//#pragma GCC target ("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,avx2,bmi2,lzcnt,tune=native") //#pragma GCC target ("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native") //#pragma GCC target ("sse4") #pragma GCC optimize("O3") //#pragma GCC optimize ("tree-vectorize") //#pragma GCC optimize("unroll-loops") #ifndef NDEBUG #define NDEBUG #endif #include #include #include #include #include using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; using u128 = __uint128_t; using f64 = double; class U64Mont { private: static u64 _ni(u64 n) noexcept { // n * ni == 1 (mod 2**64) // // n is odd number, n = 2*k+1, n >= 1, n < 2**64, k is non-negative integer, k >= 0, k < 2**63 // ni0 := n; // = 2*k+1 = (1+(2**3)*((k*(k+1)/2)**1))/(2*k+1) u64 ni = n; // ni1 := ni0 * (2 - (n * ni0)); // = (1-(2**6)*((k*(k+1)/2)**2))/(2*k+1) // ni2 := ni1 * (2 - (n * ni1)); // = (1-(2**12)*((k*(k+1)/2)**4))/(2*k+1) // ni3 := ni2 * (2 - (n * ni2)); // = (1-(2**24)*((k*(k+1)/2)**8))/(2*k+1) // ni4 := ni3 * (2 - (n * ni3)); // = (1-(2**48)*((k*(k+1)/2)**16))/(2*k+1) // ni5 := ni4 * (2 - (n * ni4)); // = (1-(2**96)*((k*(k+1)/2)**32))/(2*k+1) // // (n * ni5) mod 2**64 = ((2*k+1) * ni5) mod 2**64 = 1 mod 2**64 for (int i = 0; i < 5; ++i) { ni = ni * (2 - n * ni); } assert(n * ni == 1); // n * ni == 1 (mod 2**64) return ni; } static u64 _n1(u64 n) noexcept { // == n - 1 return n - 1; } static u64 _nh(u64 n) noexcept { // == (n + 1) / 2 return (n >> 1) + 1; } static u64 _r(u64 n) noexcept { // == 2**64 (mod n) return (-n) % n; } static u64 _rn(u64 n) noexcept { // == -1 * (2**64) (mod n) return n - _r(n); } static u64 _r2(u64 n) noexcept { // == 2**128 (mod n) return (u64)((-((u128)n)) % ((u128)n)); } static u32 _k(u64 n) noexcept { // == trailing_zeros(n - 1) // https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html#Other-Builtins return __builtin_ctzll(_n1(n)); } static u64 _d(u64 n) noexcept { // == (n - 1) >> trailing_zeros(n - 1) // n == 2**k * d + 1 return _n1(n) >> _k(n); } public: const u64 n; // == n const u64 ni; // n * ni == 1 (mod 2**64) const u64 n1; // == n - 1 const u64 nh; // == (n + 1) / 2 const u64 r; // == 2**64 (mod n) const u64 rn; // == -1 * (2**64) (mod n) const u64 r2; // == 2**128 (mod n) const u64 d; // == (n - 1) >> trailing_zeros(n - 1) // n == 2**k * d + 1 const u32 k; // == trailing_zeros(n - 1) U64Mont(u64 n) noexcept : n(n), ni(_ni(n)), n1(_n1(n)), nh(_nh(n)), r(_r(n)), rn(_rn(n)), r2(_r2(n)), d(_d(n)), k(_k(n)) { assert((n & 1) == 1); } u64 add(u64 a, u64 b) const noexcept { // add(a, b) == a + b (mod n) assert(a < n); assert(b < n); unsigned long long t, u; // https://gcc.gnu.org/onlinedocs/gcc/Integer-Overflow-Builtins.html#Integer-Overflow-Builtins bool fa = __builtin_uaddll_overflow(a, b, &t); bool fs = __builtin_usubll_overflow(t, n, &u); return (fa ? u : (fs ? t : u)); } u64 sub(u64 a, u64 b) const noexcept { // sub(a, b) == a - b (mod n) assert(a < n); assert(b < n); unsigned long long t; // https://gcc.gnu.org/onlinedocs/gcc/Integer-Overflow-Builtins.html#Integer-Overflow-Builtins bool f = __builtin_usubll_overflow(a, b, &t); return (f ? t + n : t); } u64 div2(u64 ar) const noexcept { // div2(ar) == ar / 2 (mod n) assert(ar < n); return ((ar >> 1) + ((ar & 1) == 0 ? 0 : nh)); } u64 mrmul(u64 ar, u64 br) const noexcept { // mrmul(ar, br) == (ar * br) / r (mod n) // R == 2**64 // gcd(N, R) == 1 // N * ni mod R == 1 // 0 <= ar < N < R // 0 <= br < N < R // T := ar * br // t := floor(T / R) - floor(((T * ni mod R) * N) / R) // if t < 0 then return t + N else return t assert(ar < n); assert(br < n); u128 t = ((u128)ar) * ((u128)br); unsigned long long w; // https://gcc.gnu.org/onlinedocs/gcc/Integer-Overflow-Builtins.html#Integer-Overflow-Builtins bool f = __builtin_usubll_overflow((unsigned long long)(t >> 64), (unsigned long long)((((u128)(((u64)t) * ni)) * ((u128)n)) >> 64), &w); return (w + (f ? n : 0)); } u64 mr(u64 ar) const noexcept { // mr(ar) == ar / r (mod n) // R == 2**64 // gcd(N, R) == 1 // N * ni mod R == 1 // 0 <= ar < N < R // t := floor(ar / R) - floor(((ar * ni mod R) * N) / R) // if t < 0 then return t + N else return t assert(ar < n); u64 v = (u64)((((u128)(ar * ni)) * ((u128)n)) >> 64); return ((v == 0) ? 0 : (n - v)); } u64 ar(u64 a) const noexcept { // ar(a) == a * r (mod n) assert(a < n); return mrmul(a, r2); } u64 pow(u64 ar, u64 b) const noexcept { // pow(ar, b) == ((ar / r) ** b) * r (mod n) assert(ar < n); u64 tr = ((b & 1) == 0) ? r : ar; for (b >>= 1; b != 0; b >>= 1) { ar = mrmul(ar, ar); if ((b & 1) != 0) { tr = mrmul(tr, ar); } } return tr; } }; // 64bit整数平方根(lz:ケチるループ回数*2+(0~1)、内部実装) -> (floor(sqrt(iv)), remain) u64 isqrt64i(const u64 iv, u64* remain, const u32 lz) noexcept { constexpr u128 _b = ((((u128)0x0000000000000000ULL) << 64) | ((u128)0x4000000000000000ULL)); constexpr u128 _c = ((((u128)0xfffffffffffffffeULL) << 64) | ((u128)0x0000000000000000ULL)); constexpr u128 _d = ((((u128)0x0000000000000001ULL) << 64) | ((u128)0x0000000000000000ULL)); constexpr u128 _e = ((((u128)0x0000000000000000ULL) << 64) | ((u128)0xffffffffffffffffULL)); u32 n = (64 >> 1) - (lz >> 1); u32 s = (lz >> 1) << 1; u32 t = n << 1; u128 a = iv; u128 b = _b >> s; u128 c = _c >> s; u128 d = _d >> s; u128 e = _e >> s; for (u32 i = 0; i < n; ++i) { if (a >= b) { a -= b; b = ((b + b) & c) + (b & e) + d; } else { b = ((b + b) & c) + (b & e); } a <<= 2; } *remain = (u64)(a >> t); return ((u64)(b >> t)); } // 64bit整数平方根(固定ループ回数) -> (floor(sqrt(iv)), remain) u64 isqrt64f(const u64 iv, u64* remain) noexcept { return isqrt64i(iv, remain, 0); } // 64bit整数平方根(固定ループ回数) -> remain u64 isqrt64f_remain(const u64 iv) noexcept { u64 remain; isqrt64i(iv, &remain, 0); return remain; } // 64bit整数平方根(可変ループ回数) -> (floor(sqrt(iv)), remain) u64 isqrt64d(const u64 iv, u64* remain) noexcept { return isqrt64i(iv, remain, __builtin_clzll(iv)); } // Jacobi symbol: ヤコビ記号 i32 jacobi(const i64 sa, u64 n) noexcept { u64 a; i32 j; if (sa >= 0) { a = (u64)(sa); j = 1; } else { a = (u64)(-sa); j = ((n & 3) == 3) ? -1 : 1; } while (a > 0) { int ba = __builtin_ctzll(a); a >>= ba; if (((n & 7) == 3 || (n & 7) == 5) && (ba & 1) != 0) { j = -j; } if ((a & n & 3) == 3) { j = -j; } u64 t = n; n = a; a = t; a %= n; if (a > (n >> 1)) { a = n - a; if ((n & 3) == 3) { j = -j; } } } return ((n == 1) ? j : 0); } bool primetest_u64_miller_sub(const U64Mont *const mont, u64 a) noexcept { if (a >= mont->n) { a %= mont->n; if (a == 0) { return true; } } u64 tr = mont->pow(mont->ar(a), mont->d); if (tr == mont->r || tr == mont->rn) { return true; } for (u32 j = 1; j < mont->k; ++j) { tr = mont->mrmul(tr, tr); if (tr == mont->rn) { return true; } } return false; } // Miller-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... bool primetest_u64_miller_base2_sub(const U64Mont *const mont) noexcept { u64 tr = mont->pow(mont->add(mont->r, mont->r), mont->d); if (tr == mont->r || tr == mont->rn) { return true; } for (u32 j = 1; j < mont->k; ++j) { tr = mont->mrmul(tr, tr); if (tr == mont->rn) { return true; } } return false; } // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... bool primetest_u64_lucas_sub(const U64Mont *const mont) noexcept { u64 n = mont->n; i64 d = 5; for (int i = 0; i < 64; ++i) { if (jacobi(d, n) == -1) { break; } if (i == 32 && isqrt64f_remain(n) == 0) { return false; } if ((i & 1) == 0) { d = -(d + 2); } else { d = 2 - d; } assert(i < 63); } u64 qm = mont->ar((d < 0) ? ((((u64)(1 - d)) >> 2) % n) : (n - ((((u64)(d - 1)) >> 2) % n))); u64 k = (n + 1) << __builtin_clzll(n + 1); // n = u64::MAX の時の挙動は? u64 um = mont->r; u64 vm = mont->r; u64 qn = qm; u64 dm = mont->ar((d < 0) ? (n - (((u64)(-d)) % n)) : (((u64)(d)) % n)); for (k <<= 1; k > 0; k <<= 1) { um = mont->mrmul(um, vm); vm = mont->sub(mont->mrmul(vm, vm), mont->add(qn, qn)); qn = mont->mrmul(qn, qn); if ((k >> 63) != 0) { u64 uu = mont->add(um, vm); uu = mont->div2(uu); vm = mont->add(mont->mrmul(dm, um), vm); vm = mont->div2(vm); um = uu; qn = mont->mrmul(qn, qm); } } if (um == 0 || vm == 0) { return true; } u64 x = ((n + 1) & (~n)); // n = u64::MAX の時の挙動は? for (x >>= 1; x > 0; x >>= 1) { um = mont->mrmul(um, vm); vm = mont->sub(mont->mrmul(vm, vm), mont->add(qn, qn)); if (vm == 0) { return true; } qn = mont->mrmul(qn, qn); } return false; } // Baillie–PSW primarity test bool primetest_u64_bpsw_sub(const U64Mont *const mont) noexcept { return primetest_u64_miller_base2_sub(mont) && primetest_u64_lucas_sub(mont); } // Baillie–PSW primarity test bool primetest_u64_bpsw(u64 n) noexcept { if (n == 2) { return true; } if (n == 1 || (n & 1) == 0) { return false; } U64Mont mont(n); return primetest_u64_bpsw_sub(&mont); } U64Mont u64mont_new(u64 n) noexcept { return U64Mont(n); } u64 u64mont_add(const U64Mont *const mont, u64 ar, u64 br) noexcept { return mont->add(ar, br); } u64 u64mont_sub(const U64Mont *const mont, u64 ar, u64 br) noexcept { return mont->sub(ar, br); } u64 u64mont_div2(const U64Mont *const mont, u64 ar) noexcept { return mont->div2(ar); } u64 u64mont_mrmul(const U64Mont *const mont, u64 ar, u64 br) noexcept { return mont->mrmul(ar, br); } u64 u64mont_mr(const U64Mont *const mont, u64 a) noexcept { return mont->mr(a); } u64 u64mont_ar(const U64Mont *const mont, u64 a) noexcept { return mont->ar(a); } u64 u64mont_pow(const U64Mont *const mont, u64 ar, u64 b) noexcept { return mont->pow(ar, b); } int ctzll(unsigned long long v) noexcept { return __builtin_ctzll(v); } int clzll(unsigned long long v) noexcept { return __builtin_clzll(v); } const u64 bases[] = {2,325,9375,28178,450775,9780504,1795265022}; bool primetest_u64_miller_base7_sub(const U64Mont *const mont) noexcept { if (!primetest_u64_miller_base2_sub(mont)) { return false; } for (const auto& base : bases) { if (!primetest_u64_miller_sub(mont, base)) { return false; } } return true; } bool primetest_u64_miller_base7(const u64 n) noexcept { if (n == 2) { return true; } if (n < 2 || (n & 1) == 0) { return false; } U64Mont mont(n); return primetest_u64_miller_base7_sub(&mont); } int main(int, char**) { struct timespec time_monotonic_start, time_process_start, time_monotonic_end, time_process_end; clock_gettime(CLOCK_MONOTONIC, &time_monotonic_start); clock_gettime(CLOCK_PROCESS_CPUTIME_ID, &time_process_start); int n; unsigned long long x; scanf("%d", &n); for(int i = 0; i < n; ++i) { scanf("%llu", &x); //printf("%llu %d\n", x, primetest_u64_miller_base7((u64)x) ? 1 : 0); printf("%llu %d\n", x, primetest_u64_bpsw((u64)x) ? 1 : 0); } clock_gettime(CLOCK_PROCESS_CPUTIME_ID, &time_process_end); clock_gettime(CLOCK_MONOTONIC, &time_monotonic_end); f64 d_sec_monotonic = (f64)(time_monotonic_end.tv_sec - time_monotonic_start.tv_sec) + (f64)(time_monotonic_end.tv_nsec - time_monotonic_start.tv_nsec) / (1000 * 1000 * 1000); f64 d_sec_process = (f64)(time_process_end.tv_sec - time_process_start.tv_sec) + (f64)(time_process_end.tv_nsec - time_process_start.tv_nsec) / (1000 * 1000 * 1000); fprintf(stderr, "time_monotonic:%f, time_process:%f\n", d_sec_monotonic, d_sec_process); }