#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; // using lint = long long; using pint = pair; // using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } const std::vector> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}; int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); } template std::pair operator+(const std::pair &l, const std::pair &r) { return std::make_pair(l.first + r.first, l.second + r.second); } template std::pair operator-(const std::pair &l, const std::pair &r) { return std::make_pair(l.first - r.first, l.second - r.second); } template std::vector sort_unique(std::vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template IStream &operator>>(IStream &is, std::vector &vec) { for (auto &v : vec) is >> v; return is; } template OStream &operator<<(OStream &os, const std::vector &vec); template OStream &operator<<(OStream &os, const std::array &arr); template OStream &operator<<(OStream &os, const std::unordered_set &vec); template OStream &operator<<(OStream &os, const pair &pa); template OStream &operator<<(OStream &os, const std::deque &vec); template OStream &operator<<(OStream &os, const std::set &vec); template OStream &operator<<(OStream &os, const std::multiset &vec); template OStream &operator<<(OStream &os, const std::unordered_multiset &vec); template OStream &operator<<(OStream &os, const std::pair &pa); template OStream &operator<<(OStream &os, const std::map &mp); template OStream &operator<<(OStream &os, const std::unordered_map &mp); template OStream &operator<<(OStream &os, const std::tuple &tpl); template OStream &operator<<(OStream &os, const std::vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } template std::istream &operator>>(std::istream &is, std::tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template OStream &operator<<(OStream &os, const std::tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } template OStream &operator<<(OStream &os, const std::unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::pair &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; } template OStream &operator<<(OStream &os, const std::map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl #define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr) #else #define dbg(x) ((void)0) #define dbgif(cond, x) ((void)0) #endif #include #include #include using lint = __int128; struct ModIntRuntime { private: static lint md; public: // using lint = long long; static lint mod() { return md; } lint val_; static std::vector &facs() { static std::vector facs_; return facs_; } static lint &get_primitive_root() { static lint primitive_root_ = 0; if (!primitive_root_) { primitive_root_ = [&]() { std::set fac; lint v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (lint g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModIntRuntime(g).power((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return (lint)-1; }(); } return primitive_root_; } static void set_mod(const lint &m) { if (md != m) facs().clear(); md = m; get_primitive_root() = 0; } ModIntRuntime &_setval(lint v) { val_ = (v >= md ? v - md : v); return *this; } lint val() const noexcept { return val_; } ModIntRuntime() : val_(0) {} ModIntRuntime(lint v) { _setval(v % md + md); } explicit operator bool() const { return val_ != 0; } ModIntRuntime operator+(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ + x.val_); } ModIntRuntime operator-(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ - x.val_ + md); } ModIntRuntime operator*(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ * x.val_ % md); } ModIntRuntime operator/(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ * x.inv().val() % md); } ModIntRuntime operator-() const { return ModIntRuntime()._setval(md - val_); } ModIntRuntime &operator+=(const ModIntRuntime &x) { return *this = *this + x; } ModIntRuntime &operator-=(const ModIntRuntime &x) { return *this = *this - x; } ModIntRuntime &operator*=(const ModIntRuntime &x) { return *this = *this * x; } ModIntRuntime &operator/=(const ModIntRuntime &x) { return *this = *this / x; } friend ModIntRuntime operator+(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md + x.val_); } friend ModIntRuntime operator-(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md - x.val_ + md); } friend ModIntRuntime operator*(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md * x.val_ % md); } friend ModIntRuntime operator/(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md * x.inv().val() % md); } bool operator==(const ModIntRuntime &x) const { return val_ == x.val_; } bool operator!=(const ModIntRuntime &x) const { return val_ != x.val_; } bool operator<(const ModIntRuntime &x) const { return val_ < x.val_; } // To use std::map lint power(lint n) const { lint ans = 1, tmp = this->val_; while (n) { if (n & 1) ans = ans * tmp % md; tmp = tmp * tmp % md; n /= 2; } return ans; } ModIntRuntime pow(lint n) const { return power(n); } ModIntRuntime inv() const { return this->pow(md - 2); } ModIntRuntime fac() const { int l0 = facs().size(); if (l0 > this->val_) return facs()[this->val_]; facs().resize(this->val_ + 1); for (int i = l0; i <= this->val_; i++) facs()[i] = (i == 0 ? ModIntRuntime(1) : facs()[i - 1] * ModIntRuntime(i)); return facs()[this->val_]; } ModIntRuntime doublefac() const { lint k = (this->val_ + 1) / 2; return (this->val_ & 1) ? ModIntRuntime(k * 2).fac() / (ModIntRuntime(2).pow(k) * ModIntRuntime(k).fac()) : ModIntRuntime(k).fac() * ModIntRuntime(2).pow(k); } ModIntRuntime nCr(const ModIntRuntime &r) const { return (this->val_ < r.val_) ? ModIntRuntime(0) : this->fac() / ((*this - r).fac() * r.fac()); } ModIntRuntime sqrt() const { if (val_ == 0) return 0; if (md == 2) return val_; if (power((md - 1) / 2) != 1) return 0; ModIntRuntime b = 1; while (b.power((md - 1) / 2) == 1) b += 1; lint e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModIntRuntime x = power((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModIntRuntime z = b.power(m); while (y != 1) { lint j = 0; ModIntRuntime t = y; while (t != 1) j++, t *= t; z = z.power(lint(1) << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModIntRuntime(std::min(x.val_, md - x.val_)); } }; lint ModIntRuntime::md = 1; // Linear sieve algorithm for fast prime factorization // Complexity: O(N) time, O(N) space: // - MAXN = 10^7: ~44 MB, 80~100 ms (Codeforces / AtCoder GCC, C++17) // - MAXN = 10^8: ~435 MB, 810~980 ms (Codeforces / AtCoder GCC, C++17) // Reference: // [1] D. Gries, J. Misra, "A Linear Sieve Algorithm for Finding Prime Numbers," // Communications of the ACM, 21(12), 999-1003, 1978. // - https://cp-algorithms.com/algebra/prime-sieve-linear.html // - https://37zigen.com/linear-sieve/ struct Sieve { std::vector min_factor; std::vector primes; Sieve(int MAXN) : min_factor(MAXN + 1) { for (int d = 2; d <= MAXN; d++) { if (!min_factor[d]) { min_factor[d] = d; primes.emplace_back(d); } for (const auto &p : primes) { if (p > min_factor[d] or d * p > MAXN) break; min_factor[d * p] = p; } } } // Prime factorization for 1 <= x <= MAXN^2 // Complexity: O(log x) (x <= MAXN) // O(MAXN / log MAXN) (MAXN < x <= MAXN^2) template std::map factorize(T x) const { std::map ret; assert(x > 0 and x <= ((long long)min_factor.size() - 1) * ((long long)min_factor.size() - 1)); for (const auto &p : primes) { if (x < T(min_factor.size())) break; while (!(x % p)) x /= p, ret[p]++; } if (x >= T(min_factor.size())) ret[x]++, x = 1; while (x > 1) ret[min_factor[x]]++, x /= min_factor[x]; return ret; } // Enumerate divisors of 1 <= x <= MAXN^2 // Be careful of highly composite numbers https://oeis.org/A002182/list // https://gist.github.com/dario2994/fb4713f252ca86c1254d#file-list-txt (n, (# of div. of n)): // 45360->100, 735134400(<1e9)->1344, 963761198400(<1e12)->6720 template std::vector divisors(T x) const { std::vector ret{1}; for (const auto p : factorize(x)) { int n = ret.size(); for (int i = 0; i < n; i++) { for (T a = 1, d = 1; d <= p.second; d++) { a *= p.first; ret.push_back(ret[i] * a); } } } return ret; // NOT sorted } // Euler phi functions of divisors of given x // Verified: ABC212 G https://atcoder.jp/contests/abc212/tasks/abc212_g // Complexity: O(sqrt(x) + d(x)) template std::map euler_of_divisors(T x) const { assert(x >= 1); std::map ret; ret[1] = 1; std::vector divs{1}; for (auto p : factorize(x)) { int n = ret.size(); for (int i = 0; i < n; i++) { ret[divs[i] * p.first] = ret[divs[i]] * (p.first - 1); divs.push_back(divs[i] * p.first); for (T a = divs[i] * p.first, d = 1; d < p.second; a *= p.first, d++) { ret[a * p.first] = ret[a] * p.first; divs.push_back(a * p.first); } } } return ret; } // Moebius function Table, (-1)^{# of different prime factors} for square-free x // return: [0=>0, 1=>1, 2=>-1, 3=>-1, 4=>0, 5=>-1, 6=>1, 7=>-1, 8=>0, ...] https://oeis.org/A008683 std::vector GenerateMoebiusFunctionTable() const { std::vector ret(min_factor.size()); for (unsigned i = 1; i < min_factor.size(); i++) { if (i == 1) { ret[i] = 1; } else if ((i / min_factor[i]) % min_factor[i] == 0) { ret[i] = 0; } else { ret[i] = -ret[i / min_factor[i]]; } } return ret; } // Calculate [0^K, 1^K, ..., nmax^K] in O(nmax) // Note: **0^0 == 1** template std::vector enumerate_kth_pows(long long K, int nmax) const { assert(nmax < int(min_factor.size())); assert(K >= 0); if (K == 0) return std::vector(nmax + 1, 1); std::vector ret(nmax + 1); ret[0] = 0, ret[1] = 1; for (int n = 2; n <= nmax; n++) { if (min_factor[n] == n) { ret[n] = MODINT(n).pow(K); } else { ret[n] = ret[n / min_factor[n]] * ret[min_factor[n]]; } } return ret; } }; Sieve sieve(10101010); bool check(lint p, int deg, lint val) { if (p == 2) { // dbg(make_tuple(p, deg, val)); if (deg == 1) return true; if (deg == 2) return val % 4 <= 1; while (val and val % 4 == 0) val /= 4; return val % 8 == 1 or val == 0; } using mint = ModIntRuntime; mint::set_mod(p); mint x = ModIntRuntime(val); mint sq = x.sqrt(); // dbg(make_tuple(p, val, x, sq, sq * sq - x)); dbg(make_tuple((long long)p, (long long)val, (long long)(sq * sq - x).val())); return sq * sq - x == 0; } int main() { long long B, N; cin >> B >> N; dbg(make_tuple(B, N)); auto fac = sieve.factorize(B); dbg(fac); for (auto [p, deg] : fac) { if (!check(p, deg, N)) { puts("NO"); return 0; } } puts("YES"); }