Waring problem
2020 Mathematics Subject Classification: Primary: 11P05 [MSN][ZBL]
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 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 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, 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 and [HaWr]–[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
[De] | B.N. Delone, "The Peterburg school of number theory", Moscow-Leningrad (1947) (In Russian) |
[Hu] | 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) |
[Kh] | A.Ya. Khinchin, "Three pearls of number theory", Graylock (1952) (Translated from Russian) |
[Vi] | I.M. Vinogradov, "Selected works", Springer (1985) (Translated from Russian) |
[Vi2] | I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers", Interscience (1954) (Translated from Russian) |
[HaWr] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) pp. Chapt. 6 |
[Sh] | D. Shanks, "Solved and unsolved problems in number theory", Chelsea, reprint (1978) |
[Va] | R.C. Vaughan, "The Hardy–Littlewood method", Cambridge Univ. Press (1981) |
[Wo] | T. D. Wooley, "Vinogradov's mean value theorem via efficient congruencing", Annals of Math. 175 (2012), 1575--1627. |
Waring problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Waring_problem&oldid=27030