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The sum bit is Goldbach's conjecture, which is open. The difference bit seems to be open as well (e.g. according to this source).

What if we allow the alternative? More precisely, has the following (seemingly) weaker question been resolved or is it open too?

Given $n\in\mathbb{N}$, do there always exist (not necessarily distinct) primes $p_1$, $p_2$ such that $2n\in\{p_1+p_2, p_1-p_2\}$?

Damian Reding
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    This is most likely an open problem as well. – Oussema Jan 18 '21 at 17:31
  • (Seemingly) weaker does not help if there is no technique for a proof for most of these questions. – Dietrich Burde Jan 18 '21 at 17:39
  • @DietrichBurde Agreed. I'm not trying to crack Goldbach, I'm just wondering. – Damian Reding Jan 18 '21 at 17:42
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    @DietrichBurde Adding non-linear alternatives (like sum of prime and semiprime) seems to have made things easier. Or are you saying that adding linear ones won't make it easier? There are indeed not many linear ones that can be added, seeing as everything is a sum of 5-6 primes. – Damian Reding Jan 18 '21 at 18:04
  • Yes, I am saying that proving every even integer is the sum of two primes $p+q$ or a sum $p+3q$ doesn't make things easier. Or $p-q$. But I am not completely sure, of course. – Dietrich Burde Jan 18 '21 at 19:16

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