“Waring's Problem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Waring's Problem

释义

Waring's Problem

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

In Lagrange proved that , where 4 may be reduced to 3 except fornumbers of the form (as proved by Legendre ). In the early twentieth century, Dickson, Pillai, andNiven proved that . Hilbert, Hardy, and Vinogradov proved , and thiswas subsequently reduced to by Balasubramanian et al. (1986). Liouville proved (using Lagrange'sFour-Square Theorem and Liouville Polynomial Identity) that , and this was improved to 47, 45, 41, 39,38, and finally by Wieferich. See Rademacher and Toeplitz (1957, p. 56) for a simple proof. J.-J. Chen (1964)proved that .

Dickson, Pillai, and Niven also conjectured an explicit formula for for (Bell 1945), based on the relationship

(1)

If the Diophantine (i.e.,

is restricted to being an Integer) inequality

(2)

is true, then

(3)

This was given as a lower bound by Euler,

and has been verified to be correct for

. Since1957, it has been known that at most a Finite number of

exceed Euler's

lower bound.

There is also a related problem of finding the least Integer such that every Positive Integer beyond acertain point (i.e., all but a Finite number) is the Sum of th Powers. From1920-1928, Hardy and Littlewood showed that

(4)

and conjectured that

(5)

The best currently known bound is

(6)

for some constant

. Heilbronn (1936) improved Vinogradov's results to obtain

(7)

It has long been known that

. Dickson and Landau proved that the only Integers requiring nineCubes are 23 and 239, thus establishing

. Wieferich proved that only 15Integers require eight Cubes: 15, 22, 50, 114, 167, 175, 186, 212, 213, 238, 303,364, 420, 428, and 454, establishing

. The largest number known requiring seven Cubes is8042. In 1933, Hardy and Littlewood showed that

, but this was improved in 1936 to 16 or 17, and shown to beexactly 16 by Davenport (1939b). Vaughan (1986) greatly improved on the method of Hardy and Littlewood, obtaining improvedresults for

. These results were then further improved by Brüdern (1990), who gave

, and Wooley(1992), who gave

for

to 20. Vaughan and Wooley (1993) showed

Let denote the smallest number such that almost all sufficiently large Integers are thesum of th Powers. Then (Davenport 1939a), (Hardy and Littlewood 1925), (Vaughan 1986), and (Wooley 1992). If the negatives of Powers are permitted inaddition to the powers themselves, the largest number of th Powers needed to represent an arbitraryinteger are denoted and (Wright 1934, Hunter 1941, Gardner 1986). In general, these values aremuch harder to calculate than are and .

The following table gives , , , , and for .The sequence of is Sloane's A002804.


2	4	4	3	3
3	9		[4, 5]
4	19	16	[9, 10]
5	37
6	73
7	143
8	279
9	548
10	1079
11	2132
12	4223
13	8384
14	16673
15	33203
16	66190
17	132055
18	263619
19	526502
20	1051899

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.

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