use std::io::Read; fn get_word() -> String { let stdin = std::io::stdin(); let mut stdin=stdin.lock(); let mut u8b: [u8; 1] = [0]; loop { let mut buf: Vec = Vec::with_capacity(16); loop { let res = stdin.read(&mut u8b); if res.unwrap_or(0) == 0 || u8b[0] <= b' ' { break; } else { buf.push(u8b[0]); } } if buf.len() >= 1 { let ret = String::from_utf8(buf).unwrap(); return ret; } } } fn get() -> T { get_word().parse().ok().unwrap() } fn factorize(mut x: i64) -> Vec<(i64, usize)> { let mut p = 2; let mut ans = vec![]; while p * p <= x { let mut e = 0; while x % p == 0 { x /= p; e += 1; } if e > 0 { ans.push((p, e)); } p += 1; } if x > 1 { ans.push((x, 1)); } ans } // Verified by: https://yukicoder.me/submissions/706484 fn ext_gcd(a: i64, b: i64) -> (i64, i64, i64) { if b == 0 { return (a, 1, 0); } let r = a % b; let q = a / b; let (g, x, y) = ext_gcd(b, r); (g, y, x - q * y) } fn inv_mod(a: i64, b: i64) -> i64 { let (_, mut x, _) = ext_gcd(a, b); x %= b; if x < 0 { x += b; } x } // gcd(rm[i].1, rm[j].1) == 1 for i != j // Ref: https://www.creativ.xyz/ect-gcd-crt-garner-927/ // O(n^2) fn garner(rm: &[(i64, i64)], mo: i64) -> i64 { if rm.is_empty() { return 0; } let n = rm.len(); let mut x_mo = (rm[0].0 % rm[0].1) % mo; let mut mp_mo = 1; let mut coef = Vec::with_capacity(n); coef.push(rm[0].0 % rm[0].1); for i in 1..n { let (r, m) = rm[i]; let r = r % m; let mut mp_mi = 1; let mut x_mi = 0; mp_mo = mp_mo * (rm[i - 1].1 % mo) % mo; for j in 0..i { x_mi = (x_mi + mp_mi * (coef[j] % m)) % m; mp_mi = mp_mi * (rm[j].1 % m) % m; } let t = (r - x_mi + m) % m * inv_mod(mp_mi, m) % m; x_mo = (x_mo + t % mo * mp_mo) % mo; coef.push(t); } x_mo } // https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf pub struct PrimePowComb { p: i64, e: usize, pe: i64, fac: Vec, invfac: Vec, } impl PrimePowComb { fn powmod(x: i64, mut e: i64, m: i64) -> i64 { let mut sum = 1; let mut cur = x % m; while e > 0 { if e % 2 != 0 { sum = sum * cur % m; } cur = cur * cur % m; e /= 2; } sum } // O(p^e) // p must be a prime pub fn new(p: i64, e: usize) -> Self { assert!(p <= 1 << 31); let mut pe = 1i64; for _ in 0..e { pe = pe.saturating_mul(p); } assert!(pe <= 1 << 31); let pp = p as usize; let peu = pe as usize; let mut fac = vec![0; peu]; let mut invfac = vec![0; peu]; fac[0] = 1; for i in 1..peu { if i % pp == 0 { fac[i] = fac[i - 1]; } else { fac[i] = fac[i - 1] * i as i64 % pe; } } invfac[peu - 1] = Self::powmod(fac[peu - 1], pe / p * (p - 1) - 1, pe); for i in (0..peu - 1).rev() { if i % pp == pp - 1 { invfac[i] = invfac[i + 1]; } else { invfac[i] = invfac[i + 1] * (i + 1) as i64 % pe; } } PrimePowComb { p: p, e: e, pe: pe, fac: fac, invfac: invfac, } } // (a!)_p mod p^e, \prod_{1 <= i <= a, not (p | i)} i // O(1) pub fn fac_pe(&self, a: i64) -> i64 { let pe = self.pe; assert!(a < pe); self.fac[a as usize] } // 1/(a!)_p mod p^e // O(1) pub fn invfac_pe(&self, a: i64) -> i64 { let pe = self.pe; assert!(a < pe); self.invfac[a as usize] } // Find ord_p(C(a, b)). // O(log_p a)-time pub fn comb_ord(&self, mut a: i64, mut b: i64, from: usize) -> i64 { if a < b { return -1; } let p = self.p; let mut c = a - b; let mut ans = 0; let mut carry = 0; let mut pos = 0; while a > 0 { if b % p + c % p + carry >= p { if pos >= from { ans += 1; } carry = 1; } else { carry = 0; } a /= p; b /= p; c /= p; pos += 1; } ans } // Find C(a, b) mod p^e. // O(e * log_p a)-time pub fn comb(&self, mut a: i64, mut b: i64) -> i64 { if a < b { return 0; } let ord = self.comb_ord(a, b, 0); let p = self.p; let pe = self.pe; let sgn = self.comb_ord(a, b, self.e - 1); let mut c = a - b; let mut res = 1; while a > 0 { let aw = a % pe; let bw = b % pe; let cw = c % pe; res = res * self.fac_pe(aw) % pe * self.invfac_pe(bw) % pe * self.invfac_pe(cw) % pe; a /= p; b /= p; c /= p; } for _ in 0..ord { res = res * p % pe; } if p >= 3 || pe <= 4 { if sgn % 2 != 0 { res = (pe - res) % pe; } } res } } // https://yukicoder.me/problems/no/2181 (3.5) // 問題の値は \sum_{L <= n <= R} (C(2n, n) - 2) である。 // C(2n, n) mod M は https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf で計算できる。 fn main() { let l: i64 = get(); let r: i64 = get(); let m: i64 = get(); let pe = factorize(m); let mut res = vec![]; for &(p, e) in &pe { let comb = PrimePowComb::new(p, e); let pe = comb.pe; let mut ans = 0; for i in l..=r { ans = (ans + comb.comb(2 * i, i) + pe - 2) % pe; } res.push((ans, pe)); } println!("{}", garner(&res, m)); }