Goldbach's Conjecture has been verified for all numbers up to $4 \times 10^{18}$ by Oliveira e Silva, Herzog, Pardi. On the other hand, Thomas R. Nicely 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 $[4,2000]$.
