#pragma region opt #pragma GCC optimize("O3") #pragma GCC optimize("fast-math") #pragma GCC optimize("unroll-loops") #pragma GCC target("avx2") #pragma endregion opt #pragma region header #define _GNU_SOURCE #include #include #include #include #include #include #include #include #include #pragma endregion header #pragma region type /* signed integer */ typedef int8_t i8; typedef int16_t i16; typedef int32_t i32; typedef int64_t i64; typedef __int128_t i128; /* unsigned integer */ typedef uint8_t u8; typedef uint16_t u16; typedef uint32_t u32; typedef uint64_t u64; typedef __uint128_t u128; /* floating point number */ typedef float f32; typedef double f64; typedef long double f80; #pragma endregion type #pragma region macro #define MIN(a, b) (((a) < (b)) ? (a) : (b)) #define MAX(a, b) (((a) > (b)) ? (a) : (b)) #define SWAP(a, b) (((a) ^= (b)), ((b) ^= (a)), ((a) ^= (b))) #define POPCNT32(a) __builtin_popcount((a)) #define POPCNT64(a) __builtin_popcountll((a)) #define CTZ32(a) __builtin_ctz((a)) #define CLZ32(a) __builtin_clz((a)) #define CTZ64(a) __builtin_ctzll((a)) #define CLZ64(a) __builtin_clzll((a)) #define HAS_SINGLE_BIT32(a) (__builtin_popcount((a)) == (1)) #define HAS_SINGLE_BIT64(a) (__builtin_popcountll((a)) == (1)) #define MSB32(a) ((31) - __builtin_clz((a))) #define MSB64(a) ((63) - __builtin_clzll((a))) #define BIT_WIDTH32(a) ((a) ? ((32) - __builtin_clz((a))) : (0)) #define BIT_WIDTH64(a) ((a) ? ((64) - __builtin_clzll((a))) : (0)) #define LSBit(a) ((a) & (-(a))) #define CLSBit(a) ((a) & ((a) - (1))) #define BIT_CEIL32(a) ((!(a)) ? (1) : ((POPCNT32(a)) == (1) ? ((1u) << ((31) - CLZ32((a)))) : ((1u) << ((32) - CLZ32(a))))) #define BIT_CEIL64(a) ((!(a)) ? (1) : ((POPCNT64(a)) == (1) ? ((1ull) << ((63) - CLZ64((a)))) : ((1ull) << ((64) - CLZ64(a))))) #define BIT_FLOOR32(a) ((!(a)) ? (0) : ((1u) << ((31) - CLZ32((a))))) #define BIT_FLOOR64(a) ((!(a)) ? (0) : ((1ull) << ((63) - CLZ64((a))))) #define _ROTL64(x, s) (((x) << ((s) % (64))) | (((x) >> ((64) - ((s) % (64)))))) #define _ROTR64(x, s) (((x) >> ((s) % (64))) | (((x) << ((64) - ((s) % (64)))))) #define ROTL64(x, s) (((s) == (0)) ? (x) : ((((i128)(s)) < (0)) ? (_ROTR64((x), -(s))) : (_ROTL64((x), (s))))) #define ROTR64(x, s) (((s) == (0)) ? (x) : ((((i128)(s)) < (0)) ? (_ROTL64((x), -(s))) : (_ROTR64((x), (s))))) #pragma endregion macro #pragma region io static inline int read_int(void) { // -2147483648 ~ 2147483647 (> 10 ^ 9) int c, x = 0, f = 1; while (c = getchar_unlocked(), c < 48 || c > 57) if (c == 45) f = -f; while (47 < c && c < 58) { x = x * 10 + c - 48; c = getchar_unlocked(); } return f * x; } static inline i32 in_i32(void) { // -2147483648 ~ 2147483647 (> 10 ^ 9) i32 c, x = 0, f = 1; while (c = getchar_unlocked(), c < 48 || c > 57) if (c == 45) f = -f; while (47 < c && c < 58) { x = x * 10 + c - 48; c = getchar_unlocked(); } return f * x; } static inline u32 in_u32(void) { // 0 ~ 4294967295 (> 10 ^ 9) u32 c, x = 0; while (c = getchar_unlocked(), c < 48 || c > 57); while (47 < c && c < 58) { x = x * 10 + c - 48; c = getchar_unlocked(); } return x; } static inline i64 in_i64(void) { // -9223372036854775808 ~ 9223372036854775807 (> 10 ^ 18) i64 c, x = 0, f = 1; while (c = getchar_unlocked(), c < 48 || c > 57) if (c == 45) f = -f; while (47 < c && c < 58) { x = x * 10 + c - 48; c = getchar_unlocked(); } return f * x; } static inline u64 in_u64(void) { // 0 ~ 18446744073709551615 (> 10 ^ 19) u64 c, x = 0; while (c = getchar_unlocked(), c < 48 || c > 57); while (47 < c && c < 58) { x = x * 10 + c - 48; c = getchar_unlocked(); } return x; } static inline void write_int_inner(int x) { if (x >= 10) write_int_inner(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } static inline void write_int(int x) { if (x < 0) { putchar_unlocked('-'); x = -x; } write_int_inner(x); } static inline void out_i32_inner(i32 x) { if (x >= 10) out_i32_inner(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } static inline void out_i32(i32 x) { if (x < 0) { putchar_unlocked('-'); x = -x; } out_i32_inner(x); } static inline void out_u32(u32 x) { if (x >= 10) out_u32(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } static inline void out_i64_inner(i64 x) { if (x >= 10) out_i64_inner(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } static inline void out_i64(i64 x) { if (x < 0) { putchar_unlocked('-'); x = -x; } out_i64_inner(x); } static inline void out_u64(u64 x) { if (x >= 10) out_u64(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } static inline void NL(void) { putchar_unlocked('\n'); } static inline void SP(void) { putchar_unlocked(' '); } #pragma endregion io #pragma region dump void dump_int(int x) { fprintf(stderr, "\033[1;31m%d\033[0m\n", x); } void dump_i32(i32 x) { fprintf(stderr, "\033[1;32m%d\033[0m\n", x); } void dump_i64(i64 x) { fprintf(stderr, "\033[1;33m%ld\033[0m\n", x); } void dump_u32(u32 x) { fprintf(stderr, "\033[1;34m%u\033[0m\n", x); } void dump_u64(u64 x) { fprintf(stderr, "\033[1;35m%lu\033[0m\n", x); } void dump_int_array(int *a, int a_len) { fprintf(stderr, "\033[1;31m"); for (int i = 0; i < a_len; i++) { if (i == a_len - 1) fprintf(stderr, "%d\n", a[i]); else fprintf(stderr, "%d ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_i32_array(i32 *a, int a_len) { fprintf(stderr, "\033[1;32m"); for (int i = 0; i < a_len; i++) { if (i == a_len - 1) fprintf(stderr, "%d\n", a[i]); else fprintf(stderr, "%d ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_i64_array(i64 *a, int a_len) { fprintf(stderr, "\033[1;33m"); for (int i = 0; i < a_len; i++) { if (i == a_len - 1) fprintf(stderr, "%ld\n", a[i]); else fprintf(stderr, "%ld ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_u32_array(u32 *a, int a_len) { fprintf(stderr, "\033[1;34m"); for (int i = 0; i < a_len; i++) { if (i == a_len - 1) fprintf(stderr, "%u\n", a[i]); else fprintf(stderr, "%u ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_u64_array(u64 *a, int a_len) { fprintf(stderr, "\033[1;35m"); for (int i = 0; i < a_len; i++) { if (i == a_len - 1) fprintf(stderr, "%lu\n", a[i]); else fprintf(stderr, "%lu ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_int_array_range(int *a, int a_len, int l, int r) { if (r < l || r <= 0 || l >= a_len) return; if (l < 0) l = 0; if (r >= a_len) r = a_len - 1; fprintf(stderr, "\033[1;31m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%d\n", a[i]); else fprintf(stderr, "%d ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_i32_array_range(i32 *a, int a_len, int l, int r) { if (r < l || r <= 0 || l >= a_len) return; if (l < 0) l = 0; if (r >= a_len) r = a_len - 1; fprintf(stderr, "\033[1;32m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%d\n", a[i]); else fprintf(stderr, "%d ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_i64_array_range(i64 *a, int a_len, int l, int r) { if (r < l || r <= 0 || l >= a_len) return; if (l < 0) l = 0; if (r >= a_len) r = a_len - 1; fprintf(stderr, "\033[1;33m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%ld\n", a[i]); else fprintf(stderr, "%ld ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_u32_array_range(u32 *a, int a_len, int l, int r) { if (r < l || r <= 0 || l >= a_len) return; if (l < 0) l = 0; if (r >= a_len) r = a_len - 1; fprintf(stderr, "\033[1;34m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%u\n", a[i]); else fprintf(stderr, "%u ", a[i]); } fprintf(stderr, "\033[0m"); } void dump_u64_array_range(u64 *a, int a_len, int l, int r) { if (r < l || r <= 0 || l >= a_len) return; if (l < 0) l = 0; if (r >= a_len) r = a_len - 1; fprintf(stderr, "\033[1;35m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%lu\n", a[i]); else fprintf(stderr, "%lu ", a[i]); } fprintf(stderr, "\033[0m"); } void printb(u64 v) { u64 mask = (u64)1 << (sizeof(v) * CHAR_BIT - 1); do putchar(mask & v ? '1' : '0'); while (mask >>= 1); } void putb(u64 v) { putchar('0'), putchar('b'), printb(v), putchar('\n'); } #pragma endregion dump #pragma region montgomery32 typedef uint32_t m32; m32 _one_m32(u32 mod) { return -1u % mod + 1; } m32 _r2_m32(u32 mod) { return (u64)(i64)-1 % mod + 1; } m32 _inv_m32(u32 mod) { u32 u = 1, v = 0, x = 1u << 31; for (int i = 0; i < 32; i++) { if (u & 1) u = (u + mod) >> 1, v = (v >> 1) + x; else u >>= 1, v >>= 1; } return -v; } m32 _reduce_m32(u64 a, m32 inv, u32 mod) { i64 z = (a >> 32) - ((((u32)a * inv) * (u64)mod) >> 32); return z < 0 ? z + mod : z; } m32 to_m32(u32 a, m32 r2, m32 inv, u32 mod) { return _reduce_m32((u64)a * r2, inv, mod); } u32 from_m32(m32 A, m32 inv, u32 mod) { m32 t = _reduce_m32((u64)A, inv, mod) - mod; return t + (mod & -(t >> 31u)); } m32 add_m32(m32 A, m32 B, u32 mod2) { // assert(mod2 == (mod << 1)); A += B - mod2; A += mod2 & -(A >> 31u); return A; } m32 sub_m32(m32 A, m32 B, u32 mod2) { // assert(mod2 == (mod << 1)); A -= B; A += mod2 & -(A >> 31u); return A; } m32 min_m32(m32 A, u32 mod2) { // assert(mod2 == (mod << 1)); return sub_m32(0u, A, mod2); } m32 mul_m32(m32 A, m32 B, m32 inv, u32 mod) { return _reduce_m32((u64)A * B, inv, mod); } m32 pow_m32(m32 A, i64 n, m32 inv, u32 mod) { m32 ret = _one_m32(mod); while (n > 0) { if (n & 1) ret = mul_m32(ret, A, inv, mod); A = mul_m32(A, A, inv, mod); n >>= 1; } return ret; } m32 inv_m32(m32 A, m32 inv, u32 mod) { return pow_m32(A, (i64)mod - 2, inv, mod); } m32 div_m32(m32 A, m32 B, m32 inv, u32 mod) { /* assert(is_prime(mod)); */ return mul_m32(A, inv_m32(B, inv, mod), inv, mod); } m32 in_m32(m32 r2, m32 inv, u32 mod) { u32 c, a = 0; while (c = getchar_unlocked(), c < 48 || c > 57); while (47 < c && c < 58) { a = a * 10 + c - 48; c = getchar_unlocked(); } return to_m32(a, r2, inv, mod); } void out_m32(m32 A, m32 inv, u32 mod) { u32 a = from_m32(A, inv, mod); out_u32(a); } void dump_m32(m32 x, m32 inv, u32 mod) { fprintf(stderr, "\033[1;34m%u\033[0m\n", from_m32(x, inv, mod)); } void dump_m32_array(m32 *x, int x_len, m32 inv, u32 mod) { fprintf(stderr, "\033[1;37m"); for (int i = 0; i < x_len; i++) { if (i == x_len - 1) fprintf(stderr, "%u\n", x[i]); else fprintf(stderr, "%u ", x[i]); } fprintf(stderr, "\033[0m"); fprintf(stderr, "\033[1;33m"); for (int i = 0; i < x_len; i++) { if (i == x_len - 1) fprintf(stderr, "%u\n", from_m32(x[i], inv, mod)); else fprintf(stderr, "%u ", from_m32(x[i], inv, mod)); } fprintf(stderr, "\033[0m"); } void dump_m32_array_range(m32 *x, int x_len, int l, int r, m32 inv, u32 mod) { if (r < l || r <= 0 || l >= x_len) return; if (l < 0) l = 0; if (r >= x_len) r = x_len - 1; fprintf(stderr, "\033[1;31m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%u\n", x[i]); else fprintf(stderr, "%u ", x[i]); } fprintf(stderr, "\033[0m"); fprintf(stderr, "\033[1;35m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%u\n", from_m32(x[i], inv, mod)); else fprintf(stderr, "%u ", from_m32(x[i], inv, mod)); } fprintf(stderr, "\033[0m"); } #pragma endregion montgomery32 #pragma region montgomery64 typedef uint64_t m64; m64 _one_m64(u64 mod) { return (u64)-1ull % mod + 1; } m64 _r2_m64(u64 mod) { return (u128)(i128)-1 % mod + 1; } m64 _inv_m64(u64 mod) { u64 u = 1, v = 0, x = 1ull < 63; for (int i = 0; i < 64; i++) { if (u & 1) u = (u + mod) >> 1, v = (v >> 1) + x; else u >>= 1, v >>= 1; } return -v; } m64 _reduce_m64(u128 a, m64 inv, u64 mod) { i128 A = (a >> 64) - ((((u64)a * inv) * (u128)mod) >> 64); return A < 0 ? A + mod : A; } m64 to_m64(u64 a, m64 r2, m64 inv, u64 mod) { return _reduce_m64((u128)a * r2, inv, mod); } u64 from_m64(m64 A, m64 inv, u64 mod) { m64 t = _reduce_m64((u128)A, inv, mod) - mod; return t + (mod & -(t >> 63u)); } m64 add_m64(m64 A, m64 B, u64 mod2) { // assert(mod2 == (mod << 1u)); A += B - mod2; A += mod2 & -(A >> 63u); return A; } m64 sub_m64(m64 A, m64 B, u64 mod2) { // assert(mod2 == (mod << 1u)); A -= B; A += mod2 & -(A >> 63u); return A; } m64 min_m64(m64 A, u64 mod2) { // assert(mod2 == (mod << 1u)); return sub_m64(0, A, mod2); } m64 mul_m64(m64 A, m64 B, m64 inv, u64 mod) { return _reduce_m64((u128)A * B, inv, mod); } m64 pow_m64(m64 A, i64 n, m64 inv, u64 mod) { m64 ret = _one_m64(mod); while (n > 0) { if (n & 1) ret = mul_m64(ret, A, inv, mod); A = mul_m64(A, A, inv, mod); n >>= 1; } return ret; } m64 inv_m64(m64 A, m64 inv, u64 mod) { return pow_m64(A, (i64)mod - 2, inv, mod); } m64 div_m64(m64 A, m64 B, m64 inv, u64 mod) { return mul_m64(A, inv_m64(B, inv, mod), inv, mod); } m64 in_m64(m64 r2, m64 inv, u64 mod) { u64 c, a = 0; while (c = getchar_unlocked(), c < 48 || c > 57); while (47 < c && c < 58) { a = a * 10 + c - 48; c = getchar_unlocked(); } return to_m64(a, r2, inv, mod); } void out_m64(m64 A, m64 inv, u64 mod) { u64 a = from_m64(A, inv, mod); out_u64(a); } void dump_m64(m64 A, m64 inv, u64 mod) { fprintf(stderr, "\033[1;36m%lu\033[0m\n", from_m64(A, inv, mod)); } void dump_m64_array(m64 *A, int A_len, m64 inv, u64 mod) { fprintf(stderr, "\033[1;35m"); for (int i = 0; i < A_len; i++) { if (i == A_len - 1) fprintf(stderr, "%lu\n", from_m64(A[i], inv, mod)); else fprintf(stderr, "%lu ", from_m64(A[i], inv, mod)); } fprintf(stderr, "\033[0m"); } void dump_m64_array_range(m64 *A, int A_len, int l, int r, m64 inv, u64 mod) { if (r < l || r <= 0 || l >= A_len) return; if (l < 0) l = 0; if (r >= A_len) r = A_len - 1; fprintf(stderr, "\033[1;35m"); for (int i = l; i <= r; i++) { if (i == r) fprintf(stderr, "%lu\n", from_m64(A[i], inv, mod)); else fprintf(stderr, "%lu ", from_m64(A[i], inv, mod)); } fprintf(stderr, "\033[0m"); } #pragma endregion montgomery64 #pragma region miller_rabin_primary_test bool is_prime32(u32 n) { u32 m = n - 1; m32 r2 = _r2_m32(n); m32 inv = _inv_m32(n); m32 one = _one_m32(n); m32 rev = to_m32(m, r2, inv, n); u32 d = m >> CTZ32(m); const u32 base[] = { 2u, 7u, 61u }; for (int i = 0; i < 3; i++) { if (n <= base[i]) break; m32 a = to_m32(base[i], r2, inv, n); u32 t = d; m32 y = pow_m32(a, t, inv, n); while (t != m && y != one && y != rev) { y = mul_m32(y, y, inv, n); t <<= 1; } if (y != rev && (!(t & 1))) return false; } return true; } bool is_prime64(u64 n) { u64 m = n - 1; m64 r2 = _r2_m64(n); m64 inv = _inv_m64(n); m64 one = _one_m64(n); m64 rev = to_m64(m, r2, inv, n); u64 d = m >> CTZ64(m); const u64 base[] = { 2ul, 325ul, 9375ul, 28178ul, 450775ul, 9780504ul, 1795265022ul }; for (int i = 0; i < 7; i++) { if (n <= base[i]) break; m64 a = to_m64(base[i], r2, inv, n); u64 t = d; m64 y = pow_m64(a, t, inv, n); while (t != m && y != one && y != rev) { y = mul_m64(y, y, inv, n); t <<= 1; } if (y != rev && (!(t & 1))) return false; } return true; } bool is_prime(u64 n) { if (n <= 3ul) return n == 2ul || n == 3ul; if (!(n & 1)) return false; if (n < ((u32)1u << 31)) return is_prime32((u32)n); return is_prime64(n); } #pragma endregion miller_rabin_primary_test void Main(void) { int n = read_int(); while (n--) { u64 x = in_u64(); out_u64(x); SP(); write_int(is_prime(x)); NL(); } } int main(void) { Main(); return 0; }