Vinogradov's theorem
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Vinogradov's theorem states that any sufficiently large odd integer can be written as a sum of three prime numbers. Named after Ivan Matveyevich Vinogradov.
[edit] Statement of Vinogradov's theorem
Let A be a positive real number. Then
where
,
using the von Mangoldt function Λ, and
[edit] A consequence
If N is odd, then G(N) is roughly 1, hence 
for all sufficiently large N. By showing that the contribution made to r(N) by proper prime powers is 
, one sees that
This means in particular that any sufficiently large odd integer can be written as a sum of three primes, thus showing Goldbach's weak conjecture for all but finitely many cases.
