// -*- coding:utf-8-unix -*- // モンゴメリ剰余乗算 (Montgomery modular multiplication) pub trait MontTrait { fn new(n: T) -> Self; fn add(&self, a: T, b: T) -> T; fn sub(&self, a: T, b: T) -> T; fn div2(&self, ar: T) -> T; fn mrmul(&self, ar: T, br: T) -> T; fn mr(&self, ar: T) -> T; fn ar(&self, a: T) -> T; fn pow(&self, ar: T, b: T) -> T; } pub struct Mont { n: T, // n is odd, and n > 2 ni: T, // n * ni == 1 (mod 2**64) nh: T, // == (n + 1) / 2 r: T, // == 2**64 (mod n) rn: T, // == -(2**64) (mod n) r2: T, // == 2**128 (mod n) d: T, // == (n - 1) >> (n - 1).trailing_zeros() k: BitCountType, // == (n - 1).trailing_zeros() } impl MontTrait for Mont { #[inline] fn new(n: u64) -> Self { 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; let r = n.wrapping_neg() % n; // == 2**64 (mod n) let rn = n - r; let r2 = ((n as u128).wrapping_neg() % (n as u128)) as u64; // == 2**128 (mod n) 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 { // == 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 { u } else { if fs { t } else { u } } } #[inline] fn sub(&self, a: u64, b: u64) -> u64 { // == 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 { // == ar / 2 (mod n) debug_assert!(ar < self.n); let t = ar >> 1; if (ar & 1) == 0 { t } else { t + self.nh } } #[inline] fn mrmul(&self, ar: u64, br: u64) -> u64 { // == (ar * br) / r (mod n) debug_assert!(ar < self.n); debug_assert!(br < self.n); let (n, ni) = (self.n, self.ni); let t: u128 = (ar as u128) * (br as u128); let (t, f) = ((t >> 64) as u64).overflowing_sub((((((t as u64).wrapping_mul(ni)) as u128) * (n as u128)) >> 64) as u64); if f { t.wrapping_add(n) } else { t } } #[inline] fn mr(&self, ar: u64) -> u64 { // == ar / r (mod n) debug_assert!(ar < self.n); let (n, ni) = (self.n, self.ni); let (t, f) = (((((ar.wrapping_mul(ni)) as u128) * (n as u128)) >> 64) as u64).overflowing_neg(); if f { t.wrapping_add(n) } else { t } } #[inline] fn ar(&self, a: u64) -> u64 { // == 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 { // == ((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); } } } } // 64bit整数平方根(固定ループ回数) -> (floor(sqrt(iv)), remain) #[inline] #[allow(unused)] fn isqrt64f(iv: u64) -> (u64, u64) { isqrt64i(iv, 0) } // 64bit整数平方根(可変ループ回数) -> (floor(sqrt(iv)), remain) #[inline] #[allow(unused)] fn isqrt64d(iv: u64) -> (u64, u64) { isqrt64i(iv, iv.leading_zeros()) } // 64bit整数平方根(lz:ケチるループ回数*2+(0~1)、内部実装) -> (floor(sqrt(iv)), remain) #[inline] fn isqrt64i(iv: u64, lz: u32) -> (u64, u64) { let n = (64 >> 1) - (lz >> 1); let s = (lz >> 1) << 1; let t = n << 1; let (mut a, mut b, c, d, e) = ( iv as u128, 0x0000_0000_0000_0000_4000_0000_0000_0000 >> s, 0xffff_ffff_ffff_fffe_0000_0000_0000_0000 >> s, 0x0000_0000_0000_0001_0000_0000_0000_0000 >> s, 0x0000_0000_0000_0000_ffff_ffff_ffff_ffff >> s, ); for _ in 0..n { let f = ((b + b) & c) + (b & e); if a >= b { a -= b; b = f + d; } else { b = f; } a <<= 2; } ((b >> t) as u64, (a >> t) as u64) } // Jacobi symbol: ヤコビ記号 #[inline] fn jacobi(a: i64, mut n: u64) -> i32 { let (mut a, mut j): (u64, i32) = if a >= 0 { (a as u64, 1) } else if (n & 3) == 3 { ((-a) as u64, -1) } else { ((-a) as u64, 1) }; while a > 0 { let ba = a.trailing_zeros(); a >>= ba; if ((n & 7) == 3 || (n & 7) == 5) && (ba & 1) != 0 { j = -j; } if (a & n & 3) == 3 { j = -j; } let t = a; a = n; n = t; a %= n; if a > (n >> 1) { a = n - a; if (n & 3) == 3 { j = -j; } } } if n == 1 { j } else { 0 } } #[inline] fn primetest_base2(mont: &Mont) -> bool { // Mirrer-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... let (r, rn, d, k) = (mont.r, mont.rn, mont.d, mont.k); let mut br = mont.pow(mont.add(mont.r, mont.r), d); if br == r || br == rn { return true; } for _ in 1..k { br = mont.mrmul(br, br); if br == rn { return true; } } false } #[inline] fn primetest_lucas(mont: &Mont) -> bool { // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... let n = mont.n; let mut d: i64 = 5; for i in 0.. { debug_assert!(i < 64); match jacobi(d, n) { -1 => break, 0 => if ((d.abs()) as u64) < n { return false; }, _ => {}, } if i == 32 && isqrt64f(n).1 == 0 { return false; } if (i & 1) == 1 { d = 2 - d; } else { d = -(d + 2); } } let qm = mont.ar(if d < 0 {((1 - d) as u64) / 4 % n} else {n - ((d - 1) as u64) / 4 % n}); let mut k = (n + 1) << (n + 1).leading_zeros(); let mut um = mont.r; let mut vm = mont.r; let mut qn = qm; let dm: u64 = mont.ar(if d < 0 { n - (((-d) as u64) % n) } else { (d as u64) % n }); k <<= 1; while k > 0 { um = mont.mrmul(um, vm); vm = mont.sub(mont.mrmul(vm, vm), mont.add(qn, qn)); qn = mont.mrmul(qn, qn); if (k >> 63) != 0 { let uu = mont.div2(mont.add(um, vm)); vm = mont.div2(mont.add(mont.mrmul(dm, um), vm)); um = uu; qn = mont.mrmul(qn, qm); } k <<= 1; } if um == 0 || vm == 0 { return true; } let mut x = (n + 1) & (!n); x >>= 1; while x > 0 { um = mont.mrmul(um, vm); vm = mont.sub(mont.mrmul(vm, vm), mont.add(qn, qn)); if vm == 0 { return true; } qn = mont.mrmul(qn, qn); x >>= 1; } false } // Baillie–PSW primarity test pub fn primetest_bpsw(n: u64) -> bool { if n == 2 { return true; } if n == 1 || (n & 1) == 0 { return false; } let mont = Mont::::new(n); // Mirrer-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... primetest_base2(&mont) && // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... primetest_lucas(&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 = lines.next().unwrap().unwrap().parse::().unwrap(); let mut s = String::with_capacity(n * 22); for _ in 0..n { let xs = lines.next().unwrap().unwrap(); let r = if primetest_bpsw(xs.parse::().unwrap()) { " 1\n" } else { " 0\n" }; s += &xs; s += r; } out.write_all(s.as_bytes()).unwrap(); eprint!("{}us\n", start_time.elapsed().as_micros()); }