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The Wiki article on the Goldbach conjecture (where $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$) 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. Surely it is not difficult to prove for the primorials, or am I mistaken as to the sheer complexity of the task?

Reposted to MO here

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  • Wait, what did you define as $G(n)$? I thought it referenced the conjecture;s truth for $n$, or maybe the wikipedia page. How can it have a maximum? – Thomas Andrews Nov 18 '15 at 01:45
  • $G(n)=#$ ways even $n$ can be represented by prime additions. – martin Nov 18 '15 at 01:47
  • Why should the proof be easy for primorials ? – Peter Nov 18 '15 at 09:33
  • @Peter 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. – martin Nov 18 '15 at 12:53
  • By the way, do we know a (conditional or not) lower bound for $c$? – Sylvain Julien Nov 19 '15 at 15:59
  • @SylvainJulien If I were to guess, I'd say not, but then that's just a guess. I don't have a copy of Multiplicative number theory handy, but I should think it would be be a good place to look. – martin Nov 19 '15 at 16:58

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The upper limit for G(n), n=2m, with m odd is: $$\pi(m)-\omega(m)-1$$ This is reached by 30, as 15 has 6 primes less than it, 1 of which is 2 which Goldbach ignores, and 2 are distinct prime factors of 15 (and hence 30), which can't participate as distributivity of multiplication over addition of negatives (subtraction for numbers) shows. That forces p to divide or be divisible by q in p+q=2m. The others below 15 lead to Goldbach partitions of 30, 7+23=11+19=13+17=30 . But, NO this doesn't prove it for primorials.