Numbers Written With Only 1s and 0s I just watched this video
that talked about the smallest number that can be written in some base(s) using only the digits 1 and 0. Afterwards, I wondered if there was a way to find the number of integers that can be written using only 1s and 0s in bases 3 and also base 4.
I am not sure how to start on this problem. Hints and solutions are appreciated.
 A: There seem to be infinitely many. There are 10000 up to 
 1650485993748049082320245840.  See OEIS sequence A258981.  
EDIT:
This is not a proof, but a heuristic.  Of the $3^d$ numbers with  $\le d$ base $3$ digits, 
$2^d$ have only $0$'s and $1$'s in base $3$.  Thus the probability that such a number $n$ has only $0$'s and $1$'s in base $3$ is on the order of 
$(2/3)^{\log_3(n)} = n^{\log_3(2/3)}$.  Similarly, the probability that $n$ has only $0$'s and $1$'s in base $4$ is on the order of 
$n^{\log_4(1/2)}$.  Assuming these events are approximately independent,
the probability that $n$ falls in both categories should be on the order
of $n^{\log_3(2/3) + \log_4(1/2)}$: call this $n^r$, where $r \approx -.86907$.  Since $r > -1$ the expected number of $n$ in both categories is approximately $\sum_{n=1}^\infty n^r = \infty$.
EDIT: By contrast, if we ask for only $0$'s and $1$'s in bases $4$ and $5$, in the above analysis we would get $r = \log_4(1/2) + \log_5(2/5) \approx -1.069323$, and $\sum_{n=1}^\infty n^r \approx 15.007$.  We should expect this sequence (which seems not to be in the OEIS) to be finite.  The only solutions less than $10^{400}$, if my program is right, are $0, 1, 5, 16400, 16401, 16405, 82000, 82001, 82005$.  I have no proof that there are no other solutions, but it seems quite unlikely.
