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≤k≤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.

**Answer**

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.

**Attribution***Source : Link , Question Author : Farewell , Answer Author : Farewell*