Vinogradov's theorem

In mathematics, in the field of number theory, Vinogradov's theorem states that any sufficiently large odd integer can be written as a sum of three prime numbers. It is named after Ivan Matveyevich Vinogradov who proved it in the 1930s.

tatement of Vinogradov's theorem

Let "A" be a positive real number. Then:$r(N)=\{1over\; 2\}G(N)N^2+Oleft(N^2log^\{-A\}N\; ight),$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 $Lambda$, and:$G(N)=left(prod\_\{pmid\; N\}left(1-\{1over\{left(p-1\; ight)\}^2\}\; ight)\; ight)left(prod\_\{p\; mid\; N\}left(1+\{1over\{left(p-1\; ight)\}^3\}\; ight)\; ight).$

A consequence

If "N" is odd, then "G"("N") is roughly 1, hence $N^2=Oleft(r(N)\; ight)$ for all sufficiently large "N". By showing that the contribution made to "r"("N") by proper prime powers is $Oleft(N^\{3over\; 2\}log^2N\; ight)$, one sees that:$N^2log^\{-3\}N=Oleft(hbox\{number\; of\; ways\; N\; can\; be\; written\; as\; a\; sum\; of\; three\; primes\}\; ight).$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.

ee also

*Bombieri–Vinogradov theorem

References

*
* Chapter 8.

External links

*MathWorld|urlname=VinogradovsTheorem|title=Vinogradov's Theorem