// -*- coding:utf-8-unix -*- // モンゴメリ剰余乗算 (Montgomery modular multiplication) pub struct Mont { n: T, // n is odd, and n > 2 ni: T, // n * ni == 1 (mod 2**64 {@T=u64}) r: T, // == 2**64 {@T=u64} (mod n) rn: T, // == -1 * (2**64 {T=u64}) (mod n) r2: T, // == 2**128 {@T=u64} (mod n) d: T, // == (n - 1) >> (n - 1).trailing_zeros() k: BitCounts, // == (n - 1).trailing_zeros() } pub trait MontTrait { fn n(&self) -> T; fn ni(&self) -> T; fn r(&self) -> T; fn rn(&self) -> T; fn r2(&self) -> T; fn d(&self) -> T; fn k(&self) -> BitCounts; // 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; // mrmul(ar, br) == (ar * br) / r (mod n) // R == 2**64 {@T=u64} // 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 {@T=u64} // 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; } impl MontTrait for Mont { #[inline] fn n(&self) -> u64 { self.n } #[inline] fn ni(&self) -> u64 { self.ni } #[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**3)*((k*(k+1)/2)**1))/(2*k+1) let mut ni = n; // ni1 := ni0 * ((n * ni0 - 3) * (n * ni0) + 3); // = (1+(2**9)*((k*(k+1)/2)**3))/(2*k+1) // ni2 := ni1 * ((n * ni1 - 3) * (n * ni1) + 3); // = (1+(2**27)*((k*(k+1)/2)**9))/(2*k+1) // ni3 := ni2 * ((n * ni2 - 3) * (n * ni2) + 3); // = (1+(2**81)*((k*(k+1)/2)**27))/(2*k+1) // // (n * ni3) mod 2**64 = ((2*k+1) * ni3) mod 2**64 = 1 mod 2**64 for _ in 0..3 { let t = n.wrapping_mul(ni); ni = t.wrapping_sub(3).wrapping_mul(t).wrapping_add(3).wrapping_mul(ni); } debug_assert_eq!(n.wrapping_mul(ni), 1); // n * ni == 1 (mod 2**64) let r: u64 = n.wrapping_neg() % n; // == 2**64 (mod n) let rn: u64 = n - r; // == -1 * (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, r, rn, r2, d, k }.mr(r), 1); // r / r == 1 (mod n) debug_assert_eq!(Self { n, ni, r, rn, r2, d, k }.mrmul(1, r2), r); // r2 / r == r (mod n) Self { n, ni, 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 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 }; loop { b >>= 1; if b == 0 { return t; } ar = self.mrmul(ar, ar); if (b & 1) != 0 { t = self.mrmul(t, ar); } } } } #[inline] fn prime_test_miller_sub(mont: &Mont, mut base: u64) -> bool { // Mirrer-Rabin primality test if base >= mont.n() { base %= mont.n(); if base == 0 { return true; } } let mut tr = mont.pow(mont.ar(base), mont.d()); if tr == mont.r() || tr == mont.rn() { return true; } for _ in 1..(mont.k()) { tr = mont.mrmul(tr, tr); if tr == mont.rn() { return true; } } false } pub fn miller_rabin(n: u64) -> bool { if n == 2 { return true; } if n < 2 || (n & 1) == 0 { return false; } let mont = Mont::::new(n); [2,325,9375,28178,450775,9780504,1795265022].iter().all(|&base| prime_test_miller_sub(&mont, base)) } 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 miller_rabin(x) { "1" } else { "0" }); } eprint!("{}us\n", start_time.elapsed().as_micros()); }