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 $[4,2000]$.