Problem:
Score : 500 points

Problem Statement

For integers b (b \geq 2) and n (n \geq 1), let the function f(b,n) be defined as follows:

f(b,n) = n, when n < b

f(b,n) = f(b,\,{\rm floor}(n / b)) + (n \ {\rm mod} \ b), when n \geq b

Here, {\rm floor}(n / b) denotes the largest integer not exceeding n / b,
and n \ {\rm mod} \ b denotes the remainder of n divided by b.

Less formally, f(b,n) is equal to the sum of the digits of n written in base b.
For example, the following hold:

f(10,\,87654)=8+7+6+5+4=30

f(100,\,87654)=8+76+54=138

You are given integers n and s.
Determine if there exists an integer b (b \geq 2) such that f(b,n)=s.
If the answer is positive, also find the smallest such b.

Constraints

1 \leq n \leq 10^{11}

1 \leq s \leq 10^{11}

n,\,s are integers.

Input

The input is given from Standard Input in the following format:

n
s

Output

If there exists an integer b (b \geq 2) such that f(b,n)=s, print the smallest such b.
If such b does not exist, print -1 instead.

Sample Input 1

87654
30

Sample Output 1

10

Sample Input 2

87654
138

Sample Output 2

100

Sample Input 3

87654
45678

Sample Output 3

-1

Sample Input 4

31415926535
1

Sample Output 4

31415926535

Sample Input 5

1
31415926535

Sample Output 5

-1

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports, unused_macros, unknown_lints)]

use std::collections::*;
use std::f64::*;
use std::cmp::*;
use std::cmp::Ordering::*;

macro_rules! dump {($($e:expr), +) => {println!(concat!($(stringify!($e), " = {:?}\n"), +), $($e), +)}}
macro_rules! max {($e0:expr, $($e:expr), +) => {{let mut r = $e0; $(r = max(r, $e);)+ r}}}
macro_rules! min {($e0:expr, $($e:expr), +) => {{let mut r = $e0; $(r = min(r, $e);)+ r}}}
macro_rules! freq {($v:expr) => {{let mut h = HashMap::new(); for x in $v {*h.entry(x).or_insert(0) += 1} h}}}

fn main() {
  let n = get::val::<usize>();
  let s = get::val::<usize>();

  if s == n {println!("{}", s+1); return}

  let fsqrt = |x| (x as f64).sqrt().floor() as usize;
  let csqrt = |x| (x as f64).sqrt().ceil() as usize;

  for b in 2..fsqrt(n)+1 {
    if f(b,n) == s {println!("{}",b); return}
  }

  let mut ans = std::usize::MAX; 
  for p in 1..csqrt(n) {
    let b = (n - s) / p + 1;
    if fsqrt(n) < b && b <= n && p < b && f(b,n) == s {ans = min(ans, b)}
  }

  let ans = if ans == std::usize::MAX {format!("-1")} else {format!("{}",ans)};
  println!("{}", ans)
}

fn f(b: usize, n: usize) -> usize {
  if n < b {n} else {f(b, n/b) + n%b} 
}

#[allow(dead_code)]
mod get {
  use std::io::*;
  use std::str::*;
  use std::process::*;

  pub fn val<T: FromStr>() -> T {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    if buf.is_empty() {println!("No input"); exit(0)}
    buf.trim_right().parse::<T>().ok().unwrap()
  }

  pub fn vals<T: FromStr>(n: usize) -> Vec<T> {
    let mut vec: Vec<T> = vec![];
    for _ in 0 .. n {
      vec.push(val());
    }
    vec
  }

  pub fn tuple<T1: FromStr, T2: FromStr>() -> (T1, T2) {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    if buf.is_empty() {println!("No input"); exit(0)}
    let mut it = buf.trim_right().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    (x, y)
  }

  pub fn tuples<T1: FromStr, T2: FromStr>(n: usize) -> Vec<(T1, T2)> {
    let mut vec: Vec<(T1, T2)> = vec![];
    for _ in 0 .. n {
      vec.push(tuple());
    }
    vec
  }

  pub fn tuple3<T1: FromStr, T2: FromStr, T3: FromStr>() -> (T1, T2, T3) {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    if buf.is_empty() {println!("No input"); exit(0)}
    let mut it = buf.trim_right().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    let z = it.next().unwrap().parse::<T3>().ok().unwrap();
    (x, y, z)
  }

  pub fn tuple3s<T1: FromStr, T2: FromStr, T3: FromStr>(n: usize) -> Vec<(T1, T2, T3)> {
    let mut vec: Vec<(T1, T2, T3)> = vec![];
    for _ in 0 .. n {
      vec.push(tuple3());
    }
    vec
  }

  pub fn vec<T: FromStr>() -> Vec<T> {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    if buf.is_empty() {println!("No input"); exit(0)}
    buf.trim_right().split_whitespace().map(|t| t.parse::<T>().ok().unwrap()).collect()
  }

  pub fn mat<T: FromStr>(h: usize) -> Vec<Vec<T>> {
    let mut mat = vec![];
    for _ in 0 .. h {
      mat.push(vec());
    }
    mat
  }

  pub fn chars() -> Vec<char> {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    if buf.is_empty() {println!("No input"); exit(0)}
    buf.trim_right().chars().collect()
  }
}

#[allow(dead_code)]
mod put {
  use std::string::*;

  pub fn vec<T: ToString>(vec: &Vec<T>, sep: &str) {
    let out = vec.iter().map(|e| e.to_string()).collect::<Vec<_>>().as_slice().join(sep);
    println!("{}", out);
  }

  pub fn mat<T: ToString>(mat: &Vec<Vec<T>>, sep: &str) {
    for v in mat {
      vec(v, sep);
    }
  }
}