# Goldbach conjecture and the representation number

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 $$p' + q'$$?

• Conjecturally yes, but it's not been proved, as far as I know. – Sylvain JULIEN Sep 4 '20 at 7:32
• – Stanley Yao Xiao Sep 4 '20 at 8:39
• Please always include a high-level tag like "nt.number-theory". – GH from MO Sep 4 '20 at 11:00
• @GHfromMO I forgot it. Thank you for your editing. – P.-S. Park Sep 4 '20 at 21:13
• Strictly formally speaking, if (p,q) is a solution, (q,p) is another solution. – Gérard Lang Sep 5 '20 at 10:18

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.

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

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.

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?

• I don't see how this addresses the question that was asked – Yemon Choi Sep 5 '20 at 1:57
• @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. – LAGRIDA Sep 5 '20 at 8:11
• 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 – Yemon Choi Sep 5 '20 at 16:20
• 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. – LAGRIDA Sep 5 '20 at 17:26