// -*- coding:utf-8-unix -*- // モンゴメリ剰余乗算 (Montgomery modular multiplication) pub trait UMontTrait { fn n(&self) -> T; fn ni(&self) -> T; fn nh(&self) -> T; fn r(&self) -> T; fn rn(&self) -> T; fn r2(&self) -> T; fn d(&self) -> T; fn k(&self) -> u32; // create constant structures for montgomery modular arithmetic fn new(n: T) -> Self; // addmod(a, b) == a + b (mod n) fn add(&self, a: T, b: T) -> T; // submod(a, b) == a - b (mod n) fn sub(&self, a: T, b: T) -> T; // div2mod(ar) == ar / 2 (mod n) fn div2(&self, ar: T) -> T; // 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 fn mrmul(&self, ar: T, br: T) -> T; // 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 fn mr(&self, ar: T) -> T; // ar(a) == a * r (mod n) fn ar(&self, a: T) -> T; // powir(ar, b) == ((ar / r) ** b) * r (mod n) fn pow(&self, ar: T, b: T) -> T; // Miller-Rabin primality test fn prime_test_once(&self, base: T) -> bool; } pub struct U64Mont { n: u64, // n is odd, and n > 2 ni: u64, // n * ni == 1 (mod 2**64) nh: u64, // == (n + 1) / 2 r: u64, // == 2**64 (mod n) rn: u64, // == -(2**64) (mod n) r2: u64, // == 2**128 (mod n) d: u64, // == (n - 1) >> (n - 1).trailing_zeros() k: u32, // == (n - 1).trailing_zeros() } impl UMontTrait for U64Mont { #[inline] fn n(&self) -> u64 { self.n } #[inline] fn ni(&self) -> u64 { self.ni } #[inline] fn nh(&self) -> u64 { self.nh } #[inline] fn r(&self) -> u64 { self.r } #[inline] fn rn(&self) -> u64 { self.rn } #[inline] fn r2(&self) -> u64 { self.r2 } #[inline] fn d(&self) -> u64 { self.d } #[inline] fn k(&self) -> u32 { self.k } #[inline] fn new(n: u64) -> Self { // create constant structures for montgomery modular arithmetic debug_assert_eq!(n & 1, 1); // // 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**2)*((k*(k+1))**1))/(2*k+1) let mut ni = n; // ni1 := ni0 * (2 - (n * ni0)); // = (1-(2**4)*((k*(k+1))**2))/(2*k+1) // ni2 := ni1 * (2 - (n * ni1)); // = (1-(2**8)*((k*(k+1))**4))/(2*k+1) // ni3 := ni2 * (2 - (n * ni2)); // = (1-(2**16)*((k*(k+1))**8))/(2*k+1) // ni4 := ni3 * (2 - (n * ni3)); // = (1-(2**32)*((k*(k+1))**16))/(2*k+1) // ni5 := ni4 * (2 - (n * ni4)); // = (1-(2**64)*((k*(k+1))**32))/(2*k+1) // // (n * ni5) mod 2**64 = ((2*k+1) * ni5) mod 2**64 = 1 mod 2**64 for _ in 0..5 { ni = ni.wrapping_mul(2u64.wrapping_sub(n.wrapping_mul(ni))); } debug_assert_eq!(n.wrapping_mul(ni), 1); // n * ni == 1 (mod 2**64) let nh = (n >> 1) + 1; // == (n + 1) / 2 let r: u64 = n.wrapping_neg() % n; // == 2**64 (mod n) let rn: u64 = n - r; // == -(2**64) (mod n) let r2: u64 = ((n as u128).wrapping_neg() % (n as u128)) as u64; // == 2**128 (mod n) // n == 2**k * d + 1 let mut d = n - 1; let k = d.trailing_zeros(); d >>= k; debug_assert_eq!(Self { n, ni, nh, r, rn, r2, d, k }.mr(r), 1); // r / r == 1 (mod n) debug_assert_eq!(Self { n, ni, nh, r, rn, r2, d, k }.mrmul(1, r2), r); // r2 / r == r (mod n) Self { n, ni, nh, r, rn, r2, d, k } } #[inline] fn add(&self, a: u64, b: u64) -> u64 { // addmod(a, b) == a + b (mod n) debug_assert!(a < self.n()); debug_assert!(b < self.n()); let (t, fa) = a.overflowing_add(b); let (u, fs) = t.overflowing_sub(self.n()); if fa || !fs { u } else { t } } #[inline] fn sub(&self, a: u64, b: u64) -> u64 { // submod(a, b) == a - b (mod n) debug_assert!(a < self.n()); debug_assert!(b < self.n()); let (t, f) = a.overflowing_sub(b); if f { t.wrapping_add(self.n()) } else { t } } #[inline] fn div2(&self, ar: u64) -> u64 { // div2mod(ar) == ar / 2 (mod n) debug_assert!(ar < self.n()); if (ar & 1) == 0 { ar >> 1 } else { (ar >> 1) + self.nh() } } #[inline] fn mrmul(&self, ar: u64, br: u64) -> u64 { // 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 debug_assert!(ar < self.n()); debug_assert!(br < self.n()); let t: u128 = (ar as u128) * (br as u128); let (t, f) = ((t >> 64) as u64).overflowing_sub((((((t as u64).wrapping_mul(self.ni())) as u128) * (self.n() as u128)) >> 64) as u64); if f { t.wrapping_add(self.n()) } else { t } } #[inline] fn mr(&self, ar: u64) -> u64 { // 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 debug_assert!(ar < self.n()); let (t, f) = (((((ar.wrapping_mul(self.ni())) as u128) * (self.n() as u128)) >> 64) as u64).overflowing_neg(); if f { t.wrapping_add(self.n()) } else { t } } #[inline] fn ar(&self, a: u64) -> u64 { // ar(a) == a * r (mod n) debug_assert!(a < self.n()); self.mrmul(a, self.r2()) } #[inline] fn pow(&self, mut ar: u64, mut b: u64) -> u64 { // powir(ar, b) == ((ar / r) ** b) * r (mod n) debug_assert!(ar < self.n()); let mut t = if (b & 1) == 0 { self.r() } else { ar }; b >>= 1; while b != 0 { ar = self.mrmul(ar, ar); if (b & 1) != 0 { t = self.mrmul(t, ar); } b >>= 1; } t } #[inline] fn prime_test_once(&self, base: u64) -> bool { // Miller-Rabin primality test debug_assert!(base > 1); let (n, r, d, k) = (self.n(), self.r(), self.d(), self.k()); let b = if base < n { base } else { base % n }; if b == 0 { return true; } let mut br = self.pow(self.ar(b), d); if br == r || br == self.rn { return true; } for _ in 1..k { br = self.mrmul(br, br); if br == self.rn { return true; } } false } } #[inline] fn prime_test_miller_7bases(mont: &U64Mont) -> bool { // Deterministic variants of the Miller-Rabin primality test // http://miller-rabin.appspot.com/ mont.prime_test_once(2) && mont.prime_test_once(325) && mont.prime_test_once(9375) && mont.prime_test_once(28178) && mont.prime_test_once(450775) && mont.prime_test_once(9780504) && mont.prime_test_once(1795265022) } pub fn prime_test_64_miller(n: u64) -> bool { if n == 2 { return true; } if n == 1 || (n & 1) == 0 { return false; } let mont = U64Mont::new(n); prime_test_miller_7bases(&mont) } fn main() { use std::io::{BufRead,Write}; let start_time = std::time::Instant::now(); let out = std::io::stdout(); let mut out = std::io::BufWriter::new(out.lock()); macro_rules! puts {($($format:tt)*) => (let _ = write!(out,$($format)*););} let input = std::io::stdin(); let mut lines = std::io::BufReader::new(input.lock()).lines(); let n: usize = lines.next().unwrap().unwrap().parse().unwrap(); for _ in 0..n { let x: u64 = lines.next().unwrap().unwrap().parse().unwrap(); puts!("{} {}\n", x, if prime_test_64_miller(x) { "1" } else { "0" }); } eprint!("{}us\n", start_time.elapsed().as_micros()); }