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:question: nt/factor.hpp

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#pragma once

#include "random/base.hpp"
#include "nt/primetest.hpp"

template <typename mint>
ll rho(ll n, ll c) {
  assert(n > 1);
  const mint cc(c);
  auto f = [&](mint x) { return x * x + cc; };
  mint x = 1, y = 2, z = 1, q = 1;
  ll g = 1;
  const ll m = 1LL << (__lg(n) / 5);
  for (ll r = 1; g == 1; r <<= 1) {
    x = y;
    FOR(r) y = f(y);
    for (ll k = 0; k < r && g == 1; k += m) {
      z = y;
      FOR(min(m, r - k)) y = f(y), q *= x - y;
      g = gcd(q.val(), n);
    }
  }
  if (g == n) do {
      z = f(z);
      g = gcd((x - z).val(), n);
    } while (g == 1);
  return g;
}

ll find_prime_factor(ll n) {
  assert(n > 1);
  if (primetest(n)) return n;
  FOR(100) {
    ll m = 0;
    if (n < (1 << 30)) {
      using mint = Mongomery_modint_32<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    } else {
      using mint = Mongomery_modint_64<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    }
    if (primetest(m)) return m;
    n = m;
  }
  assert(0);
  return -1;
}

// ソートしてくれる
vc<pair<ll, int>> factor(ll n) {
  assert(n >= 1);
  vc<pair<ll, int>> pf;
  FOR(p, 2, 100) {
    if (p * p > n) break;
    if (n % p == 0) {
      ll e = 0;
      do { n /= p, e += 1; } while (n % p == 0);
      pf.eb(p, e);
    }
  }
  while (n > 1) {
    ll p = find_prime_factor(n);
    ll e = 0;
    do { n /= p, e += 1; } while (n % p == 0);
    pf.eb(p, e);
  }
  sort(all(pf));
  return pf;
}

vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) {
  vc<pair<ll, int>> res;
  while (n > 1) {
    int p = lpf[n];
    int e = 0;
    while (n % p == 0) {
      n /= p;
      ++e;
    }
    res.eb(p, e);
  }
  return res;
}
#line 2 "nt/factor.hpp"

#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                     chrono::high_resolution_clock::now().time_since_epoch())
                     .count())
        * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "mod/mongomery_modint.hpp"

// odd mod.
// x の代わりに rx を持つ
template <int id, typename U1, typename U2>
struct Mongomery_modint {
  using mint = Mongomery_modint;
  inline static U1 m, r, n2;
  static constexpr int W = numeric_limits<U1>::digits;

  static void set_mod(U1 mod) {
    assert(mod & 1 && mod <= U1(1) << (W - 2));
    m = mod, n2 = -U2(m) % m, r = m;
    FOR(5) r *= 2 - m * r;
    r = -r;
    assert(r * m == U1(-1));
  }
  static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; }

  U1 x;
  Mongomery_modint() : x(0) {}
  Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){};
  U1 val() const {
    U1 y = reduce(x);
    return y >= m ? y - m : y;
  }
  mint &operator+=(mint y) {
    x = ((x += y.x) >= m ? x - m : x);
    return *this;
  }
  mint &operator-=(mint y) {
    x -= (x >= y.x ? y.x : y.x - m);
    return *this;
  }
  mint &operator*=(mint y) {
    x = reduce(U2(x) * y.x);
    return *this;
  }
  mint operator+(mint y) const { return mint(*this) += y; }
  mint operator-(mint y) const { return mint(*this) -= y; }
  mint operator*(mint y) const { return mint(*this) *= y; }
  bool operator==(mint y) const {
    return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
  }
  bool operator!=(mint y) const { return not operator==(y); }
  mint pow(ll n) const {
    assert(n >= 0);
    mint y = 1, z = *this;
    for (; n; n >>= 1, z *= z)
      if (n & 1) y *= z;
    return y;
  }
};

template <int id>
using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>;
template <int id>
using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>;
#line 3 "nt/primetest.hpp"

bool primetest(const u64 x) {
  assert(x < u64(1) << 62);
  if (x == 2 or x == 3 or x == 5 or x == 7) return true;
  if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false;
  if (x < 121) return x > 1;
  const u64 d = (x - 1) >> lowbit(x - 1);

  using mint = Mongomery_modint_64<202311020>;

  mint::set_mod(x);
  const mint one(u64(1)), minus_one(x - 1);
  auto ok = [&](u64 a) -> bool {
    auto y = mint(a).pow(d);
    u64 t = d;
    while (y != one && y != minus_one && t != x - 1) y *= y, t <<= 1;
    if (y != minus_one && t % 2 == 0) return false;
    return true;
  };
  if (x < (u64(1) << 32)) {
    for (u64 a: {2, 7, 61})
      if (!ok(a)) return false;
  } else {
    for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
      if (!ok(a)) return false;
    }
  }
  return true;
}
#line 5 "nt/factor.hpp"

template <typename mint>
ll rho(ll n, ll c) {
  assert(n > 1);
  const mint cc(c);
  auto f = [&](mint x) { return x * x + cc; };
  mint x = 1, y = 2, z = 1, q = 1;
  ll g = 1;
  const ll m = 1LL << (__lg(n) / 5);
  for (ll r = 1; g == 1; r <<= 1) {
    x = y;
    FOR(r) y = f(y);
    for (ll k = 0; k < r && g == 1; k += m) {
      z = y;
      FOR(min(m, r - k)) y = f(y), q *= x - y;
      g = gcd(q.val(), n);
    }
  }
  if (g == n) do {
      z = f(z);
      g = gcd((x - z).val(), n);
    } while (g == 1);
  return g;
}

ll find_prime_factor(ll n) {
  assert(n > 1);
  if (primetest(n)) return n;
  FOR(100) {
    ll m = 0;
    if (n < (1 << 30)) {
      using mint = Mongomery_modint_32<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    } else {
      using mint = Mongomery_modint_64<20231025>;
      mint::set_mod(n);
      m = rho<mint>(n, RNG(0, n));
    }
    if (primetest(m)) return m;
    n = m;
  }
  assert(0);
  return -1;
}

// ソートしてくれる
vc<pair<ll, int>> factor(ll n) {
  assert(n >= 1);
  vc<pair<ll, int>> pf;
  FOR(p, 2, 100) {
    if (p * p > n) break;
    if (n % p == 0) {
      ll e = 0;
      do { n /= p, e += 1; } while (n % p == 0);
      pf.eb(p, e);
    }
  }
  while (n > 1) {
    ll p = find_prime_factor(n);
    ll e = 0;
    do { n /= p, e += 1; } while (n % p == 0);
    pf.eb(p, e);
  }
  sort(all(pf));
  return pf;
}

vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) {
  vc<pair<ll, int>> res;
  while (n > 1) {
    int p = lpf[n];
    int e = 0;
    while (n % p == 0) {
      n /= p;
      ++e;
    }
    res.eb(p, e);
  }
  return res;
}
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