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
 |
(1)
|
Here 
is the floor function. If, for integers 
, the inequality
![frac[(3/2)^n]<=1-(3/4)^n](/images/equations/WaringsProblem/NumberedEquation2.svg) |
(2)
|
is true, where 
is the fractional part of 
, then
 |
(3)
|
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 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)
 |
(4)
|
The expected values of 
imply the following conjectural bounds for 
(Vaughan and Wooley 2002)
 |
(5)
|
Heilbronn (1936) improved results by Vinogradov to obtain
![G(n)<=6nlnn+[4+3ln(3+2/n)]n+3.](/images/equations/WaringsProblem/NumberedEquation6.svg) |
(6)
|
If 
,
then
 |
(7)
|
(Karatsuba 1985). For each fixed 
,
 |
(8)
|
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. 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.
 |  |  |  |  |  |
| 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 |  |
."
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Acta Math. 71, 123-143, 1939a.Davenport, H. "On Waring's
Problem for Fourth Powers." Ann. Math. 40, 731-747, 1939b.Deshouillers,
J.-M.; Hennecart, F.; and Landreau, B. "7 373 170 279 850." Math. Comput. 69,
421-439, 2000.Dickson, L. E. "Waring's Problem and Related
Results." Ch. 25 in 
in Waring's Problem." Izv. Akad. Nauk SSSR Ser.
Mat. 49, 935-947 and 1119, 1985.Khinchin, A. Y. "An
Elementary Solution of Waring's Problem." Ch. 3 in 
." Math. Comput. 55,
815-820, 1990.Landau, E. "Über eine Anwendung der Primzahlen
auf das Waringsche Problem in der elementaren Zahlentheorie." Math. Ann. 66,
102-105, 1909.Linnik, U. V. "On the Representation of Large
Numbers as Sums of Seven Cubes." Mat. Sbornik N.S. 12, 218-224,
1943.Mahler, K. "On the Fractional Parts of the Powers of a Rational
Number (II)." Mathematika 4, 122-124, 1957.Rademacher,
H. and Toeplitz, O. 
." Math. Comput. 18, 144-146, 1964.