Vinogradov's Theorem

Vinogradov's theorem , named after Ivan Matveevich Vinogradov , says that any sufficiently large odd number can be represented as the sum of three prime numbers . The so far unproven (ternary) Goldbach conjecture claims that this is true for all odd numbers greater than 5.

Winogradow proved this theorem in 1937. Before that, Hardy and Littlewood had proven in 1923 that assuming the validity of the generalized Riemann Hypothesis (GRH), all but a finite number of odd numbers can be represented as the sum of three prime numbers. Winogradov's proof, however, did not presuppose the validity of the GRH.

In Vinogradov's original proof, however, “sufficiently large” still means a limit of and in the best known refinement of the sentence , far beyond the possibilities of a computer search for the remaining cases. ${\displaystyle n>10^{6800000}}$ ${\displaystyle n>10^{1346}}$

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

Exact formulation

Let be the number of representations of a natural number as the sum of three prime numbers. Then the sentence says that ${\displaystyle r(N)}$ ${\displaystyle N}$

${\displaystyle r(N)={\frac {N^{2}}{2{(\log N)}^{3}}}G(N)+O\left({\frac {N^{2}}{{(\log N)}^{4}}}\right)}$

With

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

(the product on the left goes over the prime divisors of and the product on the right goes over the remaining prime numbers). ${\displaystyle N}$

For straight is , for odd is and asymptotically of the order . For sufficiently large odd it follows that . → See also the trigonometric polynomial for the proof method used by Winogradow (a variant of the circle method ) . ${\displaystyle N}$ ${\displaystyle G(N)=0}$ ${\displaystyle N}$ ${\displaystyle G(N)\geq 1}$ ${\displaystyle {\mathcal {O}}\left(1\right)}$ ${\displaystyle N}$ ${\displaystyle r(N)\geq 1}$

Web links

• Eric W. Weisstein : Vinogradov's Theorem . In: MathWorld (English).
• Kumchev, Tolev: An invitation to additive prime number theory . (PDF) 2005

Individual evidence

1. ↑ In: Dokl.Akad.Nauka SSSR , Volume 15, 1937, p. 291 and in The Method of trigonometrical sums in the theory of numbers , 1947
2. ^ MC Liu, TZ Wang: On the Vinogradov bound in the three primes Goldbach conjecture . In: Acta Arithmetica , Volume 105, 2002, p. 133.