As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (), that every "sufficiently large" integer is a sum of no more than 7 positive cubes (; although it is not known if 7 can be reduced), and that every integer is a sum of at most 5 signed cubes ( ; although it is not known if 5 can be reduced to 4).

It is known that every n can be written is the form

An elliptic curve of the form for n an integer is known as a Mordell curve.

The 3.1.2 equation

is a case of Fermat's last theorem with n = 3. In fact, this particular case was known not to have any solutions long before the general validity of Fermat's last theorem was established. Thue showed that a Diophantine equation of the form

for A, B, and l integers, has only finite many solutions (Hardy 1999, pp. 78-79).

Miller and Woollett (1955) and Gardiner et al. (1964) investigated integer solutions of

i.e., numbers representable as the sum of three (positive or negative) cubic numbers.

The general rational solution to the 3.1.3 equation

was found by Euler and Vieta (Dickson 1966, pp. 550-554; Hardy 1999, pp. 20-21). Hardy and Wright (1979, pp. 199-201) give a solution which can be based on the identities

This is equivalent to the general 3.2.2 solution found by Ramanujan (Dickson 1966, pp. 500 and 554; Berndt 1994, pp. 54 and 107; Hardy 1999, p. 11, 68, and 237). The smallest integer solutions are

(Fredkin 1972; Madachy 1979, pp. 124 and 141; Dutch). Other general solutions have been found by Binet (1841) and Schwering (1902), although Ramanujan's formulation is the simplest. No general solution giving all positive integral solutions is known (Dickson 1966, pp. 550-561). Y. Kohmoto has found a 3.1.3⁹ solution,

3.1.4 equations include

3.1.5 equations include

and a 3.1.6 equation is given by

The 3.2.2 equation

has a known parametric solution (Dickson 1966, pp. 550-554; Guy 1994, p. 140), and 10 solutions with sum ,

(Sloane's A001235; Moreau 1898). The first number (Madachy 1979, pp. 124 and 141) in this sequence, the so-called Hardy-Ramanujan number, is associated with a story told about Ramanujan by G. H. Hardy, but was known as early as 1657 (Berndt and Bhargava 1993). The smallest number representable in n ways as a sum of cubes is called the nth taxicab number.

Ramanujan gave a general solution to the 3.2.2 equation as

where

(Berndt 1994, p. 107). Another form due to Ramanujan is

	(49)

Hardy and Wright (1979, Theorem 412) prove that there are numbers that are expressible as the sum of two cubes in n ways for any n (Guy 1994, pp. 140-141). The proof is constructive, providing a method for computing such numbers: given rationals numbers r and s, compute

Then

The denominators can now be cleared to produce an integer solution. If is picked to be large enough, the v and w will be positive. If is still larger, the will be large enough for v and w to be used as the inputs to produce a third pair, etc. However, the resulting integers may be quite large, even for n = 2. E.g., starting with , the algorithm finds

giving

The numbers representable in three ways as a sum of two cubes (a 3.2³ equation) are

(Guy 1994, Sloane's A003825). Wilson (1997) found 32 numbers representable in four ways as the sum of two cubes (a 3.2⁴ equation). The first is

	(63)

The smallest known numbers so representable are 6963472309248, 12625136269928, 21131226514944, 26059452841000, ... (Sloane's A003826). Wilson also found six five-way sums,

	(64)
	(65)
	(66)
	(67)
	(68)
	(69)

and a single six-way sum

	(70)

A solution to the 3.4.4 equation is

(Madachy 1979, pp. 118 and 133).

3.6.6 equations also exist:

	(72)
	(73)

(Madachy 1979, p. 142; Chen Shuwen).

In 1756-1757, Euler (1761, 1849, 1915) gave a parametric solution to

as

although relatively prime solutions require the use of fractional values of n (Dickson 1966, p. 578). To avoid this, Euler also gave the solutions

for , and

for (Dickson 1966, p. 579).

 

Cannonball Problem, Cubic Number, Hardy-Ramanujan Number, Multigrade Equation, Super-d Number, Taxicab Number, Trimorphic Number, Waring's Problem