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# Waring’s problem

Waring asked whether it is possible to represent every natural number as a sum of bounded number of nonnegative $k$’th powers, that is, whether the set $\{\,n^{k}\mid n\in\mathbb{Z_{+}}\,\}$ is an additive basis. He was led to this conjecture by Lagrange’s theorem which asserted that every natural number can be represented as a sum of four squares.

Hilbert [1] was the first to prove the conjecture for all $k$. In his paper he did not give an explicit bound on $g(k)$, the number of powers needed, but later it was proved that

$g(k)=2^{k}+\left\lfloor\left(\frac{3}{2}\right)^{k}\right\rfloor-2$ |

except possibly finitely many exceptional $k$, none of which are known.

Wooley[4], improving the result of Vinogradov[3], proved that the number of $k$’th powers needed to represent all *sufficiently large* integers is

$G(k)\leq k(\ln k+\ln\ln k+O(1)).$ |

# References

- 1 David Hilbert. Beweis für Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n$-ter Potenzen (Waringsches Problem). Math. Ann., pages 281–300, 1909. Available electronically from GDZ.
- 2 Robert C. Vaughan. The Hardy-Littlewood method. Cambridge University Press, 1981. Zbl 0868.11046.
- 3 I. M. Vinogradov. On an upper bound for $G(n)$. Izv. Akad. Nauk SSSR. Ser. Mat., 23:637–642, 1959. Zbl 0089.02703.
- 4 Trevor D. Wooley. Large improvements in Waring’s problem. Ann. Math, 135(1):131–164, 1992. Zbl 0754.11026. Available online at JSTOR.

## Mathematics Subject Classification

11P05*no label found*11B13

*no label found*

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