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Title pretty much says it. Do there exist primes $p,q$ such that $p+q=2n$, where $p<q$ and $n\geq 4$ and $p,q,n\in\mathbb P$?

This is obviously a special case of Goldbach. I'm wondering whether anyone has figured out a proof that at least one non-trivial (where trivial is $p=q=n$) solution exists for those cases. I'm guessing not, but it can't hurt to ask!

e.g. $n=7 \implies 3+11=14\\ n=19 \implies 7+31=38$

Trevor
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  • It’s not really a special case of Goldbach, because Goldbach allows $p=q,$ so when $n$ is prime, $2n=n+n$ solves the Goldbach question for the case of $2n.$ – Thomas Andrews Jan 27 '20 at 23:03
  • I’m sorry, but I don’t understand the quantification. In particular, are you starting with $n$ and wondering about the existence of $p$ and $q$? – Lubin Jan 28 '20 at 01:07
  • @Lubin Exactly. – Trevor Jan 28 '20 at 01:47
  • Green-Tao says there are infinitely many cases it's true, but not if that infinity contains each prime $n$ as a middle term . –  Jan 29 '20 at 01:48

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