Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question is thus: which even positive integers are known to be both prime gaps (at least once) and the sum of 2 primes? Can we prove that the set of such integers has asymptotic density $1$ among all even positive integers?  

Thanks in advance.



Which even numbers are known to be both prime gaps and the sum of 2 primes?
Goldbach's Conjecture has been verified for all numbers up to $4 \times 10^{17}$ by [Tomás Oliveira e Silva](http://sweet.ua.pt/tos/goldbach.html). On the other hand, [Thomas R. Nicely](http://www.trnicely.net/gaps/gaplist.html) has shown that all even numbers up to $2000$ occur as prime gaps. This includes a list of all first known occurrences as part of an exhaustive search of primes up to $5 \times 10^{16}$. 


**Answer.** It includes all even numbers in $[4,2000]$. 


