Cube (algebra)

In integers

A cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 60³ are (sequence A000578 in the OEIS):

Geometrically speaking, a positive integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27.

The difference between the cubes of consecutive integers can be expressed as follows:

n³ − (n − 1)³ = 3(n − 1)n + 1.

or

(n + 1)³ − n³ = 3(n + 1)n + 1.

There is no minimum perfect cube, since the cube of a negative integer is negative. For example, −4 × −4 × −4 = −64.

Base ten

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number (8 × 8) and a cube number (4 × 4 × 4). This happens if and only if the number is a perfect sixth power (in this case 2⁶).

The last digits of each 3rd power are:

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:

If the number x is divisible by 3, its cube has digital root 9; that is,

If it has a remainder of 1 when divided by 3, its cube has digital root 1; that is,

If it has a remainder of 2 when divided by 3, its cube has digital root 8; that is,

Waring's problem for cubes

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 2³ + 2³ + 1³ + 1³ + 1³ + 1³ + 1³ + 1³ + 1³.

Fermat's last theorem for cubes

The equation x³ + y³ = z³ has no non-trivial (i.e. xyz ≠ 0) solutions in integers. In fact, it has none in Eisenstein integers.

Both of these statements are also true for the equation x³ + y³ = 3z³.

Sum of first n cubes

The sum of the first n cubes is the nth triangle number squared:

1 3 + 2 3 + ⋯ + n 3 = ( 1 + 2 + ⋯ + n ) 2 = ( n ( n + 1 ) 2 ) 2 .

For example, the sum of the first 5 cubes is the square of the 5th triangular number,

1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 15 2

A similar result can be given for the sum of the first y odd cubes,

1 3 + 3 3 + ⋯ + ( 2 y − 1 ) 3 = ( x y ) 2

but x, y must satisfy the negative Pell equation x² − 2y² = −1. For example, for y = 5 and 29, then,

1 3 + 3 3 + ⋯ + 9 3 = ( 7 ⋅ 5 ) 2 1 3 + 3 3 + ⋯ + 57 3 = ( 41 ⋅ 29 ) 2

and so on. Also, every even perfect number, except the lowest, is the sum of the first 2p−1/2
odd cubes,

28 = 2 2 ( 2 3 − 1 ) = 1 3 + 3 3 496 = 2 4 ( 2 5 − 1 ) = 1 3 + 3 3 + 5 3 + 7 3 8128 = 2 6 ( 2 7 − 1 ) = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3

Sum of cubes of numbers in arithmetic progression

There are examples of cubes of numbers in arithmetic progression whose sum is a cube:

3 3 + 4 3 + 5 3 = 6 3 11 3 + 12 3 + 13 3 + 14 3 = 20 3 31 3 + 33 3 + 35 3 + 37 3 + 39 3 + 41 3 = 66 3

with the first one also known as Plato's number. The formula F for finding the sum of n cubes of numbers in arithmetic progression with common difference d and initial cube a³,

F ( d , a , n ) = a 3 + ( a + d ) 3 + ( a + 2 d ) 3 + ⋯ + ( a + d n − d ) 3

is given by

F ( d , a , n ) = ( n / 4 ) ( 2 a − d + d n ) ( 2 a 2 − 2 a d + 2 a d n − d 2 n + d 2 n 2 )

A parametric solution to

F ( d , a , n ) = y 3

is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are known for integer d > 1, such as d = 2, 3, 5, 7, 11, 13, 37, 39, etc.

Cubes as sums of successive odd integers

In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …, the first one is a cube (1 = 1³); the sum of the next two is the next cube (3 + 5 = 2³); the sum of the next three is the next cube (7 + 9 + 11 = 3³); and so forth.

In rational numbers

Every positive rational number is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes.

In real numbers, other fields, and rings

In real numbers, the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) monotonically increase. Also, its codomain is the entire real line: the function x ↦ x³ : R → R is a surjection (takes all possible values). Only three numbers are equal to their own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x³ > x. If x < −1 or 0 < x < 1, then x³ < x. All aforementioned properties pertain also to any higher odd power (x⁵, x⁷, …) of real numbers. Equalities and inequalities are also true in any ordered ring.

Volumes of similar Euclidean solids are related as cubes of their linear sizes.

In complex numbers, the cube of a purely imaginary number is also purely imaginary. For example, i³ = −i.

The derivative of x³ equals 3x².

Cubes occasionally have the surjective property in other fields, such as in F_p for such prime p that p ≠ 1 (mod 3), but not necessarily: see the counterexample with rationals above. Also in F₇ only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to the own cubes: x³ − x = x(x − 1)(x + 1).

History

Determination of the cubes of large numbers was very common in many ancient civilizations. Mesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the Old Babylonian period (20th to 16th centuries BC). Cubic equations were known to the ancient Greek mathematician Diophantus. Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. Methods for solving cubic equations and extracting cube roots appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. The Indian mathematician Aryabhata wrote an explanation of cubes in his work Aryabhatiya. In 2010 Alberto Zanoni found a new algorithm to compute the cube of a long integer in a certain range, faster than squaring-and-multiplying.