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A problem in number theory formulated in 1770 by E. Waring in the following form: Any natural number is a sum of 4 squares, of 9 cubes and of 19 fourth-powers. In other words, for all $k\geq2$ there exists a $s=s(k)$, depending only on $k$, such that every natural number is the sum of $s$ $k$-th powers of non-negative integers. D. Hilbert in 1909 was the first to give a general solution of Waring's problem with a very rough estimate of the value of $s$ as a function of $k$; this is why the problem is sometimes known as the Hilbert–Waring problem. Let $J_{s,k}(N)$ be the number of solutions of the equation

\begin{equation}\label{war}x_1^k+\cdots+x_s^k=N\end{equation}

in non-negative integers. Hilbert's theorem then states that there exists a $s=s(k)$ for which $J_{s,k}(N)\geq1$ for any $N\geq1$. G.H. Hardy and J.E. Littlewood, who applied the [[Circle method|circle method]] to the Waring problem, demonstrated in 1928 that for $s\geq(k-2)2^{k-1}+5$ the value of $J_{s,k}(N)$ is given by an asymptotic formula of the type

\begin{equation}\label{asym}J_{s,k}(N)=AN^{s/k-1}+O(N^{s/k-1-\gamma}),\end{equation}

where $A=A(N)\geq c_0>0$, while $c_0$ and $\gamma>0$ are constants. Consequently, if $N\geq N_0(k)$, equation \ref{war} has a solution.

An elementary proof of Waring's problem was given in 1942 by Yu. V. Linnik. There exist many different generalizations of Waring's problem (the variables run through a certain subset of the set of natural numbers; the number $N$ is represented by polynomials $f_1(x_1),\ldots,f_s(x_s)$ rather than by monomials $x_1^k,\ldots,x_s^k$; equation (1) is replaced by a congruence, etc.).

Research on Waring's problem has mainly focused on sharpening estimates for the following three questions: 
# Find the smallest $s$ such that \ref{war} has solutions for all sufficiently large $N$;
# Find the smallest $s$ such that \ref{war} has solutions for all $N$;
# Find the smallest $s$ such that the number of solutions to \ref{war}, $J_{s,k}(N)$, is given by the asymptotic formula \ref{asym}.

These quantities are known as $G(k)$, $g(k)$, and $\tilde{G}(k)$ respectively. Clearly, $\tilde{G}(k)\geq G(k)$ and $g(k)\geq G(k)$. The progress on bounds for these quantities is detailed below.

== Solvable for $N$ sufficiently large ==
Let $G(k)$ be the smallest integer such that equation \ref{war} is solvable for $s\geq G(k)$ and $N$ sufficiently large depending on $k$.

It is known that $G(k)\geq k+1$. It was proved in 1934 by I.M. Vinogradov, using his own method, that

$$G(k)\leq 3k(\ln k+9).$$

Moreover, many results are available concerning $G(k)$ for small values of $k$: $G(4)=16$ (H. Davenport, 1939); $G(3)=7$ (Yu.V. Linnik, 1942).

== Solvable for all $N$ ==

Let $g(k)$ be the smallest integer such that equation \ref{war} is solvable for $s\geq g(k)$ and $N\geq1$.

It was shown in 1936 by L. Dickson and S. Pillai, who also used the [[Vinogradov method|Vinogradov method]], that

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

for all $k>6$  for which

$$\left(\frac{3}{2}\right)^k-\left[\left(\frac{3}{2}\right)^k\right]\leq1-\left(\frac{1}{2}\right)^k\left\{\left[\left(\frac{3}{2}\right)^k\right]+2\right\}.$$

The last condition was demonstrated in 1957 by K. Mahler for all sufficiently large $k$. 

It is known that $g(2)=4$ (J.L. Lagrange, 1770), $g(3)=9$ (A. Wieferich, A. Kempner, 1912), $g(4)=19$ (R. Balusabramanian, J. Deshouillers, F. Dress, 1986), $g(5)=37$ (Chen-Jingrun, 1964). See also [[Circle method|Circle method]] and {{Cite|HaWr}}–{{Cite|Sh}}.

== Asymptotic formula ==
Let $\tilde{G}(k)$ be the smallest integer such that the asymptotic formula \ref{asym} applies to $J_{s,k}(N)$ if $s\geq \tilde{G}(k)$. The result of Hardy and Littlewood mentioned above shows that

$$\tilde{G}(k)\leq(k-2)2^{k-1}+5.$$

The first substantial improvement for large values of $k$ was obtained by Vinogradov, who showed that 

$$\tilde{G}(k)\leq 4k^2\ln k.$$

The current best bound for large values of $k$ was obtained by Wooley who showed that

$$\tilde{G}(k)\leq 2k^2-k^{4/3}+O(k).$$

==References==
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|valign="top"|{{Ref|De}}||valign="top"| B.N. Delone, "The St Petersburg school of number theory", Moscow-Leningrad (1947) (In Russian) {{ZBL|0033.10403}} Translated, American Mathematical Society (2005) {{ISBN|0-8218-3457-6}} {{ZBL|1074.11002}}
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|valign="top"|{{Ref|Hu}}||valign="top"| L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie", ''Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'', '''1''' : 2 (1959) (Heft 13, Teil 1)
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|valign="top"|{{Ref|Kh}}||valign="top"| A.Ya. Khinchin,   "Three pearls of number theory" , Graylock  (1952) Translation from the second, revised Russian ed. [1948] {{ZBL|0048.27202}} Reprinted Dover (2003) {{ISBN|0486400263}}
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|valign="top"|{{Ref|Vi}}||valign="top"| I.M. Vinogradov, "Selected works", Springer (1985) (Translated from Russian)
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|valign="top"|{{Ref|Vi2}}||valign="top"| I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers", Interscience (1954) (Translated from Russian)
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|valign="top"|{{Ref|HaWr}}||valign="top"| G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) pp. Chapt. 6
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|valign="top"|{{Ref|Sh}}||valign="top"| D. Shanks, "Solved and unsolved problems in number theory", Chelsea, reprint (1978)
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|valign="top"|{{Ref|Va}}||valign="top"| R.C. Vaughan, "The Hardy–Littlewood method", Cambridge Univ. Press (1981)
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|valign="top"|{{Ref|Wo}}||valign="top"| T. D. Wooley, "Vinogradov's mean value theorem via efficient congruencing", Annals of Math. 175 (2012), 1575--1627.
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