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

7$\begingroup$ 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. $\endgroup$– Geoff RobinsonMay 14, 2013 at 7:02

6$\begingroup$ Your question seems suitable for a blog. Since Terry Tao already has a blog discussion on this topic, I've voted to close. $\endgroup$– Ryan BudneyMay 14, 2013 at 11:39

9$\begingroup$ It would be nice to have the possibility to downvote (or upvote) the actions of closing some of the questions. $\endgroup$– Cristi StoicaMay 14, 2013 at 13:05

30$\begingroup$ 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... $\endgroup$– H A HelfgottMay 14, 2013 at 16:58

6$\begingroup$ It seems to me (an algebraic number theorist, mainly), that analytic number theory, which appeared to me at a little bit sleepy when I was in grad school (at the ENS), is now experiencing an impressive blossom a little bit similar that the one that algebraic number theory experiences after Wiles' proof of Fermat. I mean, in the last ten years, the proof of the existence or arbitrary long sequence in primes, of infinitely many bounded gaps between primes, of the weak Goldbach conjecture, and so many other beautiful things. It is very heartening, as was Wiles' announcement. $\endgroup$– JoëlMay 19, 2013 at 16:12
3 Answers
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/heuristiclimitationsofthecirclemethod/

$\begingroup$ Note that the blog post was written a year ago. It is still relevant, though (particularly the comments by Helfgott). $\endgroup$ May 15, 2013 at 15:45

$\begingroup$ Does this also show every odd number $n > 5$ is of form $n=2p+q$ where $p$ and $q$ are some primes? $\endgroup$– TurboJun 26, 2013 at 15:36

4$\begingroup$ 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. $\endgroup$ Dec 24, 2013 at 10:22

$\begingroup$ This link does not work anymore, can give alternative source? $\endgroup$– MichaelOct 14, 2020 at 10:33
It needs to be iterated once again, that Vinogradov showed in 1937 that all large enough odd numbers are sum of three primes. The current contribution of Helfgott merely aims at bridging the gap between large enough and all number.
This is an interesting problem. However whereas Vinogradov's proof introduced the fundamentally new idea of bilinear forms, Helfgott contribution is on a much smaller scale. While it contributes to the particular subfield of analytic number theory concerned with explicit estimates, it most likely does not contribute to the larger field, and instead uses idea that were around for a long time.

8$\begingroup$ "It needs to be iterated once again,..." Why? And, you do not answer the question. $\endgroup$– user9072May 14, 2013 at 10:15

13$\begingroup$ But pooper is at least living up to his or her name! $\endgroup$ May 14, 2013 at 11:42

36$\begingroup$ Not to be an antipooper, but, actually, most of the ideas and improvements in my proof are qualitative rather than quantitative. I'm not an explicit person by training, and I have no doubt that careful specialists could improve on some of the constants within the proof by being more clever than I was. One of several "morals" of the story (not really new, but more people should be aware of it) is the close relationship between bilinear forms, the circle method and the large sieve. Of course, Vinogradov was working before the development of the large sieve. $\endgroup$ May 14, 2013 at 16:57

27$\begingroup$ I actually disagree (in good faith I think) with pooper: I do think analytic number theorists who don't care a fig about explicit constants would find several things in what I've done to be of interest. I just gave a talk highlighting the more conceptual bits. At any rate, I'll make the obvious point that not just Vinogradov but also Hardy and Littlewood deserve credit as initiators here. $\endgroup$ May 14, 2013 at 20:43

17$\begingroup$ Er, yes. See section 1.2, "History", of majarcs.pdf, and section 1.2, "History", of minarcs.pdf, as well as the first and second paragraphs of each paper. $\endgroup$ May 15, 2013 at 11:53