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Let $g(2n)$ be the number of representations of $2n=p+q$ with primes $p$ and $q$. Many people have asked whether $g(2n) \ge 2$ when $2n = p+q$ for some primes $p$ and $q$. That is, does $g(2n) \ge 1$ imply $g(2n) \ge 2$? From the famous Goldbach Comet, it looks probable although it was not yet proved.

Now, what can we say about the following weaker problem?

For any sufficiently large prime $p$, is there a prime $q$ such that $p+q$ has another representation $p' + q'$?

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    $\begingroup$ Conjecturally yes, but it's not been proved, as far as I know. $\endgroup$ – Sylvain JULIEN Sep 4 '20 at 7:32
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    $\begingroup$ See en.m.wikipedia.org/wiki/… $\endgroup$ – Stanley Yao Xiao Sep 4 '20 at 8:39
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    $\begingroup$ Please always include a high-level tag like "nt.number-theory". $\endgroup$ – GH from MO Sep 4 '20 at 11:00
  • $\begingroup$ @GHfromMO I forgot it. Thank you for your editing. $\endgroup$ – P.-S. Park Sep 4 '20 at 21:13
  • $\begingroup$ Strictly formally speaking, if (p,q) is a solution, (q,p) is another solution. $\endgroup$ – Gérard Lang Sep 5 '20 at 10:18
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To your first question: we don't know. To your second question: we know much more, namely if $N$ is a large odd number, then the number of representations $N=p_1+p_2-p_3$ with each $p_j$ a prime from $[2N,3N]$, has order of magnitude $N^2/(\log N)^3$. This can be proved in essentially the same way as we prove that $N$ can be written as a sum of three primes in that many ways. See also Harald Helfgott's response here.

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Here is a graph showing the number of representations of $2n$ as a sum of two primes.

enter image description here

It suggests that something much stronger than what you ask about is true. And there are heuristics that predict what is shown. but not proofs.

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Let $n \in 2\mathbb{N}^*$ large enough.

$$n = p+q, \ (p,q)\in\mathbb{P}^2 \iff (p, n-p) \in \mathbb{P}^2$$

You search for the quantative version of Goldbach's conjecure, Hardy and Littlewood in there 1923 paper "Some problems of ‘Partitio numerorum’; III : On the expression of a number as a sum of primes", conjecture that : $$G(n) \sim 2 C_2 \displaystyle {\small \Big( \prod_{\substack{p | n \\ \text{p prime} \\ 3 \leqslant p}} {\normalsize \dfrac{p-1}{p-2}} \Big)} \dfrac{n}{\log(n)^2}.$$

Where $G(n) = \#\{(p, n-p) \in \mathbb{P}^2 \, | \, p \leqslant n\}$, and : $C_2 = \displaystyle{\small \prod_{\substack{3 \leq p \\ \text{p prime}}} \left({\normalsize 1-\dfrac{1}{(p-1)^2}}\right)}$.

This conjecture agree perfectely with numeric checks, but unfortunately not proven up to now (and no hope to prove it soon).

You can see my try here : is there a link with the probabilistic model for prime numbers?

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  • $\begingroup$ I don't see how this addresses the question that was asked $\endgroup$ – Yemon Choi Sep 5 '20 at 1:57
  • $\begingroup$ @YemonChoi, he ask if $g(2n) \ge 1 \implies g(2n) \ge 2$, and i say that $g(2n)$ is more much bigger (conjectured), please delete your comment you made a mistake. $\endgroup$ – LAGRIDA Sep 5 '20 at 8:11
  • $\begingroup$ The OP mentions that conjecture, but he or she is asking about "the following weaker problem". See the answer of GHfromMO. Please read carefully the question being asked $\endgroup$ – Yemon Choi Sep 5 '20 at 16:20
  • $\begingroup$ I understund exactly his 2 questions, the second about fixing a prime number $p$, then for every other prime number $3 \leqslant q \leqslant p$, we have $n_{p,q}=p+q$ is an even number and returning to Goldbach's conjecture on it's quantative form we have approximatly $G(n_{p, q})$ representation of $n_{p,q}$ as sum of 2 primes. $\endgroup$ – LAGRIDA Sep 5 '20 at 17:26

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