Does the base 10 expansion of 2^n always contain the digit 7 if n is large enough?
As of 1994, this was an open question (page 196 here). I don’t know whether this has since been resolved.
The following Python code suggests that the conjecture may be true for n ≥ 72.
def digits(n): s = set() while n > 0: s.add(n%10) n /= 10 return s for i in range(71, 10000): p = 2**i if 7 not in digits(p): print i, p
Update: It appears that 2^n contains every digit for n > 168. See this comment.
Related post: Open question turned into exercise