# Research on the structure of a non-Goldbach number?

Has there been any research into the structure of a non-Goldbach number? This seems like it would be a profitable area for proof by contradiction, so I assume that someone has already done it. (i.e. either the structure proves impossible, or it would allow one to calculate a non-goldbach number)

If so, I'd be interested in sampling some of it. I have googled and turned up nothing, hence this question.

(Apologies if this question is at the wrong level for this site).

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This is far too vague for MO. –  Andy Putman May 14 at 20:16
I don't think any condition is known to imply in a nontrivial way that a number is a sum of two primes. So I would be delighted if I was wrong, but I guess there are no results of the type you are looking for. –  Johan Wästlund May 14 at 20:18
Questions about whether anything is known about a very specific topic are fine, but the topic in question has to be much more specific and better thought out than your question. –  Andy Putman May 14 at 20:25
@AndyPutnam My question seems pretty specific to me, although I'll admit that it's not very deeply thought out - the point of looking for research is to shortcut repeating unnecessarily elementary thought. What would be a sufficiently specific question of this type? –  user33996 May 14 at 20:32
What's the difference between studying non-Goldbach integers and the Goldbach conjecture? (to me--none). –  Wlodzimierz Holsztynski May 14 at 23:52
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The number of even integers that are not sums of two primes (non-Goldbach integers) is small in the sense that for $n \leq X$ at most $O(X^{1 - \delta})$ integers are non-Goldbach. This can be thought of as a stronger form of Vingoradov's three-prime theorem that every large enough odd number is a sum of three primes (since the former implies the later). For an old very old survey paper see http://www.ams.org/journals/bull/1949-55-03/S0002-9904-1949-09180-2/S0002-9904-1949-09180-2.pdf . For a more recent article I would suggest http://dmle.cindoc.csic.es/pdf/MATEMATICAIBEROAMERICANA_1985_01_01_03.pdf which is a paper of Heath-Brown.