } else { std::cout << "NO" << std::endl; } }