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: Time Limit Exceeded)


n = int(input())
s = int(input())
rtn = int(n**0.5)+1

def f(b,n):
    if n<b:
        return n
    return f(b,n//b) + n % b

if n < s or (n // 2)+1 < s < n :
    print("-1")
elif s == n:
    print(n+1)
elif s == n//2 +1:
    if n % 2 == 1:
        print(n//2 + 1)
    else:
        print("-1")
elif (n//3) + 2 < s <= n//2:
    print(n-s+1)
elif s == 1:
    def check(n,b):
        if n == 1:
            return b
        else:
            if n % b !=0 :
                return 0
            else:
                return check(n // b,b)
    for b in range(2,rtn):
        r = check(n,b)
        if r:
            print(r)
            exit()
    print(n)
else:
    x = n // s
    if s - (n //(x+1))  <= x:
        x = x + 1
    if 5 < n/(x -1) - n/x :
        for b in range(n//x, n//(x-1)):
            if f(b,n) == s:
                print(b)
                exit()
        for i in range(x-1,2,-1):
            for b in range(n//i + (n//x)//(n//i), n//i + (n//x)//(n//i)  + (2*n//x - n//(x+1))//(x-i)):
                if f(b,n) == s:
                    print(b)
                    exit()
    else:
        for b in range(2,(n//3)+3):
            if f(b,n) == s:
                print(b)
                exit()
    print(n-s+1)