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 Python (3.4.3) (judge verdict: Wrong Answer)

import itertools
import math 
 
n,s = [int(input()) for _ in range(2)]
 
def n_base_sum(n,b):
    if(int(n/b)):
        return n_base_sum(int(n/b),b)+n%b
    else:
        return n%b
 
def search_p(n,s):
    for p in range(math.ceil(math.sqrt(n))+1,0,-1):
        b = (n-s)//p + 1
        if b >1:
            q = n % b
            if p+q ==s:
                return b
        
    return -1
 
ans =-1
flag = False
answers = []
if s == n:
    ans = n+1
    flag =True
    answers.append(ans)
for b in  range(2,math.ceil(math.sqrt(n))+1):
    if s== n_base_sum(n,b):
        ans = b
        flag = True
        answers.append(ans)
        break
ans = search_p(n,s)
if ans != -1:
    flag =True
    answers.append(ans)
 
 
if not flag:
    print(-1)
else:
    print(int(min(answers)))