  In this note we use recent developments in sieve theory to highlight the
interplay between Goldbach and de Polignac numbers. Assuming that the primes
have level of distribution greater than $1/2$, we show that at least one of two
nice properties holds. Either consecutive Goldbach numbers lie within a finite
distance from one another or else the set of de Polignac numbers has full
density in $2 \mathbb{N}$. Using very similar techniques we give a conditional
proof that the set of limit points of the sequence of normalised prime gaps
$(p_{n+1}-p_n)/ \log p_n$ has density at least $2/3$ in the positive reals.
