# Are there an infinite number of prime numbers where removing any number of digits leaves a prime?

Suppose for the purpose of this question that number $1$ is a prime number.

Consider the prime number $311$. If we remove one $1$ from the number we arrive at the number $31$ which is also prime. If we removed $3$ instead of $1$ we would arrived at the number $11$ which is prime. If we remove $3$ from the number $31$ we arrive at the prime number $1$ and if we remove $1$ from $31$ we arrive at the number $3$ which is prime. And if we removed number $1$ from the number $11$ we would arrive at the prime number $1$.

So what property does this number $311$ have?

It has the property that if we remove one digit in any way we want we arrive at the prime number and if we remove two digits in any way we want we also arrive at the prime number. (Of course, if the original prime number had $n$ digits then it should be prime whatever $k$ digits we remove, for $1\le k\le n-1$.)

The question would be:

Are there an infinite number of prime numbers so that removing any number of digits leaves us with a prime number?

It seems to me that big enough prime numbers will rarely have the property that for every possible removal of some of the digits we will arrive at the prime number, but I see no reason that forbids an infinite number of these prime numbers.

It is clear that we cannot have digits $0,4,6,8,9$ in those prime numbers. There can be at most one $2$ because $22$ is composite,, at most one $3$ because $33$ is composite,at most one $5$ because $55$ is composite, at most one $7$ because $77$ is composite, at most two $1$`s because $111$ is composite, so such prime number can have at most $6$ digits so there is a finite number of such prime numbers.