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?

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    $\begingroup$ Yes. In the proof of Chen's theorem, he actually shows something a bit stronger than "product of at most two primes": each of the prime factors is bounded below by a small power of $p$, i.e. $p^c$ for some fractional constant $c>0$. $\endgroup$ – Erick Wong May 24 '16 at 7:59
  • $\begingroup$ This answer on MO cites a recent reference that one may take $c=3/11$. $\endgroup$ – Erick Wong May 24 '16 at 8:06

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