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 ~$10^{72227}$

Only some extended hints:
The numbers are necessarily of the form $N=\frac{10^n-1}9+a10^k$ with $0\le k<n$ and $a+1$ prime (i.e. $a\in\{1,2,4,6\}$) to meet the product criterion. Of course, $n$ must be prime to meet the length criterion. Next, the digit sum is $n+a$, which rules out $a=1$ except for small cases.
If $a=2$ we must have $n\equiv -1\pmod 6$. Then $N\equiv n-k \pmod 3$, so we need $k\not\equiv -1\pmod 3 $. Also , $\frac{10^n-1}9\equiv 2\pmod 7$, ${}\equiv 9\pmod {13}$, ${}\equiv 11\pmod{37}$, which rules out another fraction $\frac 17$, $\frac 1{13}$, $\frac1{37}$, respectively, of possible $k$ values.
Similarly, if $a=4$ we must have $n\equiv 1\pmod 6$. Then $N\equiv n+k\pmod 3$, so we need $k\not\equiv -1\pmod 3$ again. The argumentation with the primes $7,13,37$ dividing $111111$ applies as well in a similar way. 
More generally, for any (small) prime $p>5$ we can obtain restrictions for $n,k\pmod{p-1}$. This makes it easy to construct $N$ which has no small prime divisors (and of course fulfills the sum, product, and length criterion). There's still a long way to go to make $N$ prime, of course. 
Heuristically, the chance of $N$ being prime is $\sim \frac1n$; and since we have $\sim n$ choices for $k$, we'd expect a certain more or less constant amount of supreme primes per $n$ ...
