What are the main ideas of Harald Helfgott's proof that all odd $n \geq 5$ is the sum of 3 primes?
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.
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.
|show 4 more comments|