Define $r(n)$ to be the reverse of a positive integer , that is the number emerging if the decimal expansion is written down in reverse order. Emerging leading zeros are of course omitted , but this is irrelevant for my question since for primes this cannot occur.
An emirp is a prime $p$ with the property that $r(p)$ is prime as well. In particular , a palindrome prime $p$ is an emirp.
First question : Probably , there are infinite many emirps. Can this be proven or is this an open problem ? If we rule out palindrome primes , can we still expect infinite many emirps ? Or even prove that there are still infinite many ?
An emirp-twin is a twin prime pair $(p/q)$ such that $r(p)$ and $r(q)$ are prime as well. Let us also demand that neither $p$ nor $q$ is a palindrome prime to avoid duplicates in the quartuples $(p,q,r(p),r(q))$ A large example is $$p=10^{100}+30717259$$
Second question : Can we expect infinite many emirp-twins ?