TITLE: Finitely many Supreme Primes?
QUESTION [16 upvotes]: A challenge on codegolf.stackexchange is to find the highest "supreme" prime: https://codegolf.stackexchange.com/questions/35441/find-the-largest-prime-whose-length-sum-and-product-is-prime
A supreme prime has the following properties:

the number itself is prime
the number of digits is prime
the sum of digits is prime
the product of digits is prime

Are there finitely many "supreme" primes? Are there infinitely many? Currently the highest one found is ~$10^{72227}$

REPLY [7 votes]: Heuristically, a random $n$-digit number has probability approximately  $c/n$ of being prime, where $c$ is a positive constant.  Since $p$ is $3$ or $5$ or $7$ and there are $n$ possible locations for it, there are $3n$  $n$-digit candidates.  So the expected number of primes of this form among the $n$-digit candidates should be approximately constant.  This would indicate that there should be infinitely many supreme primes.  Of course it is not a proof.  It would be very surprising if this could be proven.