Difference between revisions of "Goldbach-Waring problem"
From Encyclopedia of Mathematics
(TeX) |
(Category:Number theory) |
||
Line 7: | Line 7: | ||
====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> | ||
+ | |||
+ | [[Category:Number theory]] |
Revision as of 18:53, 18 October 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
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