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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096660/v0966601.png" /> of solutions of the equation
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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 $I(N)$ of solutions of the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096660/v0966602.png" /></td> </tr></table>
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$$p_1+p_2+p_3=N$$
  
 
in prime numbers, demonstrated by I.M. Vinogradov in 1937:
 
in prime numbers, demonstrated by I.M. Vinogradov in 1937:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096660/v0966603.png" /></td> </tr></table>
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$$I(N)=\frac{N^2}{2r^3}S(N)+O\left(\frac{N^2}{r^{3.5-\epsilon}}\right),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096660/v0966604.png" /> is odd, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096660/v0966605.png" /> and
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where $N$ is odd, $r=\log N$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096660/v0966606.png" /></td> </tr></table>
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$$S(N)=\prod_p\left(1+\frac{1}{(p-1)^3}\right)\prod_{p|N}\left(1-\frac{1}{p^2-3p+3}\right)>0.6.$$
  
 
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=32823
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article