TITLE: Primes satisfy $pq+2=rt$
QUESTION [7 upvotes]: Are there infinitely many primes $p, q, r, t$ such as $pq+2=rt$
$3\times{11}+2=5\times{7}$
$5\times{11}+2=3\times{19}$
I guess this may be an open question?

REPLY [2 votes]: The question is whether there are infinitely many integers $n$ such that both $n,n+2$ have exactly two prime factors. Whether this is occurs is an open question.
This is essentially a problem in sieve theory. The parity problem is a fundamental obstruction that prevents us from solving many of these sorts of problems.
Essentially, the parity problem prevents us from distinguishing numbers with an even or odd number of prime factors.