*Asked on MSE without response [here](https://math.stackexchange.com/questions/1534467/goldbach-for-certain-classes-of-n).*


$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.


The [Wiki article on the Goldbach conjecture](https://en.wikipedia.org/wiki/Goldbach's_conjecture) states that



> 
> In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers were expressible as the sum of two primes. More precisely, they showed that there exist positive constants $c$ and $C$ such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most $C N^{1-c}$ exceptions. In particular, the set of even integers which are not the sum of two primes has density zero.
> 
> 
> 


Has the Goldbach conjecture been proven for any *specific* classes on $n$? By exhautive search,it has been proven for $4\leq n \leq 10^{18},$ but my question is whether it has been proven for *eg* primorial multiples, where $G(n)$ generally reaches it's maximum. I realise probabilistic arguments are difficult to make rigorous, but for the primorials, they are of course the most likely to have multiple additive prime partitions. Surely it is not difficult to prove for the primorials, or am I mistaken as to the sheer complexity of the task?



Goldbach for certain classes of $n$
Well, "specific" is a vague term. For example, any number of the form $p+q$ (with $p$ and $q$ primes), such as $p+3$, is trivially a sum of two primes. But the answer to your (vague) question is certainly no, at least I don't know any reasonable characterization (independent of the primes) that would yield a representation of the form $p+q$ (with $p$ and $q$ primes). Similarly, I don't know of any reasonable condition on $n$ that would ease to show that $n=p+q$, besides the obvious assumption that $n$ is even.


BTW, the excellent theorem of Montgomery-Vaughan has been quantified, e.g. János Pintz proved it some years ago with $c=1/3$ (although I don't think he published this result).


**P.S.** Also, surely, Goldbach has not been proved for primorials, nor do I know of any serious attempt that focuses on primorials.


