Combining primes for getting primes?

Question

I was thinking about what would happen when we combine two prime numbers $p$ and $q$ into one number $:pq:$ . Like if $p=5$ and $q=3$ , then $:pq:=53$ . Then if $p=7$ and $q= 11$ then $:pq:=711$ and so on for other $p$ and $q$. It seems so that there is a fairly good chance that the newly obtained number is also a prime. Further examples which we can see are (in no specific order) $$331 ,353 , 223 , 233 ,719 \cdots$$ This motivates me to put forward the following question- Does there exist infinitely many prime numbers $p$ and $q$ such that $:pq:$ is a prime?

Answer 1

Yes, there are infinitely many such primes. I don't have a theorem with proof but I think pigs would fly before someone could prove me wrong on this.

Go to the OEIS and search for "23, 37, 53, 73, 113, 137, 173, 193, 197". They have a list of ten thousand such primes, though of course that doesn't prove there are infinitely many of them. Much more telling is that the entry's keyword field doesn't have the keyword "fini", which they use to mark sequences they know to be finite.

Denote by $\mathcal{L}$ how many base 10 digits an odd prime $q \neq 5$ has. Then we need to find a prime $p$ such that $10^\mathcal{L}p + q$ is also prime. Given that there are infinitely many primes, it seems highly improbable to me that none of them would satisfy this requirement.

A slightly more interesting question would be: for every odd prime $q \neq 5$ does there exist at least one prime $p$ such that $10^\mathcal{L}p + q$ is also prime?