My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood.  His mother won't indulge his requests, but I often will.  Here is the day he wanted 100:

 

But today he wanted the prime 719, which I obliged.  When deciding which digit to eat first, he went through the choices, trying to make a composite with the digits left behind.  But he quickly realized that eating any digit would leave a prime: 71, 79, 19 are all prime.  Pleased with his discovery of this prime 719, he tried to find a larger one, but couldn't.

My questions:


Do these primes have a name?
Can you think of any more of them (clearly 23 is the smallest)?
Are there an infinite number of them?
Is there likely to be a way to find them short of using a computer?


Here is a heuristic for why we should expect there are only finitely many of these numbers.  Let $P$ be the set of positive integers with your special property.

A large number $n$ is prime with probability approximately
$$ \frac{1}{\log n}. $$
For a number $n$ to be in $P$, it must remain prime with the removal of each digit.  Removing a digit from a number $n$ results in a number on the order of 
$$\frac{n}{10}.$$
A number $n$ has approximately
$$\frac{\log n}{\log 10}$$
digits.
As a result, the probability that $n$ is in $P$ can be approximated as
$$ \left( \frac{1}{\log \frac{n}{10} } \right)^{\frac{\log n}{\log 10}} = \frac{1}{n^{\frac{\log \log (n/10)}{\log 10}}}$$
In general, if the probability of $n$ being in a set $S$ is $p(n)$ then if 
$$\sum_{n=1}^{\infty} p(n)$$
is finite, we can expect $S$ to be finite; if the sum is infinite, we can expect $S$ to be infinite.  In the case of $P$, since the sum
$$\sum_{n=2}^{\infty} \frac{1}{n^{\frac{\log \log (n/10)}{\log 10}}}$$
converges (by comparison with, say, $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$) we may conclude that we should expect $P$ to be finite.

Unfortunately, this is just a heuristic, so does not constitute a proof of the finiteness of $P$, nor does it give any idea how large the largest element of $P$ might be.

Addition: I just realized that I wasn't assuming $n$ needed to be prime to be in $P$, only that it was prime after the removal of any one digit.  So, my $P$ would include 27.  Requiring only prime numbers in $P$ just makes the set smaller, so the above argument applies just as well is we require $P$ to only contain prime numbers.
