Waring’s problem

Waring asked whether it is possible to represent every natural numberMathworldPlanetmath as a sum of bounded (http://planetmath.org/BoundedInterval) number of nonnegative k’th powers, that is, whether the set {nkn+} 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


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



  • 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