# Waring’s problem

Waring asked whether it is possible to represent every natural number^{} as a sum of bounded (http://planetmath.org/BoundedInterval) number of nonnegative $k$’th powers, that is, whether the set $\{{n}^{k}\mid n\in {\mathbb{Z}}_{+}\}$ is an additive basis (http://planetmath.org/Basis2). He was led to this conjecture by Lagrange’s theorem (http://planetmath.org/LagrangesFourSquareTheorem) 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

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

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

$$G(k)\le k(\mathrm{ln}k+\mathrm{ln}\mathrm{ln}k+O(1)).$$ |

## 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 http://gdz.sub.uni-goettingen.de/en/index.htmlGDZ.
- 2 Robert C. Vaughan. The Hardy-Littlewood method. Cambridge University Press, 1981. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0868.11046Zbl 0868.11046.
- 3 I. M. Vinogradov. On an upper bound for $G(n)$. Izv. Akad. Nauk SSSR. Ser. Mat., 23:637–642, 1959. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0089.02703Zbl 0089.02703.
- 4 Trevor D. Wooley. Large improvements in Waring’s problem. Ann. Math, 135(1):131–164, 1992. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0754.11026Zbl 0754.11026. http://links.jstor.org/sici?sici=0003-486X%28199201%292%3A135%3A1%3C131%3ALIIWP%3E2.0.CO%3B2-OAvailable online at http://www.jstor.orgJSTOR.

Title | Waring’s problem |
---|---|

Canonical name | WaringsProblem |

Date of creation | 2013-03-22 13:19:46 |

Last modified on | 2013-03-22 13:19:46 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 13 |

Author | bbukh (348) |

Entry type | Theorem |

Classification | msc 11P05 |

Classification | msc 11B13 |

Related topic | LagrangesFourSquareTheorem |

Related topic | Basis2 |