# Recursively palindromic primes. Are they rare birds?

1. A prime number is a number larger than 1 which only positive divisors are itself and 1.

Examples: 3,5,11.

2. A number is palindromic in a base $b$ if when written with digits in that basis $d_1d_2\cdots d_n$, $$d_n=d_1\\ d_{n-1} = d_2\\ \cdots$$

For example : the number 191 (in base 10) is a palindromic prime, since it is a prime number and a palindromic number. Now, the sum of the digits, $1+9+1=11$ is also a palindromic prime and $1+1=2$ is also one. So 191 is example of a palindromic prime whose digit sum is a palindromic prime whose digit sum is a $\cdots$ ( and so on).

How many sequences of palindromic primes being preserved by digit sum (all the way down to a 1 digit prime) are there? Infinite or finite? If not infinite, can we calculate how many or give an upper bound?

• A prime number is a number divisible only by itself and 1- this allows 1 as a prime which goes against convention. Is this intentional or should we replace your definition with the more conventional notion of a prime? – Colm Bhandal Jun 25 '17 at 13:42
• Wow, that seems like a really tough question. The one thing that bothers me is that it is dependent on our subjective base-10 number system, rather than something more "mathematical" so to speak. Since it's based on digits in base ten, a proof for this would probably heavily involve modulo 10 arithmetic... – Colm Bhandal Jun 25 '17 at 13:45
• If $p$ is such a prime, we must have $p\mod 9\in\{2,3,5,7\}$, since $s(n)\equiv n\pmod 9$ for all $n\in\Bbb{N}$ and eventually when we reach a one-digit number it must be prime. – Mastrem Jun 25 '17 at 14:08
• Proving that there's only finitely many would involve proving that there is only a finite amount of primes of the form $p=10^k+1$, which is very, very hard. – Mastrem Jun 25 '17 at 14:22
• @mathreadler Whether there are infinitely many primes of the form $2^k+1$ is a long standing unsolved problem; I imagine it's the same for $10^k+1$; what we do know is that $k$ must be a power of $2$ – Mastrem Jun 25 '17 at 16:17