Suppose the twin prime conjecture fails. Then, by Chen's theorem, there are infinitely many primes $p$ s. t. $p+2$ is a product of exactly two primes.
It would be nice to know that as $p$ grows, so do both factors of $p+2$. In other words, if $S = \{s \in \mathbb{N}; (\exists l > s)(l\cdot s -2 \text{ is a Chen prime}) \}$, can we prove that $S$ is infinite?