Vinogradov's theorem

The set of Vinogradov, named after Ivan Matveyitch Vinogradov, states that every sufficiently large odd number as the sum of three prime numbers can be represented. The unproved ( ternary ) Goldbach 's conjecture asserts that this is greater than 5 for all odd numbers.

Vinogradov proved this theorem 1937. Previously, Hardy and Littlewood had in 1923 proved that assuming the validity of the generalized Riemann conjecture ( GRH ) all finitely many odd numbers can be represented as a sum of three primes up to. Vinogradov's proof set, however, does not require the validity of GRH.

" Sufficiently large " means in the original proof of Vinogradov is a limit of and in the best known refinement of the sentence still far beyond the means of a computer search for the remaining cases.

Yuri Vladimirovich Linnik gave further evidence in 1946 and Nikolai Grigoryevich Tschudakow 1947.

The exact formulation

Be the number of presentations of a natural number as the sum of three primes. Then stating the proposition that

With

(the left product is over the primes that divide the right over the other primes ).

For straight, and is asymptotically of the order of odd. For sufficiently large odd follows that. → Refer to the proof of Vinogradov used method ( a variant of the circle method ) even trigonometric polynomial.