// -*- coding:utf-8-unix -*-

// モンゴメリ剰余乗算 (Montgomery modular multiplication)
trait MontTrait<T> {
    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;
    fn powodd(&self, ar: T, b: T) -> T;
}
struct Mont<T, BitCountType=u32> {
    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<u64> for Mont<u64> {
    #[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**3)*((k*(k+1)/2)**1))/(2*k+1)
        let mut ni = n;
        // ni1 := ni0 * (2 - (n * ni0)); // = (1-(2**6)*((k*(k+1)/2)**2))/(2*k+1)
        // ni2 := ni1 * (2 - (n * ni1)); // = (1-(2**12)*((k*(k+1)/2)**4))/(2*k+1)
        // ni3 := ni2 * (2 - (n * ni2)); // = (1-(2**24)*((k*(k+1)/2)**8))/(2*k+1)
        // ni4 := ni3 * (2 - (n * ni3)); // = (1-(2**48)*((k*(k+1)/2)**16))/(2*k+1)
        // ni5 := ni4 * (2 - (n * ni4)); // = (1-(2**96)*((k*(k+1)/2)**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); }
        }
    }
    #[inline]
    fn powodd(&self, mut ar: u64, mut b: u64) -> u64 {
        // == ((ar / r) ** b) * r (mod n)
        debug_assert!(ar < self.n);
        debug_assert_eq!(b & 1, 1);
        let mut t = ar;
        loop {
            b >>= 1;
            if b == 0 { return t; }
            ar = self.mrmul(ar, ar);
            if (b & 1) != 0 { t = self.mrmul(t, ar); }
        }
    }
}

#[inline]
// ニュートン法による整数平方根
fn isqrt64n(x: u64) -> u64 {
    if x <= 1 { return x; }
    let k = 32 - ((x - 1).leading_zeros() >> 1);
    let mut s = 1u64 << k; // s = 2**k
    let mut t = (s + (x >> k)) >> 1; // t = (s + x/s)/2
    // whileループ回数=除算回数は最大6回を想定
    // s > floor(sqrt(x)) => floor(sqrt(x)) <= t < s
    // s == floor(sqrt(x)) => floor(sqrt(x)) <= t <= floor(sqrt(x)) + 1
    while t < s {
        s = t;
        t = (s + (x / s)) >> 1;
    }
    s
}
#[inline]
// 平方数判定 (true: 平方数である, false: 平方数ではない)
fn issq_u64_n(x: u64) -> bool {
    let sqrt = isqrt64n(x);
    sqrt * sqrt == x
}
#[inline]
// 擬平方数判定 mod64 12/64 0.1875 (true: 平方数かもしれない, false: 平方数ではない)
fn issq_u64_mod64(x: u64) -> bool {
    (0x202021202030213u64 >> (x & 63)) & 1 == 1
}
#[inline]
// 擬平方数判定 mod4095 336/4095 0.0821 (true: 平方数かもしれない, false: 平方数ではない)
fn issq_u64_mod4095(x: u64) -> bool {
    const SQTABLE_MOD4095: [u64; 64] = [0x2001002010213,0x4200001008028001,0x20000010004,0x80200082010,0x1800008200044029,0x120080000010,0x2200000080410400,0x8100041000200800,0x800004000020100,0x402000400082201,0x9004000040,0x800002000880,0x18002000012000,0x801208,0x26100000804010,0x80000080000002,0x108040040101045,0x20c00004000102,0x400000100c0010,0x1300000040208,0x804000020010000,0x1008402002400080,0x201001000200040,0x4402000000806000,0x10402000000,0x1040008001200801,0x4080000000020400,0x10083080000002,0x8220140000040000,0x800084020100000,0x80010400010000,0x1200020108008060,0x180000000,0x400002400000018,0x4241000200,0x100800000000,0x10201008400483,0xc008000208201000,0x800420000100,0x2010002000410,0x28041000000,0x4010080000024,0x400480010010080,0x200040028000008,0x100810084020,0x20c0401000080000,0x1000240000220000,0x4000020800,0x410000000480000,0x8004008000804201,0x806020000104000,0x2080002000211000,0x1001008001000,0x20000010024000,0x480200002040000,0x48200044008000,0x100000000010080,0x80090400042,0x41040200800200,0x4000020100110,0x2000400082200010,0x1008200000000040,0x2004800002,0x2002010000080];
    let p = (x % 4095) as usize;
    (SQTABLE_MOD4095[p >> 6] >> (p & 63)) & 1 == 1
}
// 平方数であるかどうか
#[inline]
fn issq_u64(x: u64) -> bool {
    issq_u64_mod64(x) &&
    issq_u64_mod4095(x) &&
    issq_u64_n(x)
}

// 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; }
        std::mem::swap(&mut n, &mut a);
        a %= n;
        if a > (n >> 1) {
            a = n - a;
            if (n & 3) == 3 { j = -j; }
        }
    }
    if n == 1 { j } else { 0 }
}

#[inline]
fn primetest_u64_base2_sub(mont: &Mont<u64>) -> 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.powodd(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_u64_miller_sub(mont: &Mont<u64>, mut base: u64) -> bool {
    // Miller-Rabin primality test
    let (n, r, rn, d, k) = (mont.n, mont.r, mont.rn, mont.d, mont.k);
    if base > n { base %= n; if base == 0 { return true; } }
    let mut tr = mont.pow(mont.ar(base), d);
    if tr == r || tr == rn { return true; }
    for _ in 1..k { tr = mont.mrmul(tr, tr); if tr == rn { return true; } }
    false
}

#[inline]
fn primetest_u64_lucas_sub(mont: &Mont<u64>) -> 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 0u32.. {
        debug_assert!(i < 64);
        match jacobi(d, n) {
            -1 => break,
            0 => if ((d.abs()) as u64) < n { return false; },
            _ => {},
        }
        if i == 8 && issq_u64(n) { return false; }
        if (i & 1) == 1 { d = 2 - d; } else { d = -(d + 2); }
    }
    let mut q = if d < 0 {((1 - d) as u64) / 4} else {n - ((d - 1) as u64) / 4};
    if q >= n { q %= n; }
    let qm = mont.ar(q);
    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
}

#[inline]
fn primetest_u64_bpsw_sub(mont: &Mont<u64>) -> 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,...
    primetest_u64_base2_sub(mont) &&
    // Lucas primality test
    // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,...
    primetest_u64_lucas_sub(mont)
}
// 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::<u64>::new(n);
    primetest_u64_bpsw_sub(&mont)
}

// Deterministic variants of the Miller-Rabin primality test http://miller-rabin.appspot.com/
const MILLER_U64_1BASES: [u64; 1] = [
    9345883071009581737,
];
const MILLER_U64_2BASES: [u64; 2] = [
    336781006125,
    9639812373923155,
];
const MILLER_U64_3BASES: [u64; 3] = [
    4230279247111683200,
    14694767155120705706,
    16641139526367750375,
];
const MILLER_U64_4BASES: [u64; 4] = [
    2,
    141889084524735,
    1199124725622454117,
    11096072698276303650,
];
const MILLER_U64_5BASES: [u64; 5] = [
    2,
    4130806001517,
    149795463772692060,
    186635894390467037,
    3967304179347715805,
];
const MILLER_U64_6BASES: [u64; 6] = [
    2,
    123635709730000,
    9233062284813009,
    43835965440333360,
    761179012939631437,
    1263739024124850375,
];
const MILLER_U64_7BASES: [u64; 7] = [
    2,
    325,
    9375,
    28178,
    450775,
    9780504,
    1795265022,
];

#[inline]
fn primetest_u64_miller_1bases_sub(mont: &Mont<u64>) -> bool {
    debug_assert!(mont.n < 341531);
    MILLER_U64_1BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_2bases_sub(mont: &Mont<u64>) -> bool {
    debug_assert!(mont.n < 1050535501);
    MILLER_U64_2BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_3bases_sub(mont: &Mont<u64>) -> bool {
    debug_assert!(mont.n < 350269456337);
    MILLER_U64_3BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_4bases_sub(mont: &Mont<u64>) -> bool {
    debug_assert!(mont.n < 55245642489451);
    MILLER_U64_4BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_5bases_sub(mont: &Mont<u64>) -> bool {
    debug_assert!(mont.n < 7999252175582851);
    MILLER_U64_5BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_6bases_sub(mont: &Mont<u64>) -> bool {
    debug_assert!(mont.n < 585226005592931977);
    MILLER_U64_6BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_7bases_sub(mont: &Mont<u64>) -> bool {
    MILLER_U64_7BASES.iter().all(|&base| primetest_u64_miller_sub(&mont, base))
}
#[inline]
fn primetest_u64_miller_mbases_sub(mont: &Mont<u64>) -> bool {
    // Deterministic variants of the Miller-Rabin primality test http://miller-rabin.appspot.com/
    match mont.n {
        0..=341530             => primetest_u64_miller_1bases_sub(mont),
        0..=1050535500         => primetest_u64_miller_2bases_sub(mont),
        0..=350269456336       => primetest_u64_miller_3bases_sub(mont),
        0..=55245642489450     => primetest_u64_miller_4bases_sub(mont),
        0..=7999252175582850   => primetest_u64_miller_5bases_sub(mont),
        0..=585226005592931976 => primetest_u64_miller_6bases_sub(mont),
        _                      => primetest_u64_miller_7bases_sub(mont),
    }
}
#[inline]
fn primetest_u64_miller_mbases_sub_b(mont: &Mont<u64>) -> bool {
    // Deterministic variants of the Miller-Rabin primality test http://miller-rabin.appspot.com/
    match mont.n {
        0..=341530             => MILLER_U64_1BASES.iter(),
        0..=1050535500         => MILLER_U64_2BASES.iter(),
        0..=350269456336       => MILLER_U64_3BASES.iter(),
        0..=55245642489450     => MILLER_U64_4BASES.iter(),
        0..=7999252175582850   => MILLER_U64_5BASES.iter(),
        0..=585226005592931976 => MILLER_U64_6BASES.iter(),
        _                      => MILLER_U64_7BASES.iter(),
    }.all(|&base| primetest_u64_miller_sub(&mont, base))
}
pub fn primetest_u64_miller_mbases(n: u64) -> bool {
    if n == 2 { return true; }
    if n == 1 || (n & 1) == 0 { return false; }
    let mont = Mont::<u64>::new(n);
    primetest_u64_miller_mbases_sub(&mont)
}
pub fn primetest_u64_miller_mbases_b(n: u64) -> bool {
    if n == 2 { return true; }
    if n == 1 || (n & 1) == 0 { return false; }
    let mont = Mont::<u64>::new(n);
    primetest_u64_miller_mbases_sub_b(&mont)
}

#[cfg(test)]
mod tests {
    use crate::*;

    #[test]
    fn test_nbits() {
        // ten least k's for which (2**n)-k is prime
        // https://primes.utm.edu/lists/2small/0bit.html
        let primes: Vec<(u32,Vec<u64>)> = vec![
            (8,vec![5,15,17,23,27,29,33,45,57,59]),
            (9,vec![3,9,13,21,25,33,45,49,51,55]),
            (10,vec![3,5,11,15,27,33,41,47,53,57]),
            (11,vec![9,19,21,31,37,45,49,51,55,61]),
            (12,vec![3,5,17,23,39,45,47,69,75,77]),
            (13,vec![1,13,21,25,31,45,69,75,81,91]),
            (14,vec![3,15,21,23,35,45,51,65,83,111]),
            (15,vec![19,49,51,55,61,75,81,115,121,135]),
            (16,vec![15,17,39,57,87,89,99,113,117,123]),
            (17,vec![1,9,13,31,49,61,63,85,91,99]),
            (18,vec![5,11,17,23,33,35,41,65,75,93]),
            (19,vec![1,19,27,31,45,57,67,69,85,87]),
            (20,vec![3,5,17,27,59,69,129,143,153,185]),
            (21,vec![9,19,21,55,61,69,105,111,121,129]),
            (22,vec![3,17,27,33,57,87,105,113,117,123]),
            (23,vec![15,21,27,37,61,69,135,147,157,159]),
            (24,vec![3,17,33,63,75,77,89,95,117,167]),
            (25,vec![39,49,61,85,91,115,141,159,165,183]),
            (26,vec![5,27,45,87,101,107,111,117,125,135]),
            (27,vec![39,79,111,115,135,187,199,219,231,235]),
            (28,vec![57,89,95,119,125,143,165,183,213,273]),
            (29,vec![3,33,43,63,73,75,93,99,121,133]),
            (30,vec![35,41,83,101,105,107,135,153,161,173]),
            (31,vec![1,19,61,69,85,99,105,151,159,171]),
            (32,vec![5,17,65,99,107,135,153,185,209,267]),
            (33,vec![9,25,49,79,105,285,301,303,321,355]),
            (34,vec![41,77,113,131,143,165,185,207,227,281]),
            (35,vec![31,49,61,69,79,121,141,247,309,325]),
            (36,vec![5,17,23,65,117,137,159,173,189,233]),
            (37,vec![25,31,45,69,123,141,199,201,351,375]),
            (38,vec![45,87,107,131,153,185,191,227,231,257]),
            (39,vec![7,19,67,91,135,165,219,231,241,301]),
            (40,vec![87,167,195,203,213,285,293,299,389,437]),
            (41,vec![21,31,55,63,73,75,91,111,133,139]),
            (42,vec![11,17,33,53,65,143,161,165,215,227]),
            (43,vec![57,67,117,175,255,267,291,309,319,369]),
            (44,vec![17,117,119,129,143,149,287,327,359,377]),
            (45,vec![55,69,81,93,121,133,139,159,193,229]),
            (46,vec![21,57,63,77,167,197,237,287,305,311]),
            (47,vec![115,127,147,279,297,339,435,541,619,649]),
            (48,vec![59,65,89,93,147,165,189,233,243,257]),
            (49,vec![81,111,123,139,181,201,213,265,283,339]),
            (50,vec![27,35,51,71,113,117,131,161,195,233]),
            (51,vec![129,139,165,231,237,247,355,391,397,439]),
            (52,vec![47,143,173,183,197,209,269,285,335,395]),
            (53,vec![111,145,231,265,315,339,343,369,379,421]),
            (54,vec![33,53,131,165,195,245,255,257,315,327]),
            (55,vec![55,67,99,127,147,169,171,199,207,267]),
            (56,vec![5,27,47,57,89,93,147,177,189,195]),
            (57,vec![13,25,49,61,69,111,195,273,363,423]),
            (58,vec![27,57,63,137,141,147,161,203,213,251]),
            (59,vec![55,99,225,427,517,607,649,687,861,871]),
            (60,vec![93,107,173,179,257,279,369,395,399,453]),
            (61,vec![1,31,45,229,259,283,339,391,403,465]),
            (62,vec![57,87,117,143,153,167,171,195,203,273]),
            (63,vec![25,165,259,301,375,387,391,409,457,471]),
            (64,vec![59,83,95,179,189,257,279,323,353,363]),
        ];
        let start_time = std::time::Instant::now();
        for (bit_ref, kvec) in primes.iter() {
            let bit = *bit_ref;
            let &lastk = kvec.iter().last().unwrap();
            for k in 1..=lastk {
                if (k & 1) == 0 { continue; }
                let n = (if bit < 64 { 1u64 << bit } else { 0u64 }).wrapping_sub(k);
                let f = kvec.binary_search(&k).is_ok();
                let mont = Mont::<u64>::new(n);
                // Deterministic variants of the Miller-Rabin primality test
                // http://miller-rabin.appspot.com/
                // n < 341531 : 1 base Miller-Rabin primality test
                if n < 341531 { assert_eq!(primetest_u64_miller_1bases_sub(&mont), f); }
                // n < 1050535501 : 2 bases Miller-Rabin primality test
                if n < 1050535501 { assert_eq!(primetest_u64_miller_2bases_sub(&mont), f); }
                // n < 350269456337 : 3 bases Miller-Rabin primality test
                if n < 350269456337 { assert_eq!(primetest_u64_miller_3bases_sub(&mont), f); }
                // n < 55245642489451 : 4 bases Miller-Rabin primality test
                if n < 55245642489451 { assert_eq!(primetest_u64_miller_4bases_sub(&mont), f); }
                // n < 7999252175582851 : 5 bases Miller-Rabin primality test
                if n < 7999252175582851 { assert_eq!(primetest_u64_miller_5bases_sub(&mont), f); }
                // n < 585226005592931977 : 6 bases Miller-Rabin primality test
                if n < 585226005592931977 { assert_eq!(primetest_u64_miller_6bases_sub(&mont), f); }
                // n for all 64bit integer : 7 bases Miller-Rabin primality test
                assert_eq!(primetest_u64_miller_7bases_sub(&mont), f);
                // n for all 64bit integer : multi bases Miller-Rabin primality test
                assert_eq!(primetest_u64_miller_mbases_sub(&mont), f);
                // Baillie–PSW primality test
                assert_eq!(primetest_u64_bpsw_sub(&mont), f);
                // Lucas test
                assert_eq!(primetest_u64_lucas_sub(&mont), primetest_u64_lucas_sub(&mont));
            }
        }
        eprint!("test_nbits {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_base2_1e7() {
        // Miller-Rabin primality test (base 2)
        // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,...
        // composite 2-SPRP list up to 2**64 ( http://miller-rabin.appspot.com/#links )
        // Pseudoprime Statistics, Tables, and Data ( http://ntheory.org/pseudoprimes.html )
        // Miller-Rabin base 2 data (up to 1e15) ( http://ntheory.org/data/spsps.txt )
        let assumed: Vec<u64> = vec![ // #SPSP-2 Miller-Rabin base 2 (up to 1e7)
            2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,65281,74665,80581,85489,88357,90751,104653,
            130561,196093,220729,233017,252601,253241,256999,271951,280601,314821,357761,390937,458989,476971,
            486737,489997,514447,580337,635401,647089,741751,800605,818201,838861,873181,877099,916327,976873,
            983401,1004653,1016801,1023121,1082401,1145257,1194649,1207361,1251949,1252697,1302451,1325843,1357441,
            1373653,1397419,1441091,1493857,1507963,1509709,1530787,1678541,1730977,1811573,1876393,1907851,1909001,
            1969417,1987021,2004403,2081713,2181961,2205967,2264369,2269093,2284453,2304167,2387797,2419385,2510569,
            2746477,2748023,2757241,2811271,2909197,2953711,2976487,3090091,3116107,3125281,3375041,3400013,3429037,
            3539101,3567481,3581761,3605429,3898129,4181921,4188889,4335241,4360621,4469471,4502485,4513841,4682833,
            4835209,4863127,5016191,5044033,5049001,5173169,5173601,5256091,5310721,5444489,5489641,5590621,5599765,
            5672041,5681809,5919187,6140161,6226193,6233977,6334351,6368689,6386993,6787327,6836233,6952037,7177105,
            7306261,7306561,7462001,7674967,7759937,7820201,7883731,8036033,8095447,8384513,8388607,8534233,8725753,
            8727391,9006401,9056501,9069229,9073513,9371251,9564169,9567673,9588151,9729301,9774181,9863461,9995671
        ];
        let start_time = std::time::Instant::now();
        let result: Vec<u64> = (3..10_000_000).filter(|&n| {
            if (n & 1) == 0 { return false; }
            let mont = Mont::<u64>::new(n);
            let res_3bases = primetest_u64_miller_3bases_sub(&mont);
            let res_base2 = primetest_u64_base2_sub(&mont);
            assert!(!res_3bases || res_base2);
            res_3bases != res_base2
        }).collect();
        assert_eq!(assumed, result);
        eprint!("test_base2_1e7 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_lucas_1e7() {
        // Lucas primality test
        // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,...
        // Pseudoprime Statistics, Tables, and Data ( http://ntheory.org/pseudoprimes.html )
        // Strong Lucas-Selfridge data (up to 1e15) ( http://ntheory.org/data/slpsps-baillie.txt )
        let assumed: Vec<u64> = vec![ // #SLPSP Strong Lucas-Selfridge (up to 1e7)
            5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,75077,97439,100127,113573,115639,130139,155819,
            158399,161027,162133,176399,176471,189419,192509,197801,224369,230691,231703,243629,253259,268349,288919,
            313499,324899,353219,366799,391169,430127,436409,455519,487199,510479,572669,611399,622169,635627,636199,
            701999,794611,835999,839159,851927,871859,875879,887879,895439,950821,960859,1033997,1106327,1241099,
            1256293,1308119,1311389,1388903,1422319,1501439,1697183,1711469,1777159,1981559,2003579,2263127,2435423,
            2461211,2518889,2566409,2624399,2662277,2666711,2690759,2738969,2782079,2828699,2942081,2952071,3109049,
            3165119,3175883,3179609,3204599,3373649,3399527,3410531,3441239,3452147,3479111,3498879,3579599,3684251,
            3694079,3700559,3706169,3735521,3774377,3776219,3785699,3802499,3813011,3865319,3892529,3900797,3903791,
            4067279,4109363,4226777,4309631,4322399,4368869,4403027,4563719,4828277,4870847,5133281,5208377,5299139,
            5396999,5450201,5479109,5514479,5720219,5762629,5807759,5879411,5942627,6001379,6003923,6296291,6562891,
            6641189,6668099,6784721,6784861,6863291,6893531,6965639,7017949,7163441,7199399,7241639,7353917,7453619,
            7621499,8112899,8159759,8221121,8234159,8361989,8372849,8518127,8530559,8555009,8574551,8581219,8711699,
            8817899,8990279,9049319,9335969,9401893,9485951,9587411,9713027,9793313,9800981,9827711,9922337,9965069
        ];
        let start_time = std::time::Instant::now();
        let result: Vec<u64> = (3..10_000_000).filter(|&n| {
            if (n & 1) == 0 { return false; }
            let mont = Mont::<u64>::new(n);
            let res_3bases = primetest_u64_miller_3bases_sub(&mont);
            let res_lucas = primetest_u64_lucas_sub(&mont);
            assert!(!res_3bases || res_lucas);
            res_3bases != res_lucas
        }).collect();
        assert_eq!(assumed, result);
        eprint!("test_lucas_1e7 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_bpsw_1e7() { // 24bit
        let start_time = std::time::Instant::now();
        for n in 3..10_000_000 {
            if (n & 1) == 0 { continue; }
            let mont = Mont::<u64>::new(n);
            let res_3bases = primetest_u64_miller_3bases_sub(&mont);
            let res_bpsw = primetest_u64_bpsw_sub(&mont);
            assert_eq!(res_3bases, res_bpsw);
        }
        eprint!("test_bpsw_1e7 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_bpsw_4e9() { // 32bit
        let start_time = std::time::Instant::now();
        for n in 4_000_000_000..4_010_000_000 {
            if (n & 1) == 0 { continue; }
            let mont = Mont::<u64>::new(n);
            let res_3bases = primetest_u64_miller_3bases_sub(&mont);
            let res_bpsw = primetest_u64_bpsw_sub(&mont);
            assert_eq!(res_3bases, res_bpsw);
        }
        eprint!("test_bpsw_4e9 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_bpsw_1e16() { // 54bit
        let start_time = std::time::Instant::now();
        for n in 10_000_000_000_000_000..10_000_000_010_000_000 {
            if (n & 1) == 0 { continue; }
            let mont = Mont::<u64>::new(n);
            let res_7bases = primetest_u64_miller_7bases_sub(&mont);
            let res_bpsw = primetest_u64_bpsw_sub(&mont);
            assert_eq!(res_7bases, res_bpsw);
        }
        eprint!("test_bpsw_1e16 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_bpsw_9e18() { // 63bit
        let start_time = std::time::Instant::now();
        for n in 9_000_000_000_000_000_000..9_000_000_000_010_000_000 {
            if (n & 1) == 0 { continue; }
            let mont = Mont::<u64>::new(n);
            let res_7bases = primetest_u64_miller_7bases_sub(&mont);
            let res_bpsw = primetest_u64_bpsw_sub(&mont);
            assert_eq!(res_7bases, res_bpsw);
        }
        eprint!("test_bpsw_9e18 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_bpsw_10e18() { // 64bit
        let start_time = std::time::Instant::now();
        for n in 10_000_000_000_000_000_000..10_000_000_000_010_000_000 {
            if (n & 1) == 0 { continue; }
            let mont = Mont::<u64>::new(n);
            let res_7bases = primetest_u64_miller_7bases_sub(&mont);
            let res_bpsw = primetest_u64_bpsw_sub(&mont);
            assert_eq!(res_7bases, res_bpsw);
        }
        eprint!("test_bpsw_10e18 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_bpsw_18e18() { // 64bit
        let start_time = std::time::Instant::now();
        for n in 18_000_000_000_000_000_000..18_000_000_000_010_000_000 {
            if (n & 1) == 0 { continue; }
            let mont = Mont::<u64>::new(n);
            let res_7bases = primetest_u64_miller_7bases_sub(&mont);
            let res_bpsw = primetest_u64_bpsw_sub(&mont);
            assert_eq!(res_7bases, res_bpsw);
        }
        eprint!("test_bpsw_18e18 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_issq_u64() {
        let start_time = std::time::Instant::now();
        for i in 0xffff_0000u64..=0xffff_ffff {
            let n = i * i;
            assert!(issq_u64(n));
        }
        eprint!("test_issq_u64 {}us\n", start_time.elapsed().as_micros());
    }

    #[test]
    fn test_issq_u64_p1() {
        let start_time = std::time::Instant::now();
        for i in 0xffff_0000u64..=0xffff_ffff {
            let n = i * i + 1;
            assert!(!issq_u64(n));
        }
        eprint!("test_issq_u64_p1 {}us\n", start_time.elapsed().as_micros());
    }

}

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 buf = String::new();
    let mut reader = std::io::BufReader::new(input.lock());
    reader.read_line(&mut buf).unwrap();
    let n = buf.trim().parse::<usize>().unwrap();
    for _ in 0..n {
        buf.clear();
        reader.read_line(&mut buf).unwrap();
        let xs = buf.trim();
        let x = xs.parse::<u64>().unwrap();
        puts!("{} {}\n", xs, if primetest_bpsw(x) { "1" } else { "0" });
    }
    eprintln!("{}us", start_time.elapsed().as_micros());
}