#define _USE_MATH_DEFINES #include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) #define ALL(v) (v).begin(),(v).end() using ll = long long; constexpr int INF = 0x3f3f3f3f; constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL; constexpr double EPS = 1e-8; constexpr int MOD = 1000000007; // constexpr int MOD = 998244353; constexpr int dy[] = {1, 0, -1, 0}, dx[] = {0, -1, 0, 1}; constexpr int dy8[] = {1, 1, 0, -1, -1, -1, 0, 1}, dx8[] = {0, -1, -1, -1, 0, 1, 1, 1}; template inline bool chmax(T &a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T &a, U b) { return a > b ? (a = b, true) : false; } struct IOSetup { IOSetup() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); std::cout << fixed << setprecision(20); } } iosetup; // https://github.com/ei1333/library/blob/0698ba3a2169a48966949e1d23b6cbba9e52e03a/math/number-theory/fast-prime-factorization.cpp namespace FastPrimeFactorization { template< typename word, typename dword, typename sword > struct UnsafeMod { UnsafeMod() : x(0) {} UnsafeMod(word _x) : x(init(_x)) {} bool operator==(const UnsafeMod &rhs) const { return x == rhs.x; } bool operator!=(const UnsafeMod &rhs) const { return x != rhs.x; } UnsafeMod &operator+=(const UnsafeMod &rhs) { if((x += rhs.x) >= mod) x -= mod; return *this; } UnsafeMod &operator-=(const UnsafeMod &rhs) { if(sword(x -= rhs.x) < 0) x += mod; return *this; } UnsafeMod &operator*=(const UnsafeMod &rhs) { x = reduce(dword(x) * rhs.x); return *this; } UnsafeMod operator+(const UnsafeMod &rhs) const { return UnsafeMod(*this) += rhs; } UnsafeMod operator-(const UnsafeMod &rhs) const { return UnsafeMod(*this) -= rhs; } UnsafeMod operator*(const UnsafeMod &rhs) const { return UnsafeMod(*this) *= rhs; } UnsafeMod pow(uint64_t e) const { UnsafeMod ret(1); for(UnsafeMod base = *this; e; e >>= 1, base *= base) { if(e & 1) ret *= base; } return ret; } word get() const { return reduce(x); } static constexpr int word_bits = sizeof(word) * 8; static word modulus() { return mod; } static word init(word w) { return reduce(dword(w) * r2); } static void set_mod(word m) { mod = m; inv = mul_inv(mod); r2 = -dword(mod) % mod; } static word reduce(dword x) { word y = word(x >> word_bits) - word((dword(word(x) * inv) * mod) >> word_bits); return sword(y) < 0 ? y + mod : y; } static word mul_inv(word n, int e = 6, word x = 1) { return !e ? x : mul_inv(n, e - 1, x * (2 - x * n)); } static word mod, inv, r2; word x; }; using uint128_t = __uint128_t; using Mod64 = UnsafeMod< uint64_t, uint128_t, int64_t >; template<> uint64_t Mod64::mod = 0; template<> uint64_t Mod64::inv = 0; template<> uint64_t Mod64::r2 = 0; using Mod32 = UnsafeMod< uint32_t, uint64_t, int32_t >; template<> uint32_t Mod32::mod = 0; template<> uint32_t Mod32::inv = 0; template<> uint32_t Mod32::r2 = 0; bool miller_rabin_primality_test_uint64(uint64_t n) { Mod64::set_mod(n); uint64_t d = n - 1; while(d % 2 == 0) d /= 2; Mod64 e{1}, rev{n - 1}; // http://miller-rabin.appspot.com/ < 2^64 for(uint64_t a : {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) { if(n <= a) break; uint64_t t = d; Mod64 y = Mod64(a).pow(t); while(t != n - 1 && y != e && y != rev) { y *= y; t *= 2; } if(y != rev && t % 2 == 0) return false; } return true; } bool miller_rabin_primality_test_uint32(uint32_t n) { Mod32::set_mod(n); uint32_t d = n - 1; while(d % 2 == 0) d /= 2; Mod32 e{1}, rev{n - 1}; for(uint32_t a : {2, 7, 61}) { if(n <= a) break; uint32_t t = d; Mod32 y = Mod32(a).pow(t); while(t != n - 1 && y != e && y != rev) { y *= y; t *= 2; } if(y != rev && t % 2 == 0) return false; } return true; } bool is_prime(uint64_t n) { if(n == 2) return true; if(n == 1 || n % 2 == 0) return false; if(n < uint64_t(1) << 31) return miller_rabin_primality_test_uint32(n); return miller_rabin_primality_test_uint64(n); } uint64_t pollard_rho(uint64_t n) { if(is_prime(n)) return n; if(n % 2 == 0) return 2; Mod64::set_mod(n); uint64_t d; Mod64 one{1}; for(Mod64 c{one};; c += one) { Mod64 x{2}, y{2}; do { x = x * x + c; y = y * y + c; y = y * y + c; d = __gcd((x - y).get(), n); } while(d == 1); if(d < n) return d; } assert(0); } vector< uint64_t > prime_factor(uint64_t n) { if(n <= 1) return {}; uint64_t p = pollard_rho(n); if(p == n) return {p}; auto l = prime_factor(p); auto r = prime_factor(n / p); copy(begin(r), end(r), back_inserter(l)); return l; } }; int main() { int q; cin >> q; while (q--) { int n; cin >> n; vector ans = FastPrimeFactorization::prime_factor(1LL * n * n + 1); int p = ans.size(); sort(ALL(ans)); REP(i, p) cout << ans[i] << " \n"[i + 1 == p]; } return 0; }