We say that an integer $n > 1$ is a Goldbach number if $n$ is the sum of two primes. The famous Goldbach conjecture says that every integer greater than $1$ is Goldbach.

Consider the following much weaker statement:

(G') The set $G$ of Goldbach numbers satisfies $\lim\inf_{n\to\infty}\frac{|G\cap\{1,\ldots,n+1\}|}{n+1} > 0$.

Is (G') true?

We say that an integer $n > 1$ is a Goldbach number if $n$ is the sum of two primes. The famous Goldbach conjecture says that every even integer greater than $2$ is Goldbach.

Consider the following much weaker statement:

(G') The set $G$ of Goldbach numbers satisfies $\lim\inf_{n\to\infty}\frac{|G\cap\{1,\ldots,n+1\}|}{n+1} > 0$.

Is (G') true?