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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

-- AtCoder My Practice
-- author: Leonardone @ NEETSDKASU
 
-- 解説読後
-- http://arc060.contest.atcoder.jp/data/arc/060/editorial.pdf
 
import qualified Data.Array as A
import qualified Data.Map as M
import qualified Data.List as L
 
main = print . solve . map read . lines =<< getContents
 
f b n
    | n < b     = n
    | otherwise = let (dv, md) = divMod n b in md + f b dv
 
solve [n, s]
    | n == s    = n + 1
    | otherwise = ans where
        ans = case L.find ((==) s . snd) . map (\i -> (i, f i n)) . takeWhile (\i -> i * i <= n) $ [2 .. ] of
            Just (i, _) -> i
            Nothing -> foldl (\r b -> if s == f b n then b else r) (-1) bs where
                bs = map ((+) 1 . fst) . filter ((==) 0 . snd) . map (divMod (n - s)) . takeWhile (\i -> i * i <= n) $ [1 .. ]