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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asked May 14, 2013 at 5:19
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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 RobinsonCommented May 14, 2013 at 7:02
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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 BudneyCommented May 14, 2013 at 11:39
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9$\begingroup$ It would be nice to have the possibility to downvote (or upvote) the actions of closing some of the questions. $\endgroup$– Cristi StoicaCommented May 14, 2013 at 13:05
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31$\begingroup$ 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 HelfgottCommented May 14, 2013 at 16:58
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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. $\endgroup$– JoëlCommented May 19, 2013 at 16:12
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3 Answers
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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/
answered May 14, 2013 at 11:26
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$\begingroup$ Note that the blog post was written a year ago. It is still relevant, though (particularly the comments by Helfgott). $\endgroup$ Commented May 15, 2013 at 15:45
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$\begingroup$ Does this also show every odd number $n > 5$ is of form $n=2p+q$ where $p$ and $q$ are some primes? $\endgroup$– TurboCommented Jun 26, 2013 at 15:36
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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. $\endgroup$ Commented Dec 24, 2013 at 10:22
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answered May 14, 2013 at 5:54
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1$\begingroup$ This link does not work anymore, can give alternative source? $\endgroup$– MichaelCommented Oct 14, 2020 at 10:33
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$\begingroup$ The link in your answer is broken. Would it be possible that you replace it by an up-to-date link -- and also elaborate a bit more, such that this is not just a link-only answer? $\endgroup$– Stefan Kohl ♦Commented Oct 9, 2023 at 18:55
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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 where-as 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 sub-field 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.
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8$\begingroup$ "It needs to be iterated once again,..." Why? And, you do not answer the question. $\endgroup$– user9072Commented May 14, 2013 at 10:15
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13$\begingroup$ But pooper is at least living up to his or her name! $\endgroup$ Commented May 14, 2013 at 11:42
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38$\begingroup$ Not to be an anti-pooper, 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. $\endgroup$ Commented May 14, 2013 at 16:57
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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. $\endgroup$ Commented May 14, 2013 at 20:43
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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. $\endgroup$ Commented May 15, 2013 at 11:53