#pragma region opt #pragma GCC target("avx2") #pragma GCC optimize("O3") #pragma endregion opt #include #pragma region type using i8 = std::int8_t; using i16 = std::int16_t; using i32 = std::int32_t; using i64 = std::int64_t; using u8 = std::uint8_t; using u16 = std::uint16_t; using u32 = std::uint32_t; using u64 = std::uint64_t; using i128 = __int128_t; using u128 = __uint128_t; using f32 = float; using f64 = double; using f80 = long double; template using vec = std::vector; template using vvec = std::vector>; template using vvvec = std::vector>>; template using pvec = std::pair, std::vector>; #pragma endregion type #pragma region MACRO for #define FOR(i,a,b) for(int i=(a), i##_len=(b); i((obj).size())) #pragma endregion MACRO container #pragma region MACRO bits #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 _ROTL32(x, s) (((x) << ((s) % (32))) | (((x) >> ((32) - ((s) % (32)))))) #define _ROTR32(x, s) (((x) >> ((s) % (32))) | (((x) << ((32) - ((s) % (32)))))) #define ROTL32(x, s) (((s) == (0)) ? (x) : ((((i64)(s)) < (0)) ? (_ROTR32((x), -(s))) : (_ROTL32((x), (s))))) #define ROTR32(x, s) (((s) == (0)) ? (x) : ((((i64)(s)) < (0)) ? (_ROTL32((x), -(s))) : (_ROTR32((x), (s))))) #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 bits #pragma region util template inline bool chmax(T& a,T b){ if (a < b) { a = b; return 1; } return 0; } template inline bool chmin(T& a,T b){ if (a > b) { a = b; return 1; } return 0; } #pragma endregion util #pragma region IO // -2147483648 ~ 2147483647 (> 10 ^ 9) i32 in_i32(void) { 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); } // -9223372036854775808 ~ 9223372036854775807 (> 10 ^ 18) i64 in_i64(void) { 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); } // 0 ~ 4294967295 (> 10 ^ 9) u32 in_u32(void) { 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); } // 0 ~ 18446744073709551615 (> 10 ^ 19) u64 in_u64(void) { 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(' '); } #pragma endregion IO #pragma region jacobi 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; } #pragma endregion jacobi #pragma region Baillie_PSW primality test int is_prime(u64 n) { using m64 = u64; { if (n <= 1) return 0; if (n <= 3) return 1; if (!(n & 1)) return 0; } const m64 one = (u64)-1ull % n + 1; const m64 r2 = (u128)(i128)-1 % n + 1; m64 N_ = n; for (int i = 0; i < 5; i++) N_ *= 2 - N_ * n; const m64 N = N_; auto reduce = [](u128 a, m64 x, u64 mod) { u64 y = (u64)(a >> 64) - (u64)(((u128)((u64)a * x) * mod) >> 64); return (i64)y < 0 ? y + mod : y; }; auto to = [&reduce](u64 a, m64 x, m64 y, u64 mod) { return reduce((u128)a * x, y, mod); }; auto add = [](m64 x, m64 y, u64 mod) { return x + y >= mod ? x + y - mod: x + y; }; auto sub = [](m64 x, m64 y, u64 mod) { return x >= y ? x - y : mod + x - y; }; auto mul = [&reduce](m64 x, m64 y, m64 z, u64 mod) { return reduce(u128(x) * y, z, mod); }; { u64 d = (n - 1) << __builtin_clzll(n - 1); m64 t = one << 1; if (t >= n) t -= n; for (d <<= 1; d; d <<= 1) { t = mul(t, t, N, n); if (d >> 63) { t <<= 1; if (t >= n) t -= n; } } if (t != one) { u64 x = (n - 1) & -(n - 1); m64 mone = n - one; for (x >>= 1; t != mone; x >>= 1) { if (x == 0) return 0; t = mul(t, t, N, n); } } } { i64 D = 5; for (int i = 0; jacobi_symbol(D, n) != -1 && i < 64; i++) { if (i == 32) { u32 k = round(sqrtl(n)); if (k * k == n) return 0; } if (i & 1) D -= 2; else D += 2; D = -D; } m64 Q = to(D < 0 ? (1 - D) / 4 % n : n - (D - 1) / 4 % n, r2, N, n); m64 u, v, Qn; u64 k = (n + 1) << __builtin_clzll(n + 1); u = one; v = one; Qn = Q; D %= (i64)n; D = to(D < 0 ? n + D : D, r2, N, n); for (k <<= 1; k; k <<= 1) { u = mul(u, v, N, n); v = sub(mul(v, v, N, n), add(Qn, Qn, n), n); Qn = mul(Qn, Qn, N, n); if (k >> 63) { u64 uu = add(u, v, n); if (uu & 1) uu += n; uu >>= 1; v = add(mul(D, u, N, n), v, n); if (v & 1) v += n; v >>= 1; u = uu; Qn = mul(Qn, Q, N, n); } } if (u == 0 || v == 0) return 1; u64 x = (n + 1) & ~n; for (x >>= 1; x; x >>= 1) { u = mul(u, v, N, n); v = sub(mul(v, v, N, n), add(Qn, Qn, n), n); if (v == 0) return 1; Qn = mul(Qn, Qn, N, n); } } return 0; } #pragma endregion Baillie_PSW primality test void Main() { // your source here int T; scanf("%d", &T); while (T--) { unsigned long long int x; scanf("%llu", &x); printf("%llu %d\n", x, is_prime(x)); } return; } int main() { std::ios_base::sync_with_stdio(false); std::cin.tie(nullptr); std::cout.tie(nullptr); std::cout << std::fixed << std::setprecision(13); std::cerr << std::fixed << std::setprecision(3); Main(); }