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

$r(N)={1\over 2}G(N)N^2+O\left(N^2\log^{-A}N\right),$

where

$r(N)=\sum_{k_1+k_2+k_3=N}\Lambda(k_1)\Lambda(k_2)\Lambda(k_3)$ ,

using the von Mangoldt function $Λ$ , and

$G(N)=\left(\prod_{p\mid N}\left(1-{1\over{\left(p-1\right)}^2}\right)\right)\left(\prod_{p\nmid N}\left(1+{1\over{\left(p-1\right)}^3}\right)\right).$

[edit] A consequence

If N is odd, then G(N) is roughly 1, hence $N^2=O\left(r(N)\right)$ for all sufficiently large N. By showing that the contribution made to r(N) by proper prime powers is $O\left(N^{3\over 2}\log^2N\right)$ , one sees that

$N^2\log^{-3}N=O\left(\hbox{number of ways N can be written as a sum of three primes}\right).$

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.