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](https://en.wikipedia.org/wiki/Goldbach%27s_comet), it looks probable although it was not yet proved.


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



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> For any sufficiently large prime $p$, is there a prime $q$ such that $p+q$ has another representation $p' + q'$?
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Goldbach conjecture and the representation number
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](https://mathoverflow.net/questions/202979/a-variant-of-the-goldbach-conjecture/202982#202982).


