// clang-format off #pragma GCC target("avx2") #pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #pragma GCC optimize("omit-frame-pointer") #pragma GCC optimize("inline") #pragma GCC option("arch=native") #pragma GCC option("no-zero-upper") #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native") #include #include #include #include #include #include #include #include #include typedef int8_t i8; typedef int16_t i16; typedef int32_t i32; typedef int64_t i64; typedef __int128_t i128; typedef uint8_t u8; typedef uint16_t u16; typedef uint32_t u32; typedef uint64_t u64; typedef __uint128_t u128; typedef float f32; typedef double f64; typedef long double f80; /// (3, 7) -> 3 #define MIN(a, b) (((a) < (b)) ? (a) : (b)) /// (4, 5) -> 5 #define MAX(a, b) (((a) > (b)) ? (a) : (b)) /// (1, 2) -> (2, 1) #define SWAP_REF(a, b) \ do { \ (a) ^= (b); \ (b) ^= (a); \ (a) ^= (b); \ } \ while(0); /// {0, 1, 2, 3, 4, ...} -> {0, 1, 1, 2, 1, ...} #define POPCNT32(a) ((a) ? __builtin_popcount((a)) : (0)) /// {0, 1, 2, 4, 8, ...} -> {32, 0, 1, 2, 3, ...} #define CTZ32(a) ((a) ? __builtin_ctz((a)) : 32) /// {0, 1, 2, 4, 8, ...} -> {32, 31, 30, 29, 28, ...} #define CLZ32(a) ((a) ? __builtin_clz((a)) : 32) /// {0, 1, 2, 3, 4, ...} -> {0, 1, 1, 2, 1, ...} #define POPCNT64(a) ((a) ? __builtin_popcountll((a)) : (0)) /// {0, 1, 2, 4, 8, ...} -> {64, 0, 1, 2, 3, ...} #define CTZ64(a) ((a) ? __builtin_ctzll((a)) : 64) /// {0, 1, 2, 4, 8, ...} -> {64, 63, 62, 61, 60, ...} #define CLZ64(a) ((a) ? __builtin_clzll((a)) : 64) /// {0, 1, 2, 4, 8, ...} -> {-1, 0, 1, 2, 3, ...} #define MSB32(a) ((a) ? ((31) - __builtin_clz((a))) : (-1)) /// {0, 1, 2, 4, 8, ...} -> {-1, 0, 1, 2, 3, ...} #define MSB64(a) ((a) ? ((63) - __builtin_clzll((a))) : (-1)) /// {1, 2, 3, 4, 5, ...} -> {1, 2, 1, 4, 1, ...} #define LSBit(a) ((a) & (-(a))) /// {1, 2, 3, 4, 5, ...} -> {0, 0, 2, 0, 4, ...} #define CLSBit(a) ((a) & ((a) - (1))) /// {1, 2, 3, 4, 5, ...} -> {1, 2, 2, 4, 4, ...} #define BIT_FLOOR32(a) ((a) ? (1u) << MSB32((a)) : (0)) /// {6, 7, 8, 9, 10, ...} -> {4, 4, 8, 8, 8, ...} #define BIT_FLOOR64(a) ((a) ? (1ull) << MSB64((a)) : (0)) /// {1, 2, 3, 4, 5, ...} -> {1, 2, 4, 4, 8, ...} #define BIT_CEIL32_REF(a) \ do { \ --(a); \ (a) |= (a) >> (1); \ (a) |= (a) >> (2); \ (a) |= (a) >> (4); \ (a) |= (a) >> (8); \ (a) |= (a) >> (16); \ (a)++; \ } while(0); /// {8, 9, 10, 11, 12, ...} -> {8, 16, 16, 16, 16, ...} #define BIT_CEIL64_REF(a) \ do { \ --(a); \ (a) |= (a) >> (1); \ (a) |= (a) >> (2); \ (a) |= (a) >> (4); \ (a) |= (a) >> (8); \ (a) |= (a) >> (16); \ (a) |= (a) >> (32); \ (a)++; \ } while(0); #define ROTL32_INNER(x, l) (((x) << (l)) | ((x) >> ((-l) & (31)))) #define ROTR32_INNER(x, r) (((x) >> (r)) | ((x) << ((-r) & (31)))) #define ROTR32(x, r) (((r) < (0)) ? (ROTL32_INNER((x), ((u64)(-r) % (32)))) : (ROTR32_INNER((x), ((r) % (32))))) #define ROTL32(x, l) ROTR32((x), (-l)) #define ROTL64_INNER(x, l) (((x) << (l)) | ((x) >> ((-l) & (63)))) #define ROTR64_INNER(x, r) (((x) >> (r)) | ((x) << ((-r) & (63)))) #define ROTR64(x, r) (((r) < (0)) ? (ROTL64_INNER((x), ((u64)(-r) % (64)))) : (ROTR64_INNER((x), ((r) % (64))))) #define ROTL64(x, l) ROTR64((x), (-l)) 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 void out_i32_inner(i32 x) { if (x >= 10) out_i32_inner(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } void out_i32(i32 x) { if (x < 0) { putchar_unlocked('-'); x = -x; } out_i32_inner(x); } 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 void out_i64_inner(i64 x) { if (x >= 10) out_i64_inner(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } void out_i64(i64 x) { if (x < 0) { putchar_unlocked('-'); x = -x; } out_i64_inner(x); } 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; } void out_u32(u32 x) { if (x >= 10) out_u32(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } 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; } void out_u64(u64 x) { if (x >= 10) out_u64(x / 10); putchar_unlocked(x - x / 10 * 10 + 48); } void NL(void) { putchar_unlocked('\n'); } void SP(void) { putchar_unlocked(' '); } void dump_int(int x) { fprintf(stderr, "\033[1;36m%d\033[0m\n", x); } void dump_i64(i64 x) { fprintf(stderr, "\033[1;36m%ld\033[0m\n", x); } void dump_u32(u32 x) { fprintf(stderr, "\033[1;36m%u\033[0m\n", x); } void dump_u64(u64 x) { fprintf(stderr, "\033[1;36m%lu\033[0m\n", x); } void dump_int_array(int *a, int a_len) { for (int i = 0; i < a_len; i++) { if (i == a_len - 1) { fprintf(stderr, "\033[1;36m%d\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%d\033[0m ", a[i]); } } } void dump_i64_array(i64 *a, int a_len) { for (int i = 0; i < a_len; i++) { if (i == a_len - 1) { fprintf(stderr, "\033[1;36m%ld\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%ld\033[0m ", a[i]); } } } void dump_u32_array(u32 *a, int a_len) { for (int i = 0; i < a_len; i++) { if (i == a_len - 1) { fprintf(stderr, "\033[1;36m%u\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%u\033[0m ", a[i]); } } } void dump_u64_array(u64 *a, int a_len) { for (int i = 0; i < a_len; i++) { if (i == a_len - 1) { fprintf(stderr, "\033[1;36m%lu\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%lu\033[0m ", a[i]); } } } void dump_int_array_range(int *a, int a_len, int l, int r) { if (a_len <= r) { r = a_len - 1; } if (l > r) { return; } for (int i = l; i <= r; i++) { if (i == r) { fprintf(stderr, "\033[1;36m%d\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%d\033[0m ", a[i]); } } } void dump_i64_array_range(i64 *a, int a_len, int l, int r) { if (a_len <= r) { r = a_len - 1; } if (l > r) { return; } for (int i = l; i <= r; i++) { if (i == r) { fprintf(stderr, "\033[1;36m%ld\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%ld\033[0m ", a[i]); } } } void dump_u32_array_range(u32 *a, int a_len, int l, int r) { if (a_len <= r) { r = a_len - 1; } if (l > r) { return; } for (int i = l; i <= r; i++) { if (i == r) { fprintf(stderr, "\033[1;36m%u\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%u\033[0m ", a[i]); } } } void dump_u64_array_range(u64 *a, int a_len, int l, int r) { if (a_len <= r) { r = a_len - 1; } if (l > r) { return; } for (int i = l; i <= r; i++) { if (i == r) { fprintf(stderr, "\033[1;36m%lu\033[0m\n", a[i]); } else { fprintf(stderr, "\033[1;36m%lu\033[0m ", a[i]); } } } void printb_32bit(u32 v) { u32 mask = (u32)1 << (sizeof(v) * CHAR_BIT - 1); do { putchar_unlocked(mask & v ? '1' : '0'); } while (mask >>= 1); } void printb_64bit(u64 v) { u64 mask = (u64)1 << (sizeof(v) * CHAR_BIT - 1); do { putchar_unlocked(mask & v ? '1' : '0'); } while (mask >>= 1); } // clang-format on static u64 lcg_state = 14534622846793005ull; u32 lcg_rand(void) { return lcg_state = 6364136223846793005ull * lcg_state + 1442695040888963407ull; } u32 lcg_range(u32 l, u32 r) { return l + lcg_rand() % (r - l + 1); } int jacobi_symbol(i64 a, u64 n) { u64 t; int j = 1; while (a) { if (a < 0) { a = -a; if ((n & 3) == 3) j = -j; } int s = __builtin_ctzll(a); a >>= s; if (((n & 7) == 3 || (n & 7) == 5) && (s & 1)) j = -j; if ((a & n & 3) == 3) j = -j; t = a, a = n, n = t; a %= n; if ((u64)(a) > n / 2) a -= n; } return n == 1 ? j : 0; } 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 N_m64(u64 mod) { m64 N = mod; for (int i = 0; i < 5; i++) { N *= 2 - N * mod; } return N; } m64 reduce_m64(u128 a, m64 N, u64 mod) { u64 y = (u64)(a >> 64) - (u64)(((u128)((u64)a * N) * mod) >> 64); return (i64)y < 0 ? y + mod : y; } m64 to_m64(u64 a, m64 r2, m64 N, u64 mod) { return reduce_m64((u128)a * r2, N, mod); } u64 from_m64(m64 A, m64 N, u64 mod) { return reduce_m64((u128)A, N, mod); } m64 add_m64(m64 A, m64 B, u64 mod) { return A + B >= mod ? A + B - mod : A + B; } m64 sub_m64(m64 A, m64 B, u64 mod) { return A >= B ? A - B : mod + A - B; } m64 min_m64(m64 A, u64 mod) { return sub_m64(0ull, A, mod); } m64 mul_m64(m64 A, m64 B, m64 N, u64 mod) { return reduce_m64((u128)A * B, N, mod); } m64 pow_m64(m64 A, i64 n, m64 one, m64 N, u64 mod) { m64 ret = one; while (n > 0) { if (n & 1) ret = mul_m64(ret, A, N, mod); A = mul_m64(A, A, N, mod); n >>= 1; } return ret; } m64 inv_m64(m64 A, m64 one, m64 N, u64 mod) { return pow_m64(A, (i64)mod - 2, one, N, mod); } m64 div_m64(m64 A, m64 B, m64 one, m64 N, u64 mod) { /* assert(is_prime(mod)); */ return mul_m64(A, inv_m64(B, one, N, mod), N, mod); } bool eq_m64(m64 A, m64 B, m64 N, u64 mod) { return from_m64(A, N, mod) == from_m64(B, N, mod); } bool neq_m64(m64 A, m64 B, m64 N, u64 mod) { return from_m64(A, N, mod) != from_m64(B, N, mod); } m64 in_m64(m64 r2, m64 N, u64 mod) { u64 c = 0; u64 a = 0; while (c = getchar(), c < 48 || c > 57) ; while (47 < c && c < 58) { a = a * 10 + c - 48; c = getchar(); } return to_m64(a, r2, N, mod); } void out_m64(m64 A, m64 N, u64 mod) { u64 a = from_m64(A, N, mod); out_u64(a); } bool is_prime(u64 n) { { if (n <= 1) return false; if (n <= 3) return true; if (!(n & 1)) return false; } u64 mod = n; m64 r2 = r2_m64(mod); m64 N = N_m64(mod); m64 one = one_m64(mod); m64 rev = to_m64(n - 1, r2, N, mod); { for (int _ = 0; _ < 15; ++_) { u32 a = lcg_range(2u, ((n - 1) > ((1ull << 32) - 1)) ? 1u << 31 : n - 1); int x = jacobi_symbol(a, n); m64 y = (x == -1) ? rev : ((x == 0) ? 0 : one); if (y == 0 || y != pow_m64(to_m64(a, r2, N, mod), (n - 1) / 2, one, N, mod)) return false; } } return true; } int main(void) { u64 Q = in_u64(); while (Q--) { u64 x = in_u64(); out_u64(x); SP(); out_u32(is_prime(x) ? 1u : 0u); NL(); } return 0; }