TITLE: Proof of the weak Goldbach Conjecture
QUESTION [23 upvotes]: What are the main ideas of Harald Helfgott's proof that all odd $n \geq 5$ is the sum of 3 primes?

REPLY [21 votes]: 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/