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Asked on MSE without response here.

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

The Wiki article on the Goldbach 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 proved for any specific classes of $n$? By exhaustive search, it has been proved for $4\leq n \leq 10^{18},$ but my question is whether it has been proved for, e.g., primorial multiples, where $G(n)$ generally reaches its 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?

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  • $\begingroup$ So you want to ask if there is a "specific" subset $E$ of the even numbers with the property: For EVERY $e\in E$ there are two primes $p,q$ such that $e=p+q$ $\endgroup$
    – Konstantinos Gaitanas
    Commented Nov 24, 2015 at 11:36
  • $\begingroup$ @KonstantinosGaitanas yes, exactly $\endgroup$
    – martin
    Commented Nov 24, 2015 at 12:06
  • $\begingroup$ Why would the primorials be easier to express? Certainly for them you cannot use a "small" prime in the sum. Thus, if anything I'd say it should be harder for them then say for a power of two or anything else with few distinct prime factors. $\endgroup$
    – user9072
    Commented Nov 24, 2015 at 14:55
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    $\begingroup$ I am afraid there just are no such results as GH said (with more authority than I could). $\endgroup$
    – user9072
    Commented Nov 24, 2015 at 22:10
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    $\begingroup$ I can't resist commenting that ltg.ed.ac.uk/~richard/goldbach.html gives a color coded plot of $G(x)$ showing that in fact numbers of the form $2^jp$ with $p$ prime have (as a rule, relative to their size) fewer representations and those with several distinct small prime factors (primordials for example) have the most. The simple reason is explained there. $\endgroup$
    – Aaron Meyerowitz
    Commented Nov 25, 2015 at 7:13

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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.

Added. János Pintz published the above mentioned result with $c=7/25$. See here.

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  • $\begingroup$ thanks, yes, I should have included the independent of the primes distinction in the definition. The last point is interesting - where did he prove this, if not published? $\endgroup$
    – martin
    Commented Nov 24, 2015 at 22:04
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    $\begingroup$ @martin: The proof is in his notes (I guess). He has many unpublished results, actually. Smaller values like $c=1/100$ appeared in the literature (by other authors), I am sure you can find them with Google or MathSciNet or Zentralblatt. $\endgroup$
    – GH from MO
    Commented Nov 24, 2015 at 22:07
  • $\begingroup$ @martin See the "Added" section in my post. $\endgroup$
    – GH from MO
    Commented Nov 2, 2023 at 3:37

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