Goldbach versus de Polignac numbers

Abstract

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.