While thinking about prime numbers, I noticed that:
$(1)$ Very few prime numbers have squares that are palindromes.
Ex: $2$, $3$, $11$, $101$, $307$
$(2)$ Even rarer are prime numbers that are palindromes whose square are palindromes.
Ex. $2$, $3$, $11$
This inspired me to ask the following questions:
$(1)$ Are $2, 3$ the only prime numbers that don't have the digit $1$ and are palindromes whose squares are also palindromes?
$(2)$ If not, then are there a finite number of these types of these prime numbers.