# Smallest Good Base Problem

## Description

LeetCode Problem 483.

Given an integer n represented as a string, return the smallest good base of n.

We call k >= 2 a good base of n, if all digits of n base k are 1’s.

Example 1:

``````1
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Input: n = "13"
Output: "3"
Explanation: 13 base 3 is 111.
``````

Example 2:

``````1
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Input: n = "4681"
Output: "8"
Explanation: 4681 base 8 is 11111.
``````

Example 3:

``````1
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Input: n = "1000000000000000000"
Output: "999999999999999999"
Explanation: 1000000000000000000 base 999999999999999999 is 11.
``````

Constraints:

• n is an integer in the range [3, 10^18].
• n does not contain any leading zeros.

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class Solution {
public:
// For a given length d, we use binary search to check whether
// there is a base k which satisfies 1+k^1+k^2+...k^d=n.
// The left limit is 1, and the right limit is pow(n,1/d)+1,
// i.e., the d th square root of n.

string smallestGoodBase(string n) {
unsigned long long tn = (unsigned long long)stoll(n);
unsigned long long x = 1;
for (int i = 62; i >= 1; i--) {
if ((x<<i) < tn) {
unsigned long long cur = mysolve(tn, i);
if (cur != 0) return to_string(cur);
}
}
return to_string(tn - 1);
}

unsigned long long mysolve(unsigned long long n,int d) {
double tn = (double) n;
unsigned long long right = (unsigned long long)(pow(tn, 1.0/d) + 1);
unsigned long long left = 1;
while (left <= right){
unsigned long long mid = left + (right - left) / 2;
unsigned long long sum = 1,cur = 1;
for (int i = 1; i <= d;i ++) {
cur *= mid;
sum += cur;
}
if (sum == n) return mid;
if (sum > n) right = mid - 1;
else left = mid + 1;
}
return 0;
}

};
``````