# 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

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

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 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 WaringsProblem 2013-03-22 13:19:46 2013-03-22 13:19:46 bbukh (348) bbukh (348) 13 bbukh (348) Theorem msc 11P05 msc 11B13 LagrangesFourSquareTheorem Basis2