It seems obvious that infinite many primes can be formed only using two distinct digits.
$67776767776667777777$ is an example for such a prime. Even if we allow only the digits $0$ and $1$, there are $2^{n-2}$ numbers with $n$ digits, for which the first and last digit is $1$. So, we can expect that some of those numbers are prime for every sufficiently large number $n$.
Can it be proven that infinite many primes contain only two distinct digits in the decimal representation ?