$(1)$ A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime.
For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime.
$(2)$ A happy number is defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers.
For example, $19$ is happy, as the associated sequence is
$1^2 + 9^2 = 82,$
$8^2 + 2^2 = 68,$
$6^2 + 8^2 = 100,$
$1^2 + 0^2 + 0^2 = 1.$
$(3)$ A palindromic prime is a prime that remains the same when its digits are reversed.
For example, $11$
A Circular happy palindromic prime is a prime number that has all three of the above properties.
According to my testing in PARI/GP and my own results, I have found the following circular happy palindromic primes:
$7, 1111111111111111111$ (19 digits)$, 11111111111111111111111$ (23 digits)
(Haven't tested complete range of $10^{24}$ in PARI/GP)
Note that 19 and 23 are happy primes themselves which explains why $1111111111111111111, 11111111111111111111111$ are also happy primes.
Questions:
$(1)$ Are $7, 1111111111111111111, 11111111111111111111111$ the only circular happy palindromic primes?
$(2)$ Are $1111111111111111111, 11111111111111111111111$ the only happy primes of the form 11111.... ?
$(3)$ If not then are there a finite number of circular happy palindromic primes? Can we prove/disprove this?