Waring's Problem

Waring proposed a generalization of Lagrange's Four-Square Theorem, stating that every Rational Integer is the sum of a fixed number $g(n)$ of $n$th Powers of Integers, where $n$ is any given Positive Integer and $g(n)$ depends only on $n$. Waring originally speculated that $g(2)=4$, $g(3)=9$, and $g(4)=19$. In 1909, Hilbert proved the general conjecture using an identity in 25-fold multiple integrals (Rademacher and Toeplitz 1957, pp. 52-61).

In Lagrange's Four-Square Theorem, Lagrange proved that $g(2)=4$, where 4 may be reduced to 3 except for numbers of the form $4^n(8k+7)$ (as proved by Legendre ). In the early twentieth century, Dickson, Pillai, and Niven proved that $g(3)=9$. Hilbert, Hardy, and Vinogradov proved $g(4)\leq 21$, and this was subsequently reduced to $g(4)=19$ by Balasubramanian et al. (1986). Liouville proved (using Lagrange's Four-Square Theorem and Liouville Polynomial Identity) that $g(5)\leq 53$, and this was improved to 47, 45, 41, 39, 38, and finally $g(5)\leq 37$ by Wieferich. See Rademacher and Toeplitz (1957, p. 56) for a simple proof. J.-J. Chen (1964) proved that $g(5)=37$.

Dickson, Pillai, and Niven also conjectured an explicit formula for $g(s)$ for $s>6$ (Bell 1945), based on the relationship

$$\left({3\over 2}\right)^n-\left\lfloor{\left({3\over 2}\righ... ...eft\lfloor{\left({3\over 2}\right)^n+2}\right\rfloor \right\}.$$ (1)

If the Diophantine (i.e., $n$ is restricted to being an Integer) inequality

$$\left\{{\left({3\over 2}\right)^n}\right\} \leq 1-\left({3\over 4}\right)^n$$ (2)

is true, then

$$g(n)=2^n+\left\lfloor{\left({3\over 2}\right)^n}\right\rfloor -2.$$ (3)

This was given as a lower bound by Euler, and has been verified to be correct for $6\leq n\leq 200,000$. Since 1957, it has been known that at most a Finite number of $n$ exceed Euler's lower bound.

There is also a related problem of finding the least Integer $n$ such that every Positive Integer beyond a certain point (i.e., all but a Finite number) is the Sum of $G(n)$ $n$th Powers. From 1920-1928, Hardy and Littlewood showed that

$$G(n)\leq (n-2)2^{n-1}+5$$ (4)

and conjectured that

$$G(k)<\cases{ 2k+1 & for $k$\ not a power of 2\cr 4k & for $k$\ a power of 2.\cr}$$ (5)

The best currently known bound is

$$G(k)<ck\ln k$$ (6)

for some constant $c$. Heilbronn (1936) improved Vinogradov's results to obtain

$$G(n)\leq 6n\ln n+\left[{4+3\ln\left({3+{2\over n}}\right)}\right]n+3.$$ (7)

It has long been known that $G(2)=4$. Dickson and Landau proved that the only Integers requiring nine Cubes are 23 and 239, thus establishing $G(3)\leq 8$. Wieferich proved that only 15 Integers require eight Cubes: 15, 22, 50, 114, 167, 175, 186, 212, 213, 238, 303, 364, 420, 428, and 454, establishing $G(3)\leq 7$. The largest number known requiring seven Cubes is 8042. In 1933, Hardy and Littlewood showed that $G(4)\leq 19$, but this was improved in 1936 to 16 or 17, and shown to be exactly 16 by Davenport (1939b). Vaughan (1986) greatly improved on the method of Hardy and Littlewood, obtaining improved results for $n\geq 5$. These results were then further improved by Brüdern (1990), who gave $G(5)\leq 18$, and Wooley (1992), who gave $G(n)$ for $n=6$ to 20. Vaughan and Wooley (1993) showed $G(8)\leq 42$.

Let $G^+(n)$ denote the smallest number such that almost all sufficiently large Integers are the sum of $G^+(n)$ $n$th Powers. Then $G^+(3)=4$ (Davenport 1939a), $G^+(4)=15$ (Hardy and Littlewood 1925), $G^+(8)=32$ (Vaughan 1986), and $G^+(16)=64$ (Wooley 1992). If the negatives of Powers are permitted in addition to the powers themselves, the largest number of $n$th Powers needed to represent an arbitrary integer are denoted ${\it eg}(n)$ and ${\it EG}(n)$ (Wright 1934, Hunter 1941, Gardner 1986). In general, these values are much harder to calculate than are $g(n)$ and $G(n)$.

The following table gives $g(n)$, $G(n)$, $G^+(n)$, ${\it eg}(n)$, and ${\it EG}(n)$ for $n\leq 20$. The sequence of $g(n)$ is Sloane's A002804.

$n$	$g(n)$	$G(n)$	$G^+(n)$	${\it eg}(n)$	${\it EG}(n)$
2	4	4		3	3
3	9	$\leq 7$	$\leq 4$	[4, 5]
4	19	16	$\leq 15$	[9, 10]
5	37	$\leq 18$
6	73	$\leq 27$
7	143	$\leq 36$
8	279	$\leq 42$	$\leq 32$
9	548	$\leq 55$
10	1079	$\leq 63$
11	2132	$\leq 70$
12	4223	$\leq 79$
13	8384	$\leq 87$
14	16673	$\leq 95$
15	33203	$\leq 103$
16	66190	$\leq 112$	$\leq 64$
17	132055	$\leq 120$
18	263619	$\leq 129$
19	526502	$\leq 138$
20	1051899	$\leq 146$

See also Euler's Conjecture, Schnirelmann's Theorem, Vinogradov's Theorem

References

Balasubramanian, R.; Deshouillers, J.-M.; and Dress, F. ``Problème de Waring pour les bicarrés 1, 2.'' C. R. Acad. Sci. Paris Sér. I Math. 303, 85-88 and 161-163, 1986.

Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 318, 1945.

Brüdern, J. ``On Waring's Problem for Fifth Powers and Some Related Topics.'' Proc. London Math. Soc. 61, 457-479, 1990.

Davenport, H. ``On Waring's Problem for Cubes.'' Acta Math. 71, 123-143, 1939a.

Davenport, H. ``On Waring's Problem for Fourth Powers.'' Ann. Math. 40, 731-747, 1939b.

Dickson, L. E. ``Waring's Problem and Related Results.'' Ch. 25 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 717-729, 1952.

Gardner, M. ``Waring's Problems.'' Ch. 18 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.

Guy, R. K. ``Sums of Squares.'' §C20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138, 1994.

Hardy, G. H. and Littlewood, J. E. ``Some Problems of Partitio Numerorum (VI): Further Researches in Waring's Problem.'' Math. Z. 23, 1-37, 1925.

Hunter, W. ``The Representation of Numbers by Sums of Fourth Powers.'' J. London Math. Soc. 16, 177-179, 1941.

Khinchin, A. Y. ``An Elementary Solution of Waring's Problem.'' Ch. 3 in Three Pearls of Number Theory. New York: Dover, pp. 37-64, 1998.

Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, 1957.

Stewart, I. ``The Waring Experience.'' Nature 323, 674, 1986.

Vaughan, R. C. ``On Waring's Problem for Smaller Exponents.'' Proc. London Math. Soc. 52, 445-463, 1986.

Vaughan, R. C. and Wooley, T. D. ``On Waring's Problem: Some Refinements.'' Proc. London Math. Soc. 63, 35-68, 1991.

Vaughan, R. C. and Wooley, T. D. ``Further Improvements in Waring's Problem.'' Phil. Trans. Roy. Soc. London A 345, 363-376, 1993a.

Vaughan, R. C. and Wooley, T. D. ``Further Improvements in Waring's Problem III. Eighth Powers.'' Phil. Trans. Roy. Soc. London A 345, 385-396, 1993b.

Wooley, T. D. ``Large Improvements in Waring's Problem.'' Ann. Math. 135, 131-164, 1992.

Wright, E. M. ``An Easier Waring's Problem.'' J. London Math. Soc. 9, 267-272, 1934.