Finitely many Supreme Primes? 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 ~$10^{72227}$

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.
