## 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 of th Powers of Integers, where is any given Positive 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's Four-Square Theorem, Lagrange proved that , where 4 may be reduced to 3 except for numbers of the form (as proved by Legendre ). In the early twentieth century, Dickson, Pillai, and Niven proved that . Hilbert, Hardy, and Vinogradov proved , and this was subsequently reduced to by Balasubramanian et al. (1986). Liouville proved (using Lagrange's Four-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 . Since 1957, 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 a certain point (i.e., all but a Finite number) is the Sum of th Powers. From 1920-1928, 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 nine Cubes are 23 and 239, thus establishing . 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 . The largest number known requiring seven Cubes is 8042. In 1933, Hardy and Littlewood showed that , 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 . 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 the sum of th Powers. Then (Davenport 1939a), (Hardy and Littlewood 1925), (Vaughan 1986), and (Wooley 1992). If the negatives of Powers are permitted in addition to the powers themselves, the largest number of th Powers needed to represent an arbitrary integer are denoted and (Wright 1934, Hunter 1941, Gardner 1986). In general, these values are much 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

References

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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.