cp-library-cpp
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$a\uparrow\uparrow b \bmod m$
(library/number/tetration_mod.hpp)
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- Last update: 2023-09-15 20:02:25+09:00
- Include:
#include "library/number/tetration_mod.hpp"
$a\uparrow\uparrow b \bmod m$
まず、$m=1$ の場合の答えは $0$ である。
$\gcd\left(\dfrac{m}{\gcd(m,a ^ i)}, a\right) = 1$ となる最小の非負整数 $i$ を $i _ 0$ とおく。また、$m _ 0 = \dfrac{m}{\gcd(m, a ^ {i _ 0})}$ とおく。
- $a\uparrow\uparrow (b - 1) \geq i _ 0$ の場合: $x\equiv a\uparrow\uparrow (b - 1) \bmod \varphi(m _ 0)$ かつ $x \geq i _ 0$ なる整数 $x$ に対して以下が成り立つので、$a\uparrow\uparrow (b - 1) \bmod \varphi(m _ 0)$ を再帰的に計算する。
- $a\uparrow\uparrow (b - 1) \lt i _ 0$ の場合: 愚直に $a\uparrow\uparrow b = a ^ {a\uparrow\uparrow (b - 1)} \bmod m$ を計算する。
$m \geq 2 \Rightarrow \varphi(\varphi(m)) \leq \dfrac{1}{2}m$ および $d \mid m \Rightarrow \varphi(d) \mid \varphi(m)$ より、2 回再帰するたびに $m$ は半分以下になるので、再帰の深さは $O(\log m)$ である。
Depends on
- Deterministic Miller Rabin
(library/number/deterministic_miller_rabin.hpp)
- 高速素因数分解
(library/number/fast_factorize.hpp)
- Internal Eratosthenes
(library/number/internal_eratosthenes.hpp)
- Montogomery
(library/number/montogomery.hpp)
- エラトステネスの篩
(library/number/sieve_of_eratosthenes.hpp)
- Type Traits
(library/type_traits/type_traits.hpp)
Verified with
Code
#ifndef SUISEN_TETRATION_MOD
#define SUISEN_TETRATION_MOD
#include <numeric>
#include <optional>
#include "library/number/fast_factorize.hpp"
#include "library/number/sieve_of_eratosthenes.hpp"
/**
* @brief $a \uparrow \uparrow b \pmod{m}$
*/
namespace suisen {
namespace internal::tetration_mod {
constexpr int max_value = std::numeric_limits<int>::max();
int saturation_pow(int a, int b) {
if (b >= 32) return max_value;
long long res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res = std::min(res * pow_a, (long long) max_value);
pow_a = std::min(pow_a * pow_a, (long long) max_value);
}
return res;
}
int saturation_tetration(int a, int b) {
assert(a >= 2);
if (b == 0) return 1;
int exponent = 1;
for (int i = 0; i < b and exponent != max_value; ++i) exponent = saturation_pow(a, exponent);
return exponent;
}
int pow_mod(int a, int b, int m) {
long long res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res = (res * pow_a) % m;
pow_a = (pow_a * pow_a) % m;
}
return res;
}
}
/**
* @brief Calculates a↑↑b mod m (= a^(a^(a^...(b times)...)) mod m)
* @param a base
* @param b number of power operations
* @param m mod
* @return a↑↑b mod m
*/
int tetration_mod(int a, int b, int m) {
using namespace internal::tetration_mod;
if (m == 1) return 0;
if (a == 0) return 1 ^ (b & 1);
if (a == 1 or b == 0) return 1;
int i0 = 0, m0 = m;
for (int g = std::gcd(m0, a); g != 1; g = std::gcd(m0, g)) {
m0 /= g, ++i0;
}
int phi = m0;
for (auto [p, q] : fast_factorize::factorize(m0)) {
phi /= p, phi *= p - 1;
}
int exponent = saturation_tetration(a, b - 1);
if (exponent == max_value) {
exponent = tetration_mod(a, b - 1, phi);
if (i0 > exponent) {
exponent += (((i0 - exponent) + phi - 1) / phi) * phi;
}
} else if (i0 <= exponent) {
exponent -= ((exponent - i0) / phi) * phi;
}
return pow_mod(a, exponent, m);
}
}
#endif // SUISEN_TETRATION_MOD#line 1 "library/number/tetration_mod.hpp"
#include <numeric>
#include <optional>
#line 1 "library/number/fast_factorize.hpp"
#include <cmath>
#include <iostream>
#include <random>
#line 8 "library/number/fast_factorize.hpp"
#include <utility>
#line 1 "library/type_traits/type_traits.hpp"
#include <limits>
#line 6 "library/type_traits/type_traits.hpp"
#include <type_traits>
namespace suisen {
template <typename ...Constraints> using constraints_t = std::enable_if_t<std::conjunction_v<Constraints...>, std::nullptr_t>;
template <typename T, typename = std::nullptr_t> struct bitnum { static constexpr int value = 0; };
template <typename T> struct bitnum<T, constraints_t<std::is_integral<T>>> { static constexpr int value = std::numeric_limits<std::make_unsigned_t<T>>::digits; };
template <typename T> static constexpr int bitnum_v = bitnum<T>::value;
template <typename T, size_t n> struct is_nbit { static constexpr bool value = bitnum_v<T> == n; };
template <typename T, size_t n> static constexpr bool is_nbit_v = is_nbit<T, n>::value;
template <typename T, typename = std::nullptr_t> struct safely_multipliable { using type = T; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 32>>> { using type = long long; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 64>>> { using type = __int128_t; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 32>>> { using type = unsigned long long; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 64>>> { using type = __uint128_t; };
template <typename T> using safely_multipliable_t = typename safely_multipliable<T>::type;
template <typename T, typename = void> struct rec_value_type { using type = T; };
template <typename T> struct rec_value_type<T, std::void_t<typename T::value_type>> {
using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T> using rec_value_type_t = typename rec_value_type<T>::type;
template <typename T> class is_iterable {
template <typename T_> static auto test(T_ e) -> decltype(e.begin(), e.end(), std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_iterable_v = is_iterable<T>::value;
template <typename T> class is_writable {
template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::ostream&>() << e, std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_writable_v = is_writable<T>::value;
template <typename T> class is_readable {
template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::istream&>() >> e, std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_readable_v = is_readable<T>::value;
} // namespace suisen
#line 11 "library/number/fast_factorize.hpp"
#line 1 "library/number/deterministic_miller_rabin.hpp"
#include <array>
#include <cassert>
#include <cstdint>
#include <iterator>
#include <tuple>
#line 10 "library/number/deterministic_miller_rabin.hpp"
#line 1 "library/number/montogomery.hpp"
#line 7 "library/number/montogomery.hpp"
namespace suisen {
namespace internal::montgomery {
template <typename Int, typename DInt>
struct Montgomery {
private:
static constexpr uint32_t bits = std::numeric_limits<Int>::digits;
static constexpr Int mask = ~Int(0);
// R = 2**32 or 2**64
// 1. N is an odd number
// 2. N < R
// 3. gcd(N, R) = 1
// 4. R * R2 - N * N2 = 1
// 5. 0 < R2 < N
// 6. 0 < N2 < R
Int N, N2, R2;
// RR = R * R (mod N)
Int RR;
public:
constexpr Montgomery() = default;
explicit constexpr Montgomery(Int N) : N(N), N2(calcN2(N)), R2(calcR2(N, N2)), RR(calcRR(N)) {
assert(N & 1);
}
// @returns t * R (mod N)
constexpr Int make(Int t) const {
return reduce(static_cast<DInt>(t) * RR);
}
// @returns T * R^(-1) (mod N)
constexpr Int reduce(DInt T) const {
// 0 <= T < RN
// Note:
// 1. m = T * N2 (mod R)
// 2. 0 <= m < R
DInt m = modR(static_cast<DInt>(modR(T)) * N2);
// Note:
// T + m * N = T + T * N * N2 = T + T * (R * R2 - 1) = 0 (mod R)
// => (T + m * N) / R is an integer.
// => t * R = T + m * N = T (mod N)
// => t = T R^(-1) (mod N)
DInt t = divR(T + m * N);
// Note:
// 1. 0 <= T < RN
// 2. 0 <= mN < RN (because 0 <= m < R)
// => 0 <= T + mN < 2RN
// => 0 <= t < 2N
return t >= N ? t - N : t;
}
constexpr Int add(Int A, Int B) const {
return (A += B) >= N ? A - N : A;
}
constexpr Int sub(Int A, Int B) const {
return (A -= B) < 0 ? A + N : A;
}
constexpr Int mul(Int A, Int B) const {
return reduce(static_cast<DInt>(A) * B);
}
constexpr Int div(Int A, Int B) const {
return reduce(static_cast<DInt>(A) * inv(B));
}
constexpr Int inv(Int A) const; // TODO: Implement
constexpr Int pow(Int A, long long b) const {
Int P = make(1);
for (; b; b >>= 1) {
if (b & 1) P = mul(P, A);
A = mul(A, A);
}
return P;
}
private:
static constexpr Int divR(DInt t) { return t >> bits; }
static constexpr Int modR(DInt t) { return t & mask; }
static constexpr Int calcN2(Int N) {
// - N * N2 = 1 (mod R)
// N2 = -N^{-1} (mod R)
// calculates N^{-1} (mod R) by Newton's method
DInt invN = N; // = N^{-1} (mod 2^2)
for (uint32_t cur_bits = 2; cur_bits < bits; cur_bits *= 2) {
// loop invariant: invN = N^{-1} (mod 2^cur_bits)
// x = a^{-1} mod m => x(2-ax) = a^{-1} mod m^2 because:
// ax = 1 (mod m)
// => (ax-1)^2 = 0 (mod m^2)
// => 2ax - a^2x^2 = 1 (mod m^2)
// => a(x(2-ax)) = 1 (mod m^2)
invN = modR(invN * modR(2 - N * invN));
}
assert(modR(N * invN) == 1);
return modR(-invN);
}
static constexpr Int calcR2(Int N, Int N2) {
// R * R2 - N * N2 = 1
// => R2 = (1 + N * N2) / R
return divR(1 + static_cast<DInt>(N) * N2);
}
static constexpr Int calcRR(Int N) {
return -DInt(N) % N;
}
};
} // namespace internal::montgomery
using Montgomery32 = internal::montgomery::Montgomery<uint32_t, uint64_t>;
using Montgomery64 = internal::montgomery::Montgomery<uint64_t, __uint128_t>;
} // namespace suisen
#line 12 "library/number/deterministic_miller_rabin.hpp"
namespace suisen::miller_rabin {
namespace internal {
constexpr uint64_t THRESHOLD_1 = 341531ULL;
constexpr uint64_t BASE_1[]{ 9345883071009581737ULL };
constexpr uint64_t THRESHOLD_2 = 1050535501ULL;
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 uint64_t BASE_7[]{ 2U, 325U, 9375U, 28178U, 450775U, 9780504U, 1795265022U };
template <auto BASE, std::size_t SIZE>
constexpr bool miller_rabin(uint64_t n) {
if (n == 2 or n == 3 or n == 5 or n == 7) return true;
if (n <= 1 or n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0) return false;
if (n < 121) return true;
const uint32_t s = __builtin_ctzll(n - 1); // >= 1
const uint64_t d = (n - 1) >> s;
const Montgomery64 mg{ n };
const uint64_t one = mg.make(1), minus_one = mg.make(n - 1);
for (std::size_t i = 0; i < SIZE; ++i) {
uint64_t a = BASE[i] % n;
if (a == 0) continue;
uint64_t Y = mg.pow(mg.make(a), d);
if (Y == one) continue;
for (uint32_t r = 0;; ++r, Y = mg.mul(Y, Y)) {
// Y = a^(d 2^r)
if (Y == minus_one) break;
if (r == s - 1) return false;
}
}
return true;
}
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
constexpr bool is_prime(T n) {
if constexpr (std::is_signed_v<T>) {
assert(n >= 0);
}
const std::make_unsigned_t<T> n_unsigned = n;
assert(n_unsigned <= std::numeric_limits<uint64_t>::max()); // n < 2^64
using namespace internal;
if (n_unsigned < THRESHOLD_1) return miller_rabin<BASE_1, 1>(n_unsigned);
if (n_unsigned < THRESHOLD_2) return miller_rabin<BASE_2, 2>(n_unsigned);
if (n_unsigned < THRESHOLD_3) return miller_rabin<BASE_3, 3>(n_unsigned);
if (n_unsigned < THRESHOLD_4) return miller_rabin<BASE_4, 4>(n_unsigned);
if (n_unsigned < THRESHOLD_5) return miller_rabin<BASE_5, 5>(n_unsigned);
if (n_unsigned < THRESHOLD_6) return miller_rabin<BASE_6, 6>(n_unsigned);
return miller_rabin<BASE_7, 7>(n_unsigned);
}
} // namespace suisen::miller_rabin
#line 1 "library/number/sieve_of_eratosthenes.hpp"
#line 6 "library/number/sieve_of_eratosthenes.hpp"
#include <vector>
#line 1 "library/number/internal_eratosthenes.hpp"
#line 6 "library/number/internal_eratosthenes.hpp"
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
#line 9 "library/number/sieve_of_eratosthenes.hpp"
namespace suisen {
template <unsigned int N>
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<int> prime_list(unsigned int max_val = N) const {
using namespace internal::sieve;
std::vector<int> 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 <unsigned int N>
std::uint8_t SimpleSieve<N>::flag[SimpleSieve<N>::siz];
template <unsigned int N>
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<std::pair<int, int>> factorize(unsigned int n) const {
assert(0 < n and n <= N);
std::vector<std::pair<int, int>> 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<int> divisors(unsigned int n) const {
assert(0 < n and n <= N);
std::vector<int> 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 N>
unsigned int Sieve<N>::pf[Sieve<N>::base_max + internal::sieve::K];
} // namespace suisen
#line 14 "library/number/fast_factorize.hpp"
namespace suisen::fast_factorize {
namespace internal {
template <typename T>
constexpr int floor_log2(T n) {
int i = 0;
while (n) n >>= 1, ++i;
return i - 1;
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T pollard_rho(const T n) {
using M = safely_multipliable_t<T>;
const T m = T(1) << (floor_log2(n) / 5);
static std::mt19937_64 rng{std::random_device{}()};
std::uniform_int_distribution<T> dist(0, n - 1);
// const Montgomery64 mg{n};
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 <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<std::pair<T, int>> factorize(T n) {
static constexpr int threshold = 1000000;
static Sieve<threshold> sieve;
std::vector<std::pair<T, int>> 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
#line 9 "library/number/tetration_mod.hpp"
/**
* @brief $a \uparrow \uparrow b \pmod{m}$
*/
namespace suisen {
namespace internal::tetration_mod {
constexpr int max_value = std::numeric_limits<int>::max();
int saturation_pow(int a, int b) {
if (b >= 32) return max_value;
long long res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res = std::min(res * pow_a, (long long) max_value);
pow_a = std::min(pow_a * pow_a, (long long) max_value);
}
return res;
}
int saturation_tetration(int a, int b) {
assert(a >= 2);
if (b == 0) return 1;
int exponent = 1;
for (int i = 0; i < b and exponent != max_value; ++i) exponent = saturation_pow(a, exponent);
return exponent;
}
int pow_mod(int a, int b, int m) {
long long res = 1, pow_a = a;
for (; b; b >>= 1) {
if (b & 1) res = (res * pow_a) % m;
pow_a = (pow_a * pow_a) % m;
}
return res;
}
}
/**
* @brief Calculates a↑↑b mod m (= a^(a^(a^...(b times)...)) mod m)
* @param a base
* @param b number of power operations
* @param m mod
* @return a↑↑b mod m
*/
int tetration_mod(int a, int b, int m) {
using namespace internal::tetration_mod;
if (m == 1) return 0;
if (a == 0) return 1 ^ (b & 1);
if (a == 1 or b == 0) return 1;
int i0 = 0, m0 = m;
for (int g = std::gcd(m0, a); g != 1; g = std::gcd(m0, g)) {
m0 /= g, ++i0;
}
int phi = m0;
for (auto [p, q] : fast_factorize::factorize(m0)) {
phi /= p, phi *= p - 1;
}
int exponent = saturation_tetration(a, b - 1);
if (exponent == max_value) {
exponent = tetration_mod(a, b - 1, phi);
if (i0 > exponent) {
exponent += (((i0 - exponent) + phi - 1) / phi) * phi;
}
} else if (i0 <= exponent) {
exponent -= ((exponent - i0) / phi) * phi;
}
return pow_mod(a, exponent, m);
}
}