Sum of three primes Can all natural numbers ($n\ge 6$) be represented as the sum of three primes?
With computer I checked up to $10000$, but couldn't prove it.
 A: It was proved by Vinogradov that every large enough odd integer is the sum of at most $3$ primes, and it seems essentially certain that apart from a few uninteresting small cases, every odd integer is the sum of $3$ primes.
Even integers are a different matter. To prove that every even integer $n$ is the sum of three primes, one would have to prove the Goldbach Conjecture, since one of the three primes must be $2$, and therefore $n-2$ must be the sum of two primes.
A: From the second edition of The Hardy-Littlewood Method by Robert C. Vaughan, the Corollary to Theorem 3.4 on page 33 is that all sufficiently large odd number is the sum of three primes. 
For even numbers and two primes, let $E(n)$ be the count of exceptions up to $n,$ meaning the count of even numbers that are not the sum of two primes. Let $A$ be a positive constant. The Corollary to Theorem 3.7 on page 36 is that there is a positive constant $C$ (where $C$ depends on $A$) such that $$  E(n) \leq C n (\log n)^{-A}.  $$ So failures are eventually uncommon. I don't see that taking $p_1 + p_2 + 2$ improves matters very much.
Vaughan also refers to a book by one of  the main investigators in this area, Additive Theory of Prime Numbers by L.-K. Hua (1965).
