# Goldbach-Waring problem

From Encyclopedia of Mathematics

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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%E2%80%93Waring_problem&oldid=22520