Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current techniques? As a non-expert it seems to me that these conjectures should be equally hard to prove, which must of course not be true since these statements are not equivalent, but they are of the same "nature" and I don't see why it is such a difference that a number is the sum of two or of three primes...

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    $\begingroup$ A timely question. arxiv.org/abs/1305.2897 $\endgroup$ – zyx May 14 '13 at 10:13
  • $\begingroup$ yes, I was just goint through this paper...that's why I came up with this question $\endgroup$ – Andy Teich May 14 '13 at 10:14

The "current methods" have almost always been refinements of Hardy-Littlewood circle method. Terry Tao has a blog post where he describes how the circle method applies to the problem, and why experts think this method alone will not yield the even Goldbach conjecture.

  • $\begingroup$ so there is no "easy" answer to my question... $\endgroup$ – Andy Teich May 14 '13 at 10:10
  • $\begingroup$ @AndyTeich To see the limitations of any method it is always necessary to know at least the basics of how the method works and is applied so that we can see where the argument breaks down. So it can't even much easier than that I'm afraid. $\endgroup$ – Ragib Zaman May 14 '13 at 10:14
  • $\begingroup$ I know some of the techniques, but still, I am a non-expert...but I was wondering if there is also some easier (more "intuitive", whatever intuitive means) explanation $\endgroup$ – Andy Teich May 14 '13 at 10:18
  • $\begingroup$ So my question is not really why these methods don't work in the other case, but more why is it that binary conjecture is harder than the ternary conjcecture... $\endgroup$ – Andy Teich May 14 '13 at 10:23
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    $\begingroup$ Section 2 of Tao's post strikes me as pretty elementary, and is roughly what I would say, but written better. $\endgroup$ – David E Speyer May 14 '13 at 18:06

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