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.

  • 5
    $\begingroup$ OEIS entry for primes $p$ such that both $p$ and $p^2$ are palindromes. It is apparently a conjecture that all such primes $>3$ consist only of zeros and ones. $\endgroup$ – Wojowu Apr 22 at 14:31

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