I know that if $p = 6k+1$ with $k$ integer, then $(2^p-p) \mod 6 \equiv 1$. I think that this means that $2^p-p$ could be prime. My question is: are there a finite or infinite number of primes which can be written like this?
At this moment I do not have complete explanation of your question but with the use of my own properties of primes which I discovered last year I made a conclusion i.e.:
If (2p - 5) and (2p + 5) are primes than (2p - a) and (2p + a) are also primes only and only if "a" is a multiple of 5(which is rare but can come) or "a" is a prime number maybe "p" also.
7 also show similar property in (2p - 7) and (2p + 7)