Finding the Smallest Good Base in JavaScript

Good Base

For an integer num, we call k (k >= 2) a good base of num, if all digits of num base k are 1.

For instance: 13 base 3 is 111, hence 3 is a good base for num = 13

Problem

We are required to write a JavaScript function that takes in string str that represents a number as the only argument. The function should return the string representation of the smallest possible number which is a good base for str.

For example, if the input to the function is −

const str = "4681";

Then the output should be −

const output = "8";

Output Explanation:

Because 4681 base 8 is 11111

Example

The code for this will be −

const str = "4681";
const smallestGoodBase = (n = '1') => {
   const N = BigInt(n), bigint2 = BigInt(2), bigint1 = BigInt(1), bigint0 = BigInt(0)
   let maxLen = countLength(N, bigint2) // result at most maxLen 1s
   const findInHalf = (length, smaller = bigint2, bigger = N) => {
      if (smaller > bigger){
         return [false];
      };
      if (smaller == bigger) {
         return [valueOf1s(smaller, length) == N, smaller]
      };
      let mid = (smaller + bigger) / bigint2;
      let val = valueOf1s(mid, length);
      if(val == N){
         return [true, mid];
      };
      if (val > N){
         return findInHalf(length, smaller, mid - bigint1);
      };
      return findInHalf(length, mid + bigint1, bigger);
   };
   for (let length = maxLen; length > 0; length--) {
      let [found, base] = findInHalf(length);
      if(found){
         return '' + base;
      }
   };
   return '' + (N - 1);
   function valueOf1s(base, lengthOf1s) {
      let t = bigint1
      for (let i = 1; i < lengthOf1s; i++) {
         t *= base
         t += bigint1
      }
      return t
   }
   function countLength(N, base) {
      let t = N, len = 0
      while (t > bigint0) {
         t /= base
         len++
      }
      return len
   }
};
console.log(smallestGoodBase(str));

Output

And the output in the console will be −

8