TITLE: Are there infinitely many primes $p$ such that $p-2$ and $p+2$ are composite?
QUESTION [9 upvotes]: Are there infinitely many primes $p$ such that $p-2$ and $p+2$ are composite?
If $p\neq3$ then either $p+2$ or $p-2$ is a multiple of three, but this does not settle the matter for both.
We know that there are infinitely many primes. But it is not known whether there are infinitely many twin primes, so something extra is needed here.

REPLY [7 votes]: Almost all primes have that property, as a result of the scarcity of twin primes.
Infiniteness of non-twin primes.