Start as you wish had you prove that for any kdigit number M there exists a number n such that the string of first k digits of 2
^{n} equals M.
Find a power of 2 whose decimal expansion begins with the 12digit string "201320132013". It need not be the smallest such number.
Bonus:
Find the smallest such number and prove it to be the smallest.
It's expected that n would be a 12 or 13digit number so that the number of possibilities (pigeonholes) would be large enough that there's a high enough probability of one or more satisfying the requirement.

Posted by Charlie
on 20130105 12:18:21 