While discussing about prime numbers with other users, I noticed that:
$(1)$ There are very few pairs of palindromic prime numbers that do not contain the digit $1$ and that have products which are palindromes.
Ex : $[2, 3], [2, 30203], [2, 30403]$
$(2)$ For the large range that was tested on PARI/GP, I noticed that all the palindromic primes that yielded palindromic products were composed only of the digits $0, 2, 3, 4$
$(3)$ The palindromic primes of these pairs are equal to $2$ or always exist in the range of $3 \times 10^k$ to $4 \times 10^k$ where $k \in \Bbb{+Z}, 0$
User Peter helped me get the following results for $a, b \lt 10^7$ on PARI/GP:
[2, 3]
[2, 30203]
[2, 30403]
[2, 32323]
[2, 32423]
[2, 3002003]
[2, 3222223]
[2, 3223223]
[2, 3233323]
[2, 3304033]
[2, 3343433]
[2, 3400043]
[2, 3424243]
[2, 3443443]
[2, 3444443]
[3, 30203]
[3, 32323]
[3, 3002003]
[3, 3222223]
[3, 3223223]
[3, 3233323]
[30203, 3002003]
Questions:
(1) Are there a finite number of pairs of $a, b$, where $a, b$ are palindromic primes that do not contain the digit $1$ and $ab$ is a palindrome?
(2) Are all the palindromic primes that yielded palindromic products composed only of the digits $0, 2, 3, 4$?
(3) Are all the palindromic primes of these pairs equal to $2$ or always exist in the range of $3 \times 10^k$ to $4 \times 10^k$ where $k \in \Bbb{+Z}, 0$?