Waring's Problem

Waring's problem asks whether, for every fixed positive integer , there is a number such that every positive integer is a sum of at most th powers of positive integers. Equivalently, zero summands may be used to obtain exactly th powers of nonnegative integers. The notation denotes the least possible number. Waring (1770, 1782) proposed this generalization of Lagrange's four-square theorem in his Meditationes algebraicae and speculated that , , and . Hilbert proved the general conjecture in 1909 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 with and nonnegative integers (as proved by Legendre; Hardy 1999, p. 12). In 1909, Wieferich proved that . In 1859, Liouville proved (using Lagrange's four-square theorem and Liouville polynomial identity) that . Hardy and Littlewood obtained the asymptotic bound (Vaughan and Wooley 2002). The exact value was established by Balasubramanian et al. (1986). For the case , in 1896, Maillet began with a proof that , in 1909 Wieferich proved , and Chen (1964) proved that .

Dickson, Pillai, and Niven studied explicit formulas for (Bell 1945, pp. 318 and 602; Vaughan and Wooley 2002). For , a sufficient condition for the ideal formula below is

Here is the floor function. If, for integers , the inequality

is true, where is the fractional part of , then

Each sufficient condition above is applied separately for a fixed in its stated range. It need only hold at that value of to give the displayed formula for there, with no assumption about other values of .

The expression on the right was given as a lower bound by J. A. Euler, son of Leonhard Euler, and has been verified to be correct for (Kubina and Wunderlich 1990, extending Stemmler 1964). Furthermore, Mahler (1957) proved that at most a finite number of exponents have exceeding Euler's lower bound. Unfortunately, the proof is nonconstructive.

For a fixed integer , the related function is the least number of th powers of positive integers needed to represent every sufficiently large positive integer as a sum of at most that many terms. Hardy and Littlewood obtained the bound (Vaughan and Wooley 2002)

The expected values of imply the following conjectural bounds for (Vaughan and Wooley 2002)

Heilbronn (1936) improved results by Vinogradov to obtain

If , then

(Karatsuba 1985). For each fixed ,

for all sufficiently large , with the threshold allowed to depend on (Wooley 1992).

It has long been known that . Deshouillers et al. (2000) conjectured that 7373170279850 is the largest integer that cannot be expressed as the sum of four nonnegative cubes.

Landau established that , and Linnik (1943) improved this to (Siksek 2016). Dickson (1939, as reported by Siksek 2016) showed that the only integers requiring nine cubes are 23 and 239. Exactly 15 integers require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454 (Wells 1986, p. 70; Siksek 2016). The largest number known requiring seven cubes is 8042. Siksek (2016) proved that all integers greater than 454 are the sum of at most seven positive cubes. The complete set of exceptional numbers requiring more than 7 positive cubes is 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, and 454 (OEIS A018888), as conjectured by Jacobi (1851).

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 upper bounds for for to 20. Vaughan and Wooley (1993ab) showed .

Let denote the least number such that the positive integers expressible as a sum of at most th powers of positive integers have natural density 1. The classical value is (Vaughan and Wooley 2002). Also, (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 OEIS A002804.