Start as you wish
had you prove that for any k-digit number M there exists a number n such that the string of first k digits of 2n
Find a power of 2 whose decimal expansion begins with the 12-digit string "201320132013". It need not be the smallest such number.
Find the smallest such number and prove it to be the smallest.
It's expected that n would be a 12- or 13-digit 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 2013-01-05 12:18:21