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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 \begin{equation*} g(k)=2^k+\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor-2 \end{equation*}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 \begin{equation*} G(k)\leq k (\ln k + \ln \ln k + O(1)). \end{equation*}
- 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.
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