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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?

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    $\begingroup$ To add to Stanley Yao Xiao's answer, the fact that $G(x)=(1/2+o(1))x$ (enough to answer your question) was first proved by Vinogradov in the 1930s. $\endgroup$
    – Thomas Bloom
    Commented Nov 3, 2023 at 8:52
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    $\begingroup$ @ThomasBloom I think that this was proved by Chudakov (1937), Estermann (1938), van der Corput (1938). $\endgroup$
    – GH from MO
    Commented Nov 3, 2023 at 13:40
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    $\begingroup$ Please make some minimal research before asking a question. For example, the answer to your question can be found at en.wikipedia.org/wiki/Goldbach%27s_conjecture $\endgroup$
    – GH from MO
    Commented Nov 3, 2023 at 13:41

1 Answer 1

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Way more than that is true. We have $G(x) = |G \cap [1,x]|$ satisfying

$$\displaystyle G(x) = \frac{x}{2} + O(x^{0.72}),$$

by a theorem of Janos Pintz. See this paper:https://arxiv.org/pdf/1804.09084.pdf

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