We will call a number N fortunate if it can be written with

We will call a number N fortunate if it can be written with four equal digits in some base b Element zopf^+. Clearly 2222 is fortunate, as is 3333_6. Why is 2000 fortunate?

Solution

A number is said to be fortunate if it can be written as 4 equal digit in any base.

Thus if we convert 2000 from base 10 to base 7,

2000_base10 = 5555_base7

Thus 2000 can be represented as 4 equal digit in base 7, hence is fortunate number.