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#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY #include "../factorize.hpp" #include "../sieve.hpp" #include <cassert> #include <cstdio> int main() { int lim = 5e7; Sieve sieve(lim); for (int x = 1; x <= lim; x++) { bool is_prime_1 = (sieve.min_factor[x] == x); bool is_prime_2 = is_prime(x); assert(is_prime_1 == is_prime_2); } puts("Hello World"); }
#line 1 "number/test/miller-rabin-5e7.test.cpp" #define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY #line 2 "random/xorshift.hpp" #include <cstdint> // CUT begin uint32_t rand_int() // XorShift random integer generator { static uint32_t x = 123456789, y = 362436069, z = 521288629, w = 88675123; uint32_t t = x ^ (x << 11); x = y; y = z; z = w; return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); } double rand_double() { return (double)rand_int() / UINT32_MAX; } #line 3 "number/factorize.hpp" #include <algorithm> #include <array> #include <cassert> #include <numeric> #include <vector> namespace SPRP { // http://miller-rabin.appspot.com/ const std::vector<std::vector<__int128>> bases{ {126401071349994536}, // < 291831 {336781006125, 9639812373923155}, // < 1050535501 (1e9) {2, 2570940, 211991001, 3749873356}, // < 47636622961201 (4e13) {2, 325, 9375, 28178, 450775, 9780504, 1795265022} // <= 2^64 }; inline int get_id(long long n) { if (n < 291831) { return 0; } else if (n < 1050535501) { return 1; } else if (n < 47636622961201) return 2; else { return 3; } } } // namespace SPRP // Miller-Rabin primality test // https://ja.wikipedia.org/wiki/%E3%83%9F%E3%83%A9%E3%83%BC%E2%80%93%E3%83%A9%E3%83%93%E3%83%B3%E7%B4%A0%E6%95%B0%E5%88%A4%E5%AE%9A%E6%B3%95 // Complexity: O(lg n) per query struct { long long modpow(__int128 x, __int128 n, long long mod) noexcept { __int128 ret = 1; for (x %= mod; n; x = x * x % mod, n >>= 1) ret = (n & 1) ? ret * x % mod : ret; return ret; } bool operator()(long long n) noexcept { if (n < 2) return false; if (n % 2 == 0) return n == 2; int s = __builtin_ctzll(n - 1); for (__int128 a : SPRP::bases[SPRP::get_id(n)]) { if (a % n == 0) continue; a = modpow(a, (n - 1) >> s, n); bool may_composite = true; if (a == 1) continue; for (int r = s; r--; a = a * a % n) { if (a == n - 1) may_composite = false; } if (may_composite) return false; } return true; } } is_prime; struct { // Pollard's rho algorithm: find factor greater than 1 long long find_factor(long long n) { assert(n > 1); if (n % 2 == 0) return 2; if (is_prime(n)) return n; long long c = 1; auto f = [&](__int128 x) -> long long { return (x * x + c) % n; }; for (int t = 1;; t++) { for (c = 0; c == 0 or c + 2 == n;) c = rand_int() % n; long long x0 = t, m = std::max(n >> 3, 1LL), x, ys, y = x0, r = 1, g, q = 1; do { x = y; for (int i = r; i--;) y = f(y); long long k = 0; do { ys = y; for (int i = std::min(m, r - k); i--;) y = f(y), q = __int128(q) * std::abs(x - y) % n; g = std::__gcd<long long>(q, n); k += m; } while (k < r and g <= 1); r <<= 1; } while (g <= 1); if (g == n) { do { ys = f(ys); g = std::__gcd(std::abs(x - ys), n); } while (g <= 1); } if (g != n) return g; } } std::vector<long long> operator()(long long n) { std::vector<long long> ret; while (n > 1) { long long f = find_factor(n); if (f < n) { auto tmp = operator()(f); ret.insert(ret.end(), tmp.begin(), tmp.end()); } else ret.push_back(n); n /= f; } std::sort(ret.begin(), ret.end()); return ret; } long long euler_phi(long long n) { long long ret = 1, last = -1; for (auto p : this->operator()(n)) ret *= p - (last != p), last = p; return ret; } } FactorizeLonglong; #line 3 "number/sieve.hpp" #include <map> #line 5 "number/sieve.hpp" // CUT begin // Linear sieve algorithm for fast prime factorization // Complexity: O(N) time, O(N) space: // - MAXN = 10^7: ~44 MB, 80~100 ms (Codeforces / AtCoder GCC, C++17) // - MAXN = 10^8: ~435 MB, 810~980 ms (Codeforces / AtCoder GCC, C++17) // Reference: // [1] D. Gries, J. Misra, "A Linear Sieve Algorithm for Finding Prime Numbers," // Communications of the ACM, 21(12), 999-1003, 1978. // - https://cp-algorithms.com/algebra/prime-sieve-linear.html // - https://37zigen.com/linear-sieve/ struct Sieve { std::vector<int> min_factor; std::vector<int> primes; Sieve(int MAXN) : min_factor(MAXN + 1) { for (int d = 2; d <= MAXN; d++) { if (!min_factor[d]) { min_factor[d] = d; primes.emplace_back(d); } for (const auto &p : primes) { if (p > min_factor[d] or d * p > MAXN) break; min_factor[d * p] = p; } } } // Prime factorization for 1 <= x <= MAXN^2 // Complexity: O(log x) (x <= MAXN) // O(MAXN / log MAXN) (MAXN < x <= MAXN^2) template <class T> std::map<T, int> factorize(T x) const { std::map<T, int> ret; assert(x > 0 and x <= ((long long)min_factor.size() - 1) * ((long long)min_factor.size() - 1)); for (const auto &p : primes) { if (x < T(min_factor.size())) break; while (!(x % p)) x /= p, ret[p]++; } if (x >= T(min_factor.size())) ret[x]++, x = 1; while (x > 1) ret[min_factor[x]]++, x /= min_factor[x]; return ret; } // Enumerate divisors of 1 <= x <= MAXN^2 // Be careful of highly composite numbers https://oeis.org/A002182/list // https://gist.github.com/dario2994/fb4713f252ca86c1254d#file-list-txt (n, (# of div. of n)): // 45360->100, 735134400(<1e9)->1344, 963761198400(<1e12)->6720 template <class T> std::vector<T> divisors(T x) const { std::vector<T> ret{1}; for (const auto p : factorize(x)) { int n = ret.size(); for (int i = 0; i < n; i++) { for (T a = 1, d = 1; d <= p.second; d++) { a *= p.first; ret.push_back(ret[i] * a); } } } return ret; // NOT sorted } // Euler phi functions of divisors of given x // Verified: ABC212 G https://atcoder.jp/contests/abc212/tasks/abc212_g // Complexity: O(sqrt(x) + d(x)) template <class T> std::map<T, T> euler_of_divisors(T x) const { assert(x >= 1); std::map<T, T> ret; ret[1] = 1; std::vector<T> divs{1}; for (auto p : factorize(x)) { int n = ret.size(); for (int i = 0; i < n; i++) { ret[divs[i] * p.first] = ret[divs[i]] * (p.first - 1); divs.push_back(divs[i] * p.first); for (T a = divs[i] * p.first, d = 1; d < p.second; a *= p.first, d++) { ret[a * p.first] = ret[a] * p.first; divs.push_back(a * p.first); } } } return ret; } // Moebius function Table, (-1)^{# of different prime factors} for square-free x // return: [0=>0, 1=>1, 2=>-1, 3=>-1, 4=>0, 5=>-1, 6=>1, 7=>-1, 8=>0, ...] https://oeis.org/A008683 std::vector<int> GenerateMoebiusFunctionTable() const { std::vector<int> ret(min_factor.size()); for (unsigned i = 1; i < min_factor.size(); i++) { if (i == 1) { ret[i] = 1; } else if ((i / min_factor[i]) % min_factor[i] == 0) { ret[i] = 0; } else { ret[i] = -ret[i / min_factor[i]]; } } return ret; } // Calculate [0^K, 1^K, ..., nmax^K] in O(nmax) // Note: **0^0 == 1** template <class MODINT> std::vector<MODINT> enumerate_kth_pows(long long K, int nmax) const { assert(nmax < int(min_factor.size())); assert(K >= 0); if (K == 0) return std::vector<MODINT>(nmax + 1, 1); std::vector<MODINT> ret(nmax + 1); ret[0] = 0, ret[1] = 1; for (int n = 2; n <= nmax; n++) { if (min_factor[n] == n) { ret[n] = MODINT(n).pow(K); } else { ret[n] = ret[n / min_factor[n]] * ret[min_factor[n]]; } } return ret; } }; // Sieve sieve((1 << 20)); #line 5 "number/test/miller-rabin-5e7.test.cpp" #include <cstdio> int main() { int lim = 5e7; Sieve sieve(lim); for (int x = 1; x <= lim; x++) { bool is_prime_1 = (sieve.min_factor[x] == x); bool is_prime_2 = is_prime(x); assert(is_prime_1 == is_prime_2); } puts("Hello World"); }