TITLE: Deleting any digit yields a prime... is there a name for this?
QUESTION [171 upvotes]: 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?

REPLY [8 votes]: Assuming the relaxation that the original number need not be prime (treated in an answer) digit repetitions give large solutions.
Playing with 2 decimal digits gave the $332$ digit composite solution 
11111111111111111111111111111111111111111111111111111111999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999