I'll try and explain what I asked. Take the number $5$, which is prime. Its binary representation, $101$, is also prime. I computed a program in python that picks all numbers from a list that verifies this ($k$ is prime and its binary representation is prime if we consider it as a decimal representation), and it keeps showing numbers up until $5000$ (which is as far as I checked).
I guess we first had to prove that there exists an infinite amount of primes numbers whose digits are onle $0's$ and $1's$, and then we would have to prove that the number of prime numbers (let's call the set of primes $A$) is the same as the number of prime numbers with digits $0's$ and $1's$ (let's call this set $B$). This second part can be easy ifthe first result is true, since $B\subset A \subset \Bbb N$. So the second result is true.
Does anyone have any idea on how to prove the first result? Or anything similar to it? Thanks in advance!
Edit: Thanks to Peter and CyclotomicField for some insight! I run a program in python that showed how many primes of this type can be found from $2$ to $1001$, $1002$ to $2001$, $2002$ to $3001$, ... until $1000001$. I drew in Mathematica the list of these numbers of primes, and got this:
It might not be much, but I thought it was interesting...