As I think much about primes and usually that ends up with just a formulation of some new/old conjectures I came to the following idea.
First, an example. Take a prime $1277$. Now increment every digit of that prime by $2$ to obtain $3499$, which is also a prime. We can do this even if some digit is $8$ or $9$, for example a number $4899$ would be by this procedure of incrementation of digits by $2$ be incremented to $6101111$.
If we have some prime $p$ that after this procedure of incrementation of digits by $2$ gives another prime $q$ then we could call $p$ a 2-incrementable prime.
It is evident that this can be further generalized to arrive at the concept of 2m-incrementable primes for every natural number $2m$.
But, are there an infinite or finite number of 2-incrementable primes?
Edit 1: Trivial note: This procedure can be specialized to some number of digits or to only one digit (another generaliization). If we do the incremention of only the last digit then we are in the domain of twin primes, that is, twin primes are 2-incrementable primes with respect to only the last digit.