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Difference between revisions of "Goldbach-Waring problem"

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A problem concerning the behaviour of the number of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445601.png" /> of the equation
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A problem concerning the behaviour of the number of solutions $I_k(N)$ 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/g/g044/g044560/g0445602.png" /></td> </tr></table>
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$$p_1^n+\ldots+p_k^n=N,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445603.png" /> are prime numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445604.png" /> (cf. [[Waring problem|Waring problem]]; [[Goldbach problem|Goldbach problem]]). The results obtained (up to 1977) are roughly the same as those obtained in Waring's problem: It has been proved that the equation is solvable (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445605.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445606.png" />, while an asymptotic formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445607.png" /> has been obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044560/g0445608.png" />. These solutions were obtained by the [[Vinogradov method|Vinogradov method]].
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where $p_1,\ldots,p_k$ are prime numbers and $n\geq1$ (cf. [[Waring problem|Waring problem]]; [[Goldbach problem|Goldbach problem]]). The results obtained (up to 1977) are roughly the same as those obtained in Waring's problem: It has been proved that the equation is solvable (i.e. $I_k(N)>0$) if $k=O(n\log n)$, while an asymptotic formula for $I_k(N)$ has been obtained for $k=O(n^2\log n)$. These solutions were obtained by the [[Vinogradov method|Vinogradov method]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>

Revision as of 11:46, 10 August 2014

A problem concerning the behaviour of the number of solutions $I_k(N)$ of the equation

$$p_1^n+\ldots+p_k^n=N,$$

where $p_1,\ldots,p_k$ are prime numbers and $n\geq1$ (cf. Waring problem; Goldbach problem). The results obtained (up to 1977) are roughly the same as those obtained in Waring's problem: It has been proved that the equation is solvable (i.e. $I_k(N)>0$) if $k=O(n\log n)$, while an asymptotic formula for $I_k(N)$ has been obtained for $k=O(n^2\log n)$. These solutions were obtained by the Vinogradov method.

References

[1] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (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:
Goldbach-Waring problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Goldbach-Waring_problem&oldid=32795
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article