Binary vs. Ternary Goldbach Conjecture 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...

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.