#include #include #include #include #include #include #include #include #include #include #include #include #include #define getchar getchar_unlocked #define putchar putchar_unlocked #define _rep(_1, _2, _3, _4, name, ...) name #define rep2(i, n) rep3(i, 0, n) #define rep3(i, a, b) rep4(i, a, b, 1) #define rep4(i, a, b, c) for (int i = int(a); i < int(b); i += int(c)) #define rep(...) _rep(__VA_ARGS__, rep4, rep3, rep2, _)(__VA_ARGS__) using namespace std; using i64 = long long; using u8 = unsigned char; using u32 = unsigned; using u64 = unsigned long long; using f80 = long double; int get_int() { int c, n; while ((c = getchar()) < '0'); n = c - '0'; while ((c = getchar()) >= '0') n = n * 10 + (c - '0'); return n; } namespace factor { using u128 = __uint128_t; template struct Mod { Mod() : n_(0) {} Mod(word n) : n_(init(n)) {} static constexpr int word_bits = sizeof(word) * 8; static void set_mod(word m) { mod = m, inv = mul_inv(m), r2 = -dword(m) % m; } static word mul_inv(word n) { word x = n; rep(_, 5) x *= 2 - n * x; return x; } static word reduce(dword w) { word y = (w >> word_bits) - ((dword(word(w) * inv) * mod) >> word_bits); return sword(y) < 0 ? y + mod : y; } static word init(word n) { return reduce(u128(n) * r2); } static word ilog2(word n) { return (n == 0) ? 0 : __lg(n); } Mod& operator += (Mod rhs) { if ((n_ += rhs.n_) >= mod) n_ -= mod; return *this; } Mod& operator -= (Mod rhs) { if (sword(n_ -= rhs.n_) < 0) n_ += mod; return *this; } Mod& operator *= (Mod rhs) { n_ = reduce(u128(n_) * rhs.n_); return *this; } bool operator == (Mod rhs) { return n_ == rhs.n_; } bool operator != (Mod rhs) { return !(*this == rhs); } Mod operator + (Mod rhs) { return Mod(*this) += rhs; } Mod operator - (Mod rhs) { return Mod(*this) -= rhs; } Mod operator * (Mod rhs) { return Mod(*this) *= rhs; } Mod operator - () { return Mod() - *this; }; Mod pow(word e) { Mod ret = Mod(1), base = *this; for (; e; e >>= 1, base *= base) { if (e & 1) ret *= base; } return ret; } word val() { return reduce(n_); } friend ostream& operator << (ostream& os, Mod& m) { return os << m.val(); } word n_; static word mod, inv, r2; }; using m64 = Mod; using m32 = Mod; template <> u32 m32::mod = 0; template <> u32 m32::inv = 0; template <> u32 m32::r2 = 0; template <> u64 m64::mod = 0; template <> u64 m64::inv = 0; template <> u64 m64::r2 = 0; template word gcd(word a, word b) { while (b) { word t = a % b; a = b; b = t; } return a; } template word brent(word n, word c) { // n must be composite and odd. const u64 s = 256; mod::set_mod(n); const mod one = mod(1), mc = mod(c); auto f = [&] (mod x) { return x * x + mc; }; mod y = one; for (u64 l = 1; ; l <<= 1) { auto x = y; rep(_, l) y = f(y); mod p = one; rep(k, 0, l, s) { auto sy = y; rep(_, min(s, l - k)) y = f(y), p *= y - x; word g = gcd(n, p.n_); if (g == 1) continue; if (g == n) for (g = 1, y = sy; g == 1; ) y = f(y), g = gcd(n, (y - x).n_); return g; } } } u64 brent(u64 n, u64 c) { if (n < (u32(1) << 31)) { return brent(n, c); } else if (n < (u64(1) << 63)) { return brent(n, c); } else { assert(0); } } template u32 composite(word n, word d, int s, const u32* bases, int base_size) { mod::set_mod(n); mod one = mod(1), minus_one = -one; rep(i, base_size) { mod b = mod(bases[i]).pow(d); if (b == one || b == minus_one) continue; int t = s - 1; for (; t > 0; --t) if ((b *= b) == minus_one) break; if (t == 0) return bases[i]; } return false; } bool miller_rabin(u64 n) { static const u32 bases[][7] { {2, 3}, {2, 299417}, {2, 7, 61}, {15, 176006322, u32(4221622697)}, {2, 2570940, 211991001, u32(3749873356)}, {2, 2570940, 880937, 610386380, u32(4130785767)}, {2, 325, 9375, 28178, 450775, 9780504, 1795265022}, }; if (n <= 1) return false; if (!(n & 1)) return n == 2; if (n <= 8) return true; u64 d = n - 1; u64 s = __builtin_ctzll(d); d >>= s; u32 base_id = 6, base_size = 7; if (n < 1373653) { base_id = 0, base_size = 2; } else if (n < 19471033) { base_id = 1, base_size = 2; } else if (n < u64(4759123141)) { base_id = 2, base_size = 3; } else if (n < u64(154639673381)) { base_id = 3, base_size = 3; } else if (n < u64(47636622961201)) { base_id = 4, base_size = 4; } else if (n < u64(3770579582154547)) { base_id = 5, base_size = 5; } if (n < (u32(1) << 31)) { return !composite(n, d, s, bases[base_id], base_size); } else if (n < (u64(1) << 63)) { return !composite(n, d, s, bases[base_id], base_size); } else assert(0); return true; } struct ExactDiv { ExactDiv() {} ExactDiv(u64 n) : n(n), i(m64::mul_inv(n)), t(u64(-1) / n) {} friend u64 operator / (u64 n, ExactDiv d) { return n * d.i; }; bool divide(u64 n) { return n / *this <= t; } u64 n, i, t; }; vector primes; void init(u32 n) { u32 sqrt_n = sqrt(n); vector isprime(n + 1, 1); rep(i, 2, sqrt_n + 1) if (isprime[i]) rep(j, i * i, n + 1, i) isprime[j] = 0; primes.clear(); rep(i, 2, n + 1) if (isprime[i]) primes.push_back(ExactDiv(i)); } u32 isqrt(u64 n) { return sqrtl(n); } u64 square(u64 n) { return n * n; } u64 ctz(u64 n) { return __builtin_ctzll(n); } bool is_prime(u64 n) { if (n <= 1) return false; if (!(n & 1)) return n == 2; rep(i, 1, primes.size()) { auto p = primes[i]; if (square(p.n) > n) return true; if (p.divide(n)) return n == p.n; } return miller_rabin(n); } u64 next_prime(u64 n) { if (n <= 1) return 2; if (n <= 2) return 3; for (n = (n + 1) / 2 * 2 + 1; !is_prime(n); n += 2); return n; } bool is_prime_power_2(u64 n) { if (n <= 1) return false; if (!(n & 1)) return (n & (n - 1)) == 0; const u32 bases[] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022}; for (u64 q = n; ; ) { int s = __builtin_ctzll(q - 1); u64 d = (q - 1) >> s; u32 a = composite(q, d, s, bases, 7); if (a % q == 0) { while (n % q == 0) n /= q; return n == 1; } m64 b = a; u64 g = gcd((b.pow(q) - b).val(), q); if (g == 1 || g == q) return false; q = g; } } u128 ipow(u128 a, int e) { u128 ret = 1; for (; e; e >>= 1, a = a * a) { if (e & 1) ret *= a; } return ret; } pair kth_integer_root(u64 n, int k) { if (k == 1) return {n, true}; if (k == 2) { u64 ret = sqrtl(n); return {ret, ret * ret == n}; } if (k == 3) { u32 ret = cbrtl(n); return {ret, u64(ret) * ret * ret == n}; } u64 ret = pow(n, 1. / k); while (ipow(ret, k) < n) ++ret; u128 a = 0; while ((a = ipow(ret, k)) > n) --ret; return {ret, a == n}; } bool is_prime_power(u64 n) { if (n <= 1) return false; if (!(n & 1)) return (n & (n - 1)) == 0; rep(i, 1, primes.size()) { auto p = primes[i]; if (square(p.n) > n) return true; if (!p.divide(n)) continue; while (p.divide(n)) n = n / p; return n == 1; } static const int ps[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 }; rep(i, 18) { for (int k = ps[i]; ;) { u64 a; bool exact; tie(a, exact) = kth_integer_root(n, k); if (a <= primes.back().n) return miller_rabin(n); if (!exact) break; n = a; } } assert(0); } using P = pair; vector

factors(u64 n) { assert(n < (u64(1) << 63)); auto ret = vector

(); if (n <= 1) { return ret; } u32 v = isqrt(n); if (u64(v) * v == n) { auto res = factors(v); for (auto& pe : res) pe.second <<= 1; return res; } if (!(n & 1)) { u32 e = ctz(n); ret.emplace_back(2, e); n >>= e; } u64 lim = square(primes[primes.size() - 1].n); rep(pi, 1, primes.size()) { auto p = primes[pi]; if (square(p.n) > n) break; if (p.divide(n)) { u32 e = 1; n = n / p; while (p.divide(n)) n = n / p, e++; ret.emplace_back(p.n, e); } } u32 s = ret.size(); while (n > lim && !miller_rabin(n)) { for (u64 c = 1; ; ++c) { u64 p = brent(n, c); if (!miller_rabin(p)) continue; u32 e = 1; n /= p; while (n % p == 0) n /= p, e += 1; ret.emplace_back(p, e); break; } } if (n > 1) ret.emplace_back(n, 1); if (ret.size() - s >= 2) sort(ret.begin() + s, ret.end()); return ret; } } // namespace factor void solve() { factor::init(512); int T = get_int(); rep(_, T) { i64 n; scanf("%lld", &n); if (n == 2) { puts("No"); } else if (n % 2 == 0) { puts("Yes"); } else { bool found = false; for (i64 two = 2; two <= n; two <<= 1) { if (factor::is_prime_power(n - two)) { found = true; break; } } puts(found ? "Yes" : "No"); } } } int main() { clock_t beg = clock(); solve(); clock_t end = clock(); fprintf(stderr, "%.3f sec\n", double(end - beg) / CLOCKS_PER_SEC); return 0; }