Goldbach's Weak Conjecture

Question

I have a few questions on GWC, as the Wikipedia's page on it appears to be somewhat incomplete.

1. Which of the following two statements is considered as the actual GWC?

   • Every odd number greater than 5 can be expressed as the sum of three primes
   • Every odd number greater than 7 can be expressed as the sum of three odd primes

   The first statement is easily implied from GSC, but the second statement doesn't appear to be so (at least not easily)... So unless I'm overlooking something here, these two statements seem far from being equivalent to each other.

2. Has it really been proved or only claimed proved by Harald Helfgott?

   • This page on Wikipedia says that it has been proved by Harald Helfgott
   • This page on Wikipedia says that it has been claimed proved by Harald Helfgott

Thanks

Answer 1

For 2.: actually he says he proved such a result, but I am not aware whether his result was validated. Here two of his works:

http://arxiv.org/pdf/1404.2224.pdf

http://arxiv.org/pdf/1312.7748.pdf

However, if $n>4$ even, then GSC implies that exists odd primes $p$ and $q$ such that $n=p+q$. Thus, if $m>7$ odd, then $n=m-3>4$ is even, then $m=p+q+3$.