Difference between revisions of "Goldbach-Waring problem"
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| − | A problem concerning the behaviour of the number of solutions | + | {{TEX|done}} |
| + | A problem concerning the behaviour of the number of solutions $I_k(N)$ of the equation | ||
| − | + | $$p_1^n+\dotsb+p_k^n=N,$$ | |
| − | where | + | where $p_1,\dotsc,p_k$ 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> | ||
| + | |||
| + | [[Category:Number theory]] | ||
Latest revision as of 13:22, 14 February 2020
A problem concerning the behaviour of the number of solutions $I_k(N)$ of the equation
$$p_1^n+\dotsb+p_k^n=N,$$
where $p_1,\dotsc,p_k$ 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=22519
Goldbach-Waring problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Goldbach-Waring_problem&oldid=22519
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article