Goldbach's Conjecture has been verified for all numbers up to $4 \times 10^{18}$ by [Oliveira e Silva, Herzog, Pardi](https://doi.org/10.1090/S0025-5718-2013-02787-1).  On the other hand, [Thomas R. Nicely](https://web.archive.org/web/20191118035255/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 $[4,2000]$.