In his Meditationes algebraicae, Waring (1770, 1782) 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 positive 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; 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 Little established , and this was subsequently reduced to 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 .
In 1936, Dickson, Pillai, and Niven also conjectured an explicit formula for for (Bell 1945, pp. 318 and 602), based on the relationship
If the Diophantine (i.e., is restricted to being an integer) inequality
is true, where is the fractional part of , then
This 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 1990). Furthermore, Mahler (1957) proved that at most a finite number of exceed Euler's lower bound. Unfortunately, the proof is nonconstructive.
There is also a related (but more difficult) 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
and conjectured that
Heilbronn (1936) improved results by Vinogradov to obtain
If , then
(Karatsuba 1985), and for large ,
for any positive (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 (1909) established that , and in 1939 Dickson showed that the only integers requiring nine cubes are 23 and 239. Wieferich proved that only 15 integers require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454 (OEIS A018889), establishing (Wells 1986, p. 70). 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 (1993ab) 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 OEIS A002804.