#define PROBLEM "https://judge.yosupo.jp/problem/tetration_mod"
#include "../../math/tetration.hpp"
#include <cassert>
#include <cstdint>
#include "../../utilities/fast_io.hpp"
#include <vector>
namespace {
uint64_t minimum(uint64_t first, uint64_t second) {
return first < second ? first : second;
}
uint64_t brute_pow_bounded(uint64_t base, uint64_t exponent, uint64_t limit) {
if (limit == 0) return 0;
if (exponent == 0) return 1 < limit ? 1 : limit;
if (base == 0) return 0;
__uint128_t result = 1;
for (uint64_t i = 0; i < exponent; i++) {
result *= base;
if (result >= limit) return limit;
}
return uint64_t(result);
}
uint64_t brute_tetration_bounded(uint64_t base, uint64_t height, uint64_t limit) {
if (limit == 0) return 0;
if (height == 0) return minimum(1, limit);
if (height == 1) return minimum(base, limit);
uint64_t exponent = brute_tetration_bounded(base, height - 1, limit);
return brute_pow_bounded(base, exponent, limit);
}
uint64_t brute_tower_bounded(
const std::vector<uint64_t>& bases,
int index,
uint64_t limit
) {
if (limit == 0) return 0;
if (index == int(bases.size())) return minimum(1, limit);
if (index + 1 == int(bases.size())) return minimum(bases[index], limit);
uint64_t exponent = brute_tower_bounded(bases, index + 1, limit);
return brute_pow_bounded(bases[index], exponent, limit);
}
void test_fixed_cases() {
assert(m1une::math::tetration_mod(2ULL, 0, 1000) == 1);
assert(m1une::math::tetration_mod(2ULL, 1, 1000) == 2);
assert(m1une::math::tetration_mod(2ULL, 4, 1000) == 536);
assert(m1une::math::tetration_mod(3ULL, 3, 100) == 87);
assert(m1une::math::tetration_mod(0ULL, 0, 1000) == 1);
assert(m1une::math::tetration_mod(0ULL, 1, 1000) == 0);
assert(m1une::math::tetration_mod(0ULL, 2, 1000) == 1);
assert(m1une::math::tetration_mod(0ULL, 3, 1000) == 0);
assert(m1une::math::tetration_mod(7ULL, 100, 1) == 0);
assert(m1une::math::tetration_bounded(2ULL, 5, 1000000) == 1000000);
std::vector<unsigned long long> empty;
assert(m1une::math::power_tower_mod(empty, 37) == 1);
assert(m1une::math::power_tower_bounded(empty, 10) == 1);
std::vector<unsigned long long> zero_zero;
zero_zero.push_back(0);
zero_zero.push_back(0);
assert(m1une::math::power_tower_mod(zero_zero, 1000) == 1);
std::vector<unsigned long long> two_zero;
two_zero.push_back(2);
two_zero.push_back(0);
assert(m1une::math::power_tower_mod(two_zero, 1000) == 1);
std::vector<unsigned long long> two_three_four;
two_three_four.push_back(2);
two_three_four.push_back(3);
two_three_four.push_back(4);
assert(m1une::math::power_tower_mod(two_three_four, 1000) == 352);
}
void test_tetration_against_bruteforce() {
const uint64_t cap = 1000000;
for (uint64_t base = 0; base <= 8; base++) {
for (uint64_t height = 0; height <= 7; height++) {
for (uint64_t limit = 0; limit <= 1000; limit++) {
uint64_t expected = brute_tetration_bounded(base, height, limit);
uint64_t actual = m1une::math::tetration_bounded(base, height, limit);
assert(actual == expected);
}
uint64_t exact = brute_tetration_bounded(base, height, cap);
if (exact == cap) continue;
for (uint64_t mod = 1; mod <= 257; mod++) {
assert(m1une::math::tetration_mod(base, height, mod) == exact % mod);
}
}
}
}
void enumerate_towers(
int length,
int position,
std::vector<uint64_t>& bases
) {
if (position == length) {
const uint64_t cap = 1000000;
for (uint64_t limit = 0; limit <= 300; limit++) {
uint64_t expected = brute_tower_bounded(bases, 0, limit);
uint64_t actual = m1une::math::power_tower_bounded(bases, limit);
assert(actual == expected);
}
uint64_t exact = brute_tower_bounded(bases, 0, cap);
if (exact == cap) return;
for (uint64_t mod = 1; mod <= 127; mod++) {
assert(m1une::math::power_tower_mod(bases, mod) == exact % mod);
}
return;
}
for (uint64_t base = 0; base <= 4; base++) {
bases[position] = base;
enumerate_towers(length, position + 1, bases);
}
}
void test_power_tower_against_bruteforce() {
for (int length = 0; length <= 5; length++) {
std::vector<uint64_t> bases(length);
enumerate_towers(length, 0, bases);
}
}
} // namespace
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_fixed_cases();
test_tetration_against_bruteforce();
test_power_tower_against_bruteforce();
int test_count;
fast_input >> test_count;
while (test_count--) {
uint64_t base, height, modulus;
fast_input >> base >> height >> modulus;
fast_output << m1une::math::tetration_mod(base, height, modulus) << '\n';
}
}
#line 1 "verify/math/tetration.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/tetration_mod"
#line 1 "math/tetration.hpp"
#include <cassert>
#include <concepts>
#include <cstdint>
#include <type_traits>
#include <vector>
#line 1 "math/prime_factorization.hpp"
#include <algorithm>
#line 7 "math/prime_factorization.hpp"
#include <numeric>
#include <utility>
#line 10 "math/prime_factorization.hpp"
namespace m1une {
namespace math {
namespace internal {
inline uint64_t multiply_mod(uint64_t a, uint64_t b, uint64_t mod) {
return static_cast<uint64_t>(static_cast<unsigned __int128>(a) * b % mod);
}
inline uint64_t power_mod(uint64_t base, uint64_t exponent, uint64_t mod) {
uint64_t result = 1;
while (exponent > 0) {
if (exponent & 1) result = multiply_mod(result, base, mod);
base = multiply_mod(base, base, mod);
exponent >>= 1;
}
return result;
}
inline uint64_t pollard_random() {
static uint64_t state = 0x123456789abcdef0ULL;
state += 0x9e3779b97f4a7c15ULL;
uint64_t value = state;
value = (value ^ (value >> 30)) * 0xbf58476d1ce4e5b9ULL;
value = (value ^ (value >> 27)) * 0x94d049bb133111ebULL;
return value ^ (value >> 31);
}
} // namespace internal
inline bool is_prime(uint64_t value) {
if (value < 2) return false;
for (uint64_t prime : {2ULL, 3ULL, 5ULL, 7ULL, 11ULL, 13ULL, 17ULL, 19ULL, 23ULL, 29ULL, 31ULL, 37ULL}) {
if (value % prime == 0) return value == prime;
}
uint64_t odd_part = value - 1;
int power_of_two = 0;
while ((odd_part & 1) == 0) {
odd_part >>= 1;
power_of_two++;
}
for (uint64_t base : {2ULL, 325ULL, 9375ULL, 28178ULL, 450775ULL, 9780504ULL, 1795265022ULL}) {
if (base % value == 0) continue;
uint64_t x = internal::power_mod(base % value, odd_part, value);
if (x == 1 || x == value - 1) continue;
bool composite = true;
for (int i = 1; i < power_of_two; i++) {
x = internal::multiply_mod(x, x, value);
if (x == value - 1) {
composite = false;
break;
}
}
if (composite) return false;
}
return true;
}
namespace internal {
inline uint64_t pollard_rho(uint64_t value) {
for (uint64_t prime : {2ULL, 3ULL, 5ULL, 7ULL, 11ULL, 13ULL, 17ULL, 19ULL, 23ULL, 29ULL, 31ULL, 37ULL}) {
if (value % prime == 0) return prime;
}
while (true) {
const uint64_t constant = pollard_random() % (value - 1) + 1;
uint64_t y = pollard_random() % (value - 1) + 1;
uint64_t x = 0;
uint64_t saved_y = 0;
uint64_t gcd = 1;
uint64_t segment_length = 1;
auto advance = [&](uint64_t current) {
return static_cast<uint64_t>(
(static_cast<unsigned __int128>(multiply_mod(current, current, value)) + constant) % value);
};
while (gcd == 1) {
x = y;
for (uint64_t i = 0; i < segment_length; i++) y = advance(y);
for (uint64_t offset = 0; offset < segment_length && gcd == 1; offset += 128) {
saved_y = y;
uint64_t product = 1;
const uint64_t block = std::min<uint64_t>(128, segment_length - offset);
for (uint64_t i = 0; i < block; i++) {
y = advance(y);
const uint64_t difference = x > y ? x - y : y - x;
product = multiply_mod(product, difference, value);
}
gcd = std::gcd(product, value);
}
segment_length <<= 1;
}
if (gcd == value) {
do {
saved_y = advance(saved_y);
const uint64_t difference = x > saved_y ? x - saved_y : saved_y - x;
gcd = std::gcd(difference, value);
} while (gcd == 1);
}
if (gcd != value) return gcd;
}
}
inline void factor_recursively(uint64_t value, std::vector<uint64_t>& factors) {
if (value == 1) return;
if (is_prime(value)) {
factors.push_back(value);
return;
}
const uint64_t divisor = pollard_rho(value);
factor_recursively(divisor, factors);
factor_recursively(value / divisor, factors);
}
} // namespace internal
inline std::vector<uint64_t> prime_factors(uint64_t value) {
assert(value >= 1);
std::vector<uint64_t> result;
internal::factor_recursively(value, result);
std::sort(result.begin(), result.end());
return result;
}
inline std::vector<std::pair<uint64_t, int>> prime_factorize(uint64_t value) {
std::vector<uint64_t> factors = prime_factors(value);
std::vector<std::pair<uint64_t, int>> result;
for (uint64_t prime : factors) {
if (result.empty() || result.back().first != prime) {
result.emplace_back(prime, 1);
} else {
result.back().second++;
}
}
return result;
}
inline std::vector<uint64_t> divisors(uint64_t value) {
std::vector<uint64_t> result = {1};
for (const auto& factor : prime_factorize(value)) {
const int current_size = int(result.size());
uint64_t power = 1;
for (int exponent = 1; exponent <= factor.second; exponent++) {
power *= factor.first;
for (int i = 0; i < current_size; i++) {
result.push_back(result[i] * power);
}
}
}
std::sort(result.begin(), result.end());
return result;
}
inline uint64_t euler_phi(uint64_t value) {
assert(value >= 1);
uint64_t result = value;
for (const auto& factor : prime_factorize(value)) {
result = result / factor.first * (factor.first - 1);
}
return result;
}
inline int mobius(uint64_t value) {
assert(value >= 1);
int result = 1;
for (const auto& factor : prime_factorize(value)) {
if (factor.second >= 2) return 0;
result = -result;
}
return result;
}
} // namespace math
} // namespace m1une
#line 11 "math/tetration.hpp"
namespace m1une {
namespace math {
namespace tetration_detail {
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
uint64_t to_uint64(T value) {
if constexpr (std::signed_integral<T>) {
assert(value >= 0);
}
return static_cast<uint64_t>(value);
}
inline uint64_t multiply_mod(uint64_t first, uint64_t second, uint64_t mod) {
return static_cast<uint64_t>(
static_cast<__uint128_t>(first) * second % mod
);
}
inline uint64_t pow_mod(uint64_t base, __uint128_t exponent, uint64_t mod) {
assert(mod >= 1);
if (mod == 1) return 0;
base %= mod;
uint64_t result = 1 % mod;
while (exponent > 0) {
if ((exponent & 1) != 0) result = multiply_mod(result, base, mod);
base = multiply_mod(base, base, mod);
exponent >>= 1;
}
return result;
}
inline uint64_t pow_bounded(uint64_t base, uint64_t exponent, uint64_t limit) {
if (limit == 0) return 0;
__uint128_t result = 1;
for (uint64_t i = 0; i < exponent; i++) {
result *= base;
if (result >= limit) return limit;
}
return static_cast<uint64_t>(result);
}
inline uint64_t exponent_threshold(uint64_t base, uint64_t limit) {
assert(base >= 2);
if (limit <= 1) return 0;
uint64_t exponent = 0;
uint64_t value = 1;
while (value < limit) {
exponent++;
if (value > limit / base) return exponent;
value *= base;
}
return exponent;
}
inline uint64_t tetration_bounded_unsigned(uint64_t base, uint64_t height, uint64_t limit) {
if (limit == 0) return 0;
if (height == 0) return limit < 1 ? limit : 1;
if (height == 1) return base < limit ? base : limit;
if (base == 0) {
const uint64_t value = (height & 1) == 0 ? 1 : 0;
return value < limit ? value : limit;
}
if (base == 1) return limit < 1 ? limit : 1;
const uint64_t threshold = exponent_threshold(base, limit);
const uint64_t exponent = tetration_bounded_unsigned(base, height - 1, threshold);
if (exponent >= threshold) return limit;
return pow_bounded(base, exponent, limit);
}
inline uint64_t tetration_mod_unsigned(uint64_t base, uint64_t height, uint64_t mod) {
assert(mod >= 1);
if (mod == 1) return 0;
if (height == 0) return 1 % mod;
if (height == 1) return base % mod;
if (base == 0) return (height & 1) == 0 ? 1 % mod : 0;
if (base == 1) return 1 % mod;
const uint64_t phi = euler_phi(mod);
uint64_t reduced_exponent = tetration_mod_unsigned(base, height - 1, phi);
__uint128_t exponent = reduced_exponent;
if (tetration_bounded_unsigned(base, height - 1, phi) >= phi) {
exponent += phi;
}
return pow_mod(base, exponent, mod);
}
inline uint64_t power_tower_bounded_unsigned(
const std::vector<uint64_t>& bases,
int index,
uint64_t limit
) {
if (limit == 0) return 0;
if (index == int(bases.size())) return limit < 1 ? limit : 1;
const uint64_t base = bases[index];
if (index + 1 == int(bases.size())) return base < limit ? base : limit;
if (base == 0) {
const uint64_t exponent = power_tower_bounded_unsigned(bases, index + 1, 1);
const uint64_t value = exponent == 0 ? 1 : 0;
return value < limit ? value : limit;
}
if (base == 1) return limit < 1 ? limit : 1;
const uint64_t threshold = exponent_threshold(base, limit);
const uint64_t exponent = power_tower_bounded_unsigned(bases, index + 1, threshold);
if (exponent >= threshold) return limit;
return pow_bounded(base, exponent, limit);
}
inline uint64_t power_tower_mod_unsigned(
const std::vector<uint64_t>& bases,
int index,
uint64_t mod
) {
assert(mod >= 1);
if (mod == 1) return 0;
if (index == int(bases.size())) return 1 % mod;
if (index + 1 == int(bases.size())) return bases[index] % mod;
const uint64_t phi = euler_phi(mod);
uint64_t reduced_exponent = power_tower_mod_unsigned(bases, index + 1, phi);
__uint128_t exponent = reduced_exponent;
if (power_tower_bounded_unsigned(bases, index + 1, phi) >= phi) {
exponent += phi;
}
return pow_mod(bases[index], exponent, mod);
}
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
std::vector<uint64_t> normalize_bases(const std::vector<T>& bases) {
std::vector<uint64_t> result;
result.reserve(bases.size());
for (T base : bases) result.push_back(to_uint64(base));
return result;
}
} // namespace tetration_detail
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
uint64_t tetration_mod(T base, uint64_t height, uint64_t mod) {
assert(mod >= 1);
return tetration_detail::tetration_mod_unsigned(
tetration_detail::to_uint64(base),
height,
mod
);
}
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
uint64_t tetration_bounded(T base, uint64_t height, uint64_t limit) {
return tetration_detail::tetration_bounded_unsigned(
tetration_detail::to_uint64(base),
height,
limit
);
}
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
uint64_t power_tower_mod(const std::vector<T>& bases, uint64_t mod) {
assert(mod >= 1);
std::vector<uint64_t> normalized = tetration_detail::normalize_bases(bases);
return tetration_detail::power_tower_mod_unsigned(normalized, 0, mod);
}
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
uint64_t power_tower_bounded(const std::vector<T>& bases, uint64_t limit) {
std::vector<uint64_t> normalized = tetration_detail::normalize_bases(bases);
return tetration_detail::power_tower_bounded_unsigned(normalized, 0, limit);
}
} // namespace math
} // namespace m1une
#line 4 "verify/math/tetration.test.cpp"
#line 1 "utilities/fast_io.hpp"
#line 5 "utilities/fast_io.hpp"
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"
namespace m1une {
namespace utilities {
struct FastOutput;
namespace internal {
// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;
// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};
template <class T>
struct is_range<T, std::void_t<
decltype(std::begin(std::declval<T&>())),
decltype(std::end(std::declval<T&>()))
>> : std::true_type {};
template <class T>
inline constexpr bool is_range_v = is_range<T>::value;
template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));
template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;
template <class T, class = void>
struct range_stored_value {
using type = range_value_t<T>;
};
template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};
template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;
// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};
template <class T, std::size_t N>
struct is_char_array<T[N]>
: std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};
template <class T>
struct is_string_like
: std::bool_constant<
std::is_same_v<std::decay_t<T>, std::string>
|| std::is_same_v<std::decay_t<T>, const char*>
|| std::is_same_v<std::decay_t<T>, char*>
|| is_char_array<std::remove_reference_t<T>>::value
> {};
template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;
// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};
template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
: std::true_type {};
template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;
template <class T, class = void>
struct has_static_mod_raw : std::false_type {};
template <class T>
struct has_static_mod_raw<
T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
: std::true_type {};
template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;
// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
std::is_integral_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>
|| std::is_same_v<std::remove_cv_t<T>, __uint128_t>;
template <class T>
inline constexpr bool is_signed_v =
std::is_signed_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>;
template <class T>
struct make_unsigned {
using type = std::make_unsigned_t<T>;
};
template <>
struct make_unsigned<__int128_t> {
using type = __uint128_t;
};
template <>
struct make_unsigned<__uint128_t> {
using type = __uint128_t;
};
template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;
} // namespace internal
struct FastInput {
static constexpr int buffer_size = 1 << 20;
private:
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _length;
int _file_descriptor;
bool _streaming;
bool refill() {
_position = 0;
if (_streaming) {
ssize_t length;
do {
length = ::read(_file_descriptor, _buffer, buffer_size);
} while (length < 0 && errno == EINTR);
if (length <= 0) {
_length = 0;
return false;
}
_length = int(length);
} else {
_length = int(std::fread(_buffer, 1, buffer_size, _stream));
}
return _length != 0;
}
template <class T>
bool read_integer_from_stream(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-') {
negative = true;
c = read_char_raw();
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
result = negative ? result * 10 - (c - '0')
: result * 10 + (c - '0');
c = read_char_raw();
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + T(c - '0');
c = read_char_raw();
}
value = negative ? T(0) - result : result;
}
return true;
}
bool prepare_number() {
if (_length - _position >= 64) return true;
const int remaining = _length - _position;
if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
_position = 0;
_length = remaining + added;
if (_length < buffer_size) _buffer[_length] = '\0';
return _length != 0;
}
public:
explicit FastInput(std::FILE* stream = stdin)
: _stream(stream),
_position(0),
_length(0),
_file_descriptor(::fileno(stream)),
_streaming([&] {
struct stat status;
return _file_descriptor >= 0
&& ::fstat(_file_descriptor, &status) == 0
&& !S_ISREG(status.st_mode);
}()) {}
FastInput(const FastInput&) = delete;
FastInput& operator=(const FastInput&) = delete;
int read_char_raw() {
if (_position == _length && !refill()) return EOF;
return _buffer[_position++];
}
bool skip_spaces() {
int c = read_char_raw();
while (c != EOF && c <= ' ') c = read_char_raw();
if (c == EOF) return false;
--_position;
return true;
}
bool read(char& value) {
if (!skip_spaces()) return false;
value = char(read_char_raw());
return true;
}
bool read(std::string& value) {
if (!skip_spaces()) return false;
value.clear();
while (true) {
const int begin = _position;
while (_position < _length &&
static_cast<unsigned char>(_buffer[_position]) > ' ') {
++_position;
}
value.append(_buffer + begin, _position - begin);
if (_position < _length) {
++_position;
return true;
}
if (!refill()) return true;
}
}
bool read(bool& value) {
int x;
if (!read(x)) return false;
value = x != 0;
return true;
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>,
bool
>
read(T& value) {
if (_streaming) return read_integer_from_stream(value);
if (!prepare_number()) return false;
int c = static_cast<unsigned char>(_buffer[_position++]);
while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);
bool negative = false;
if (c == '-') {
negative = true;
c = static_cast<unsigned char>(_buffer[_position++]);
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
const int first = c - '0';
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = negative ? result * 100 - (first * 10 + second)
: result * 100 + (first * 10 + second);
++_position;
} else {
result = negative ? result * 10 - first : result * 10 + first;
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
const unsigned first = unsigned(c - '0');
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = result * 100 + T(first * 10 + unsigned(second));
++_position;
} else {
result = result * 10 + T(first);
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = negative ? T(0) - result : result;
}
if (_position > _length) _position = _length;
return true;
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>, bool>
read(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-' || c == '+') {
negative = c == '-';
c = read_char_raw();
}
long double result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + (c - '0');
c = read_char_raw();
}
if (c == '.') {
long double place = 0.1L;
c = read_char_raw();
while ('0' <= c && c <= '9') {
result += (c - '0') * place;
place *= 0.1L;
c = read_char_raw();
}
}
if (c == 'e' || c == 'E') {
c = read_char_raw();
bool exponent_negative = false;
if (c == '-' || c == '+') {
exponent_negative = c == '-';
c = read_char_raw();
}
int exponent = 0;
while ('0' <= c && c <= '9') {
exponent = exponent * 10 + (c - '0');
c = read_char_raw();
}
long double scale = 1;
long double power = 10;
while (exponent > 0) {
if (exponent & 1) scale *= power;
power *= power;
exponent >>= 1;
}
result = exponent_negative ? result / scale : result * scale;
}
value = static_cast<T>(negative ? -result : result);
return true;
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>,
bool
>
read(T& value) {
long long x;
if (!read(x)) return false;
if constexpr (internal::has_static_mod_raw_v<T>) {
if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
value = T::raw(uint32_t(x));
} else {
value = T(x);
}
} else {
value = T(x);
}
return true;
}
template <class First, class Second>
bool read(std::pair<First, Second>& value) {
if (!read(value.first)) return false;
return read(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>,
bool
>
read(Range& range) {
using StoredValue = internal::range_stored_value_t<Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
for (auto&& value : range) {
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
bool x;
if (!read(x)) return false;
value = x;
} else {
if (!read(value)) return false;
}
}
return true;
}
template <class First, class Second, class... Rest>
bool read(First& first, Second& second, Rest&... rest) {
if (!read(first)) return false;
return read(second, rest...);
}
template <class T>
FastInput& operator>>(T& value) {
if (!read(value)) std::abort();
return *this;
}
};
struct FastOutput {
static constexpr int buffer_size = 1 << 20;
private:
inline static const auto digit_quads = [] {
std::array<char, 40000> result{};
for (int i = 0; i < 10000; i++) {
int value = i;
for (int j = 3; j >= 0; j--) {
result[4 * i + j] = char('0' + value % 10);
value /= 10;
}
}
return result;
}();
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _precision;
std::chars_format _float_format;
char _range_separator;
std::string* _capture = nullptr;
template <class T>
std::string format_cell(const T& value) {
std::string result;
struct CaptureGuard {
std::string*& target;
std::string* previous;
~CaptureGuard() { target = previous; }
} guard{_capture, _capture};
_capture = &result;
write(value);
return result;
}
template <class Matrix>
void write_aligned_matrix(const Matrix& matrix) {
std::vector<std::vector<std::string>> rows;
std::vector<std::size_t> widths;
for (const auto& row : matrix) {
auto& cells = rows.emplace_back();
std::size_t column = 0;
for (const auto& value : row) {
cells.push_back(format_cell(value));
if (column == widths.size()) widths.push_back(0);
widths[column] = std::max(widths[column], cells.back().size());
++column;
}
}
bool first = true;
for (const auto& row : rows) {
if (!first) write_char('\n');
first = false;
for (std::size_t column = 0; column < row.size(); ++column) {
if (column != 0) write_char(_range_separator);
for (std::size_t padding = row[column].size();
padding < widths[column]; ++padding) {
write_char(' ');
}
write(row[column]);
}
}
}
public:
explicit FastOutput(std::FILE* stream = stdout)
: _stream(stream),
_position(0),
_precision(6),
_float_format(std::chars_format::general),
_range_separator(' ') {
if (_stream == stdout
&& internal::standard_output_instance == nullptr) {
internal::standard_output_instance = this;
}
}
FastOutput(const FastOutput&) = delete;
FastOutput& operator=(const FastOutput&) = delete;
~FastOutput() {
flush();
if (internal::standard_output_instance == this) {
internal::standard_output_instance = nullptr;
}
}
void flush() {
if (_position != 0) {
std::fwrite(_buffer, 1, _position, _stream);
_position = 0;
}
std::fflush(_stream);
}
void write_char(char c) {
if (_capture != nullptr) {
_capture->push_back(c);
return;
}
if (_position == buffer_size) flush();
_buffer[_position++] = c;
}
void write(const char* s) {
while (*s != '\0') write_char(*s++);
}
void write(const std::string& s) {
if (_capture != nullptr) {
_capture->append(s);
return;
}
std::size_t position = 0;
while (position < s.size()) {
if (_position == buffer_size) flush();
const std::size_t copied =
std::min<std::size_t>(buffer_size - _position, s.size() - position);
std::memcpy(_buffer + _position, s.data() + position, copied);
_position += int(copied);
position += copied;
}
}
void write(char c) {
write_char(c);
}
void write(bool value) {
write_char(value ? '1' : '0');
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>>
write(T value) {
char digits[128];
auto [end, error] = std::to_chars(
digits,
digits + sizeof(digits),
value,
_float_format,
_precision
);
if (error != std::errc()) std::abort();
for (const char* pointer = digits; pointer != end; pointer++) {
write_char(*pointer);
}
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>
>
write(T value) {
using Raw = std::remove_cv_t<T>;
using Unsigned = internal::make_unsigned_t<Raw>;
Unsigned magnitude;
if constexpr (internal::is_signed_v<Raw>) {
if (value < 0) {
write_char('-');
magnitude = Unsigned(0) - Unsigned(value);
} else {
magnitude = Unsigned(value);
}
} else {
magnitude = value;
}
if (magnitude == 0) {
write_char('0');
return;
}
unsigned chunks[16];
int count = 0;
while (magnitude >= 10000) {
const Unsigned quotient = magnitude / 10000;
chunks[count++] = unsigned(magnitude - quotient * 10000);
magnitude = quotient;
}
if (_capture == nullptr && _position > buffer_size - 64) flush();
char captured[64];
char* const begin = _capture != nullptr ? captured : _buffer + _position;
char* destination = begin;
const unsigned leading = unsigned(magnitude);
const char* first = digit_quads.data() + 4 * leading;
int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
for (; skip < 4; skip++) *destination++ = first[skip];
while (count--) {
const char* digits = digit_quads.data() + 4 * chunks[count];
std::memcpy(destination, digits, 4);
destination += 4;
}
if (_capture != nullptr) {
_capture->append(begin, destination - begin);
} else {
_position += int(destination - begin);
}
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>
>
write(const T& value) {
write(value.val());
}
template <class First, class Second>
void write(const std::pair<First, Second>& value) {
write(value.first);
write_char(' ');
write(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>
>
write(const Range& range) {
using StoredValue = internal::range_stored_value_t<const Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
bool first = true;
for (const auto& value : range) {
if (!first) write_char(nested ? '\n' : _range_separator);
first = false;
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
write(static_cast<bool>(value));
} else {
write(value);
}
}
}
template <class First, class... Rest>
void print(const First& first, const Rest&... rest) {
write(first);
((write_char(' '), write(rest)), ...);
}
void println() {
write_char('\n');
}
void set_precision(int precision) {
_precision = precision;
}
void set_fixed(int precision = 6) {
_float_format = std::chars_format::fixed;
_precision = precision;
}
void set_general(int precision = 6) {
_float_format = std::chars_format::general;
_precision = precision;
}
void set_range_separator(char separator) {
_range_separator = separator;
}
template <class Matrix>
void write_aligned(const Matrix& matrix) {
using Row = internal::range_stored_value_t<const Matrix>;
using Cell = internal::range_stored_value_t<const Row>;
static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
"write_aligned requires a two-dimensional range");
static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
"write_aligned requires scalar cells");
write_aligned_matrix(matrix);
}
template <class Matrix>
void println_aligned(const Matrix& matrix) {
write_aligned(matrix);
write_char('\n');
}
template <class... Args>
void println(const Args&... args) {
print(args...);
write_char('\n');
}
template <class T>
FastOutput& operator<<(const T& value) {
write(value);
return *this;
}
};
} // namespace utilities
} // namespace m1une
#line 9 "verify/math/tetration.test.cpp"
namespace {
uint64_t minimum(uint64_t first, uint64_t second) {
return first < second ? first : second;
}
uint64_t brute_pow_bounded(uint64_t base, uint64_t exponent, uint64_t limit) {
if (limit == 0) return 0;
if (exponent == 0) return 1 < limit ? 1 : limit;
if (base == 0) return 0;
__uint128_t result = 1;
for (uint64_t i = 0; i < exponent; i++) {
result *= base;
if (result >= limit) return limit;
}
return uint64_t(result);
}
uint64_t brute_tetration_bounded(uint64_t base, uint64_t height, uint64_t limit) {
if (limit == 0) return 0;
if (height == 0) return minimum(1, limit);
if (height == 1) return minimum(base, limit);
uint64_t exponent = brute_tetration_bounded(base, height - 1, limit);
return brute_pow_bounded(base, exponent, limit);
}
uint64_t brute_tower_bounded(
const std::vector<uint64_t>& bases,
int index,
uint64_t limit
) {
if (limit == 0) return 0;
if (index == int(bases.size())) return minimum(1, limit);
if (index + 1 == int(bases.size())) return minimum(bases[index], limit);
uint64_t exponent = brute_tower_bounded(bases, index + 1, limit);
return brute_pow_bounded(bases[index], exponent, limit);
}
void test_fixed_cases() {
assert(m1une::math::tetration_mod(2ULL, 0, 1000) == 1);
assert(m1une::math::tetration_mod(2ULL, 1, 1000) == 2);
assert(m1une::math::tetration_mod(2ULL, 4, 1000) == 536);
assert(m1une::math::tetration_mod(3ULL, 3, 100) == 87);
assert(m1une::math::tetration_mod(0ULL, 0, 1000) == 1);
assert(m1une::math::tetration_mod(0ULL, 1, 1000) == 0);
assert(m1une::math::tetration_mod(0ULL, 2, 1000) == 1);
assert(m1une::math::tetration_mod(0ULL, 3, 1000) == 0);
assert(m1une::math::tetration_mod(7ULL, 100, 1) == 0);
assert(m1une::math::tetration_bounded(2ULL, 5, 1000000) == 1000000);
std::vector<unsigned long long> empty;
assert(m1une::math::power_tower_mod(empty, 37) == 1);
assert(m1une::math::power_tower_bounded(empty, 10) == 1);
std::vector<unsigned long long> zero_zero;
zero_zero.push_back(0);
zero_zero.push_back(0);
assert(m1une::math::power_tower_mod(zero_zero, 1000) == 1);
std::vector<unsigned long long> two_zero;
two_zero.push_back(2);
two_zero.push_back(0);
assert(m1une::math::power_tower_mod(two_zero, 1000) == 1);
std::vector<unsigned long long> two_three_four;
two_three_four.push_back(2);
two_three_four.push_back(3);
two_three_four.push_back(4);
assert(m1une::math::power_tower_mod(two_three_four, 1000) == 352);
}
void test_tetration_against_bruteforce() {
const uint64_t cap = 1000000;
for (uint64_t base = 0; base <= 8; base++) {
for (uint64_t height = 0; height <= 7; height++) {
for (uint64_t limit = 0; limit <= 1000; limit++) {
uint64_t expected = brute_tetration_bounded(base, height, limit);
uint64_t actual = m1une::math::tetration_bounded(base, height, limit);
assert(actual == expected);
}
uint64_t exact = brute_tetration_bounded(base, height, cap);
if (exact == cap) continue;
for (uint64_t mod = 1; mod <= 257; mod++) {
assert(m1une::math::tetration_mod(base, height, mod) == exact % mod);
}
}
}
}
void enumerate_towers(
int length,
int position,
std::vector<uint64_t>& bases
) {
if (position == length) {
const uint64_t cap = 1000000;
for (uint64_t limit = 0; limit <= 300; limit++) {
uint64_t expected = brute_tower_bounded(bases, 0, limit);
uint64_t actual = m1une::math::power_tower_bounded(bases, limit);
assert(actual == expected);
}
uint64_t exact = brute_tower_bounded(bases, 0, cap);
if (exact == cap) return;
for (uint64_t mod = 1; mod <= 127; mod++) {
assert(m1une::math::power_tower_mod(bases, mod) == exact % mod);
}
return;
}
for (uint64_t base = 0; base <= 4; base++) {
bases[position] = base;
enumerate_towers(length, position + 1, bases);
}
}
void test_power_tower_against_bruteforce() {
for (int length = 0; length <= 5; length++) {
std::vector<uint64_t> bases(length);
enumerate_towers(length, 0, bases);
}
}
} // namespace
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_fixed_cases();
test_tetration_against_bruteforce();
test_power_tower_against_bruteforce();
int test_count;
fast_input >> test_count;
while (test_count--) {
uint64_t base, height, modulus;
fast_input >> base >> height >> modulus;
fast_output << m1une::math::tetration_mod(base, height, modulus) << '\n';
}
}
verify/math/tetration.test.cpp