結果

問題 No.577 Prime Powerful Numbers
ユーザー Min_25Min_25
提出日時 2017-10-14 04:06:44
言語 C++14
(gcc 13.2.0 + boost 1.83.0)
結果
AC  
実行時間 4 ms / 2,000 ms
コード長 9,393 bytes
コンパイル時間 1,389 ms
コンパイル使用メモリ 103,568 KB
実行使用メモリ 4,384 KB
最終ジャッジ日時 2023-08-11 05:09:51
合計ジャッジ時間 2,315 ms
ジャッジサーバーID
（参考情報） 		judge11 / judge15
このコードへのチャレンジ（β）

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 2 ms
4,380 KB
testcase_01 AC 2 ms
4,380 KB
testcase_02 AC 1 ms
4,380 KB
testcase_03 AC 2 ms
4,376 KB
testcase_04 AC 1 ms
4,376 KB
testcase_05 AC 3 ms
4,376 KB
testcase_06 AC 2 ms
4,384 KB
testcase_07 AC 4 ms
4,380 KB
testcase_08 AC 2 ms
4,376 KB
testcase_09 AC 2 ms
4,376 KB
testcase_10 AC 2 ms
4,380 KB
権限があれば一括ダウンロードができます

ソースコード

diff # 

#include <cstdio>
#include <cassert>
#include <cmath>
#include <cstring>

#include <iostream>
#include <algorithm>
#include <vector>
#include <map>
#include <set>
#include <functional>
#include <stack>
#include <queue>

#include <tuple>

#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 <typename word, typename dword, typename sword>
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<u64, u128, i64>;
using m32 = Mod<u32, u64, int>;
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 <typename word>
word gcd(word a, word b) {
  while (b) { word t = a % b; a = b; b = t; }
  return a;
}

template <typename word, typename mod>
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<u32, m32>(n, c);
  } else if (n < (u64(1) << 63)) {
    return brent<u64, m64>(n, c);
  } else {
    assert(0);
  }
}

template <typename word, typename mod>
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<u32, m32>(n, d, s, bases[base_id], base_size);
  } else if (n < (u64(1) << 63)) {
    return !composite<u64, m64>(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<ExactDiv> primes;

void init(u32 n) {
  u32 sqrt_n = sqrt(n);
  vector<u8> 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<u64, m64>(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<u64, bool> 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<u64, u32>;
vector<P> factors(u64 n) {
  assert(n < (u64(1) << 63));

  auto ret = vector<P>();
  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;
}
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