# Proof of the weak Goldbach Conjecture

What are the main ideas of Harald Helfgott's proof that all odd $n \geq 5$ is the sum of 3 primes?

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Didn't Vinogradov prove it for sufficiently large odd numbers in something like 1937? So, it seems reasonable to believe that deciding the question one way or the other would be a matter of time after that. –  Geoff Robinson May 14 '13 at 7:02
For the first question, I would simply read the introduction of Helfgott's paper. –  François Brunault May 14 '13 at 7:39
Your question seems suitable for a blog. Since Terry Tao already has a blog discussion on this topic, I've voted to close. –  Ryan Budney May 14 '13 at 11:39
It would be nice to have the possibility to downvote (or upvote) the actions of closing some of the questions. –  Cristi Stoica May 14 '13 at 13:05
Geoff - the problem was that previously existing constants were larger than the number of subatomic particles in the universe multiplied by the number of microseconds since the Big Bang... –  H A Helfgott May 14 '13 at 16:58

I think this blog post of Terry Tao, as well as the comments following it (including some from Helfgott) answer this question as completely as one could reasonably hope.

https://terrytao.wordpress.com/2012/05/20/heuristic-limitations-of-the-circle-method/

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Thank you! –  Rodrigo A. Pérez May 14 '13 at 16:47
Note that the blog post was written a year ago. It is still relevant, though (particularly the comments by Helfgott). –  Timothy Chow May 15 '13 at 15:45
Does this also show every odd number $n > 5$ is of form $n=2p+q$ where $p$ and $q$ are some primes? –  J.A Jun 26 '13 at 15:36
I would say Terry Tao's post is a good explanation of why proofs based on the circle method (such as mine) will not, in and of themselves, work out for the binary Goldbach problem. My comments give some idea of my strategy for the ternary problem (as of May 2012), but I think I've explained things better elsewhere. –  H A Helfgott Dec 24 '13 at 10:22