TITLE: Infiniteness of non-twin primes.
QUESTION [118 upvotes]: Well, we all know the twin prime conjecture.
There are infinitely many primes $p$, such that $p+2$ is also prime.
Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is NOT prime.

REPLY [3 votes]: Let $p$ be an odd prime.  Then at least one of 
$$ p+2, p+4, p+6, p+8$$
must be composite since its final digit must be a $5$.  Therefore there is at least one non-twin prime near each odd prime.