Difference between revisions of "Vinogradov-Goldbach theorem"
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| − | A theorem on the representation of all sufficiently large odd numbers by a sum of three prime numbers. It is a consequence of the asymptotic formula for the number | + | {{TEX|done}} |
| + | A theorem on the representation of all sufficiently large odd numbers by a sum of three prime numbers. It is a consequence of the asymptotic formula for the number $I(N)$ of solutions of the equation | ||
| − | + | $$p_1+p_2+p_3=N$$ | |
in prime numbers, demonstrated by I.M. Vinogradov in 1937: | in prime numbers, demonstrated by I.M. Vinogradov in 1937: | ||
| − | + | $$I(N)=\frac{N^2}{2r^3}S(N)+O\left(\frac{N^2}{r^{3.5-\epsilon}}\right),$$ | |
| − | where | + | where $N$ is odd, $r=\log N$ and |
| − | + | $$S(N)=\prod_p\left(1+\frac{1}{(p-1)^3}\right)\prod_{p|N}\left(1-\frac{1}{p^2-3p+3}\right)>0.6.$$ | |
Cf. [[Vinogradov method|Vinogradov method]]; [[Goldbach problem|Goldbach problem]]. | Cf. [[Vinogradov method|Vinogradov method]]; [[Goldbach problem|Goldbach problem]]. | ||
Latest revision as of 15:21, 10 August 2014
A theorem on the representation of all sufficiently large odd numbers by a sum of three prime numbers. It is a consequence of the asymptotic formula for the number $I(N)$ of solutions of the equation
$$p_1+p_2+p_3=N$$
in prime numbers, demonstrated by I.M. Vinogradov in 1937:
$$I(N)=\frac{N^2}{2r^3}S(N)+O\left(\frac{N^2}{r^{3.5-\epsilon}}\right),$$
where $N$ is odd, $r=\log N$ and
$$S(N)=\prod_p\left(1+\frac{1}{(p-1)^3}\right)\prod_{p|N}\left(1-\frac{1}{p^2-3p+3}\right)>0.6.$$
Cf. Vinogradov method; Goldbach problem.
References
| [1] | I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) |
| [2] | L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1) |
How to Cite This Entry:
Vinogradov-Goldbach theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov-Goldbach_theorem&oldid=23105
Vinogradov-Goldbach theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov-Goldbach_theorem&oldid=23105
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article