y = x^{3} for values of 0 ≤ x ≤ 25.
In arithmetic and algebra, the cube of a number n is its third power: the result of the number multiplied by itself twice:

n^{3} = n × n × n.
It is also the number multiplied by its square:

n^{3} = n × n^{2}.
This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the onethird power.
Both cube and cube root are odd functions:

(−n)^{3} = −(n^{3}).
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 2^{3} = 8 or (x + 1)^{3}.
Contents

In integers 1

Base ten 1.1

Waring's problem for cubes 1.2

Fermat's last theorem for cubes 1.3

Sum of first n cubes 1.4

Sum of cubes of numbers in arithmetic progression 1.5

Cubes as sums of successive odd integers 1.6

In rational numbers 2

In real numbers, other fields, and rings 3

History 4

Notes 5

See also 6

References 7

External links 8
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 positive perfect cubes up to 60^{3} are (sequence A000578 in OEIS):
1^{3} = 1

11^{3} = 1331

21^{3} = 9261

31^{3} = 29791

41^{3} = 68921

51^{3} = 132651

2^{3} = 8

12^{3} = 1728

22^{3} = 10648

32^{3} = 32768

42^{3} = 74088

52^{3} = 140608

3^{3} = 27

13^{3} = 2197

23^{3} = 12167

33^{3} = 35937

43^{3} = 79507

53^{3} = 148877

4^{3} = 64

14^{3} = 2744

24^{3} = 13824

34^{3} = 39304

44^{3} = 85184

54^{3} = 157464

5^{3} = 125

15^{3} = 3375

25^{3} = 15625

35^{3} = 42875

45^{3} = 91125

55^{3} = 166375

6^{3} = 216

16^{3} = 4096

26^{3} = 17576

36^{3} = 46656

46^{3} = 97336

56^{3} = 175616

7^{3} = 343

17^{3} = 4913

27^{3} = 19683

37^{3} = 50653

47^{3} = 103823

57^{3} = 185193

8^{3} = 512

18^{3} = 5832

28^{3} = 21952

38^{3} = 54872

48^{3} = 110592

58^{3} = 195112

9^{3} = 729

19^{3} = 6859

29^{3} = 24389

39^{3} = 59319

49^{3} = 117649

59^{3} = 205379

10^{3} = 1000

20^{3} = 8000

30^{3} = 27000

40^{3} = 64000

50^{3} = 125000

60^{3} = 216000

Geometrically speaking, a positive number 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 pattern between every perfect cube from negative infinity to positive infinity is as follows,

n^{3} = (n − 1)^{3} + 3(n − 1)n + 1.
or

n^{3} = (n + 1)^{3} − 3(n + 1)n − 1.
There is no smallest perfect cube, since negative integers are included. 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.
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. 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 is divisible by 3, its cube has digital root 9;

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

If it has a remainder of 2 when divided by 3, its cube has digital root 8.
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 = 2^{3} + 2^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3}.
Fermat's last theorem for cubes
The equation x^{3} + y^{3} = z^{3} has no nontrivial (i.e. xyz ≠ 0) solutions in integers. In fact, it has none in Eisenstein integers.^{[1]}
Both of these statements are also true for the equation^{[2]} x^{3} + y^{3} = 3z^{3}.
Sum of first n cubes
The sum of the first n cubes is the nth triangle number squared:

1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2=\left(\frac{n(n+1)}{2}\right)^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+\dots+(2y1)^3 = (xy)^2
but x, y must satisfy the negative Pell equation x^22y^2 = 1. For example, for y = 5 and 29, then,

1^3+3^3+\dots+9^3 = (7\cdot 5)^2 \,

1^3+3^3+\dots+57^3 = (41\cdot 29)^2
and so on. Also, every even perfect number, except the first one, is the sum of the first 2^{(p−1)/2} odd cubes,

28 = 2^2(2^31) = 1^3+3^3

496 = 2^4(2^51) = 1^3+3^3+5^3+7^3

8128 = 2^6(2^71) = 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 a^{3},

F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+\cdots+(a+dnd)^3
is given by

F(d,a,n) = (n/4)(2ad+dn)(2a^22ad+2adnd^2n+d^2n^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.^{[3]}
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 = 1^{3}); the sum of the next two is the next cube (3+5 = 2^{3}); the sum of the next three is the next cube (7+9+11 = 3^{3}); and so forth.
In rational numbers
Every positive rational number is the sum of three positive rational cubes,^{[4]} and there are rationals that are not the sum of two rational cubes.^{[5]}
In real numbers, other fields, and rings
x³ plotted on a Cartesian plane
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 ↦ x^{3} : R → R is a surjection (takes all possible values). Only three numbers equal to the own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x^{3} > x. If x < −1 or 0 < x < 1, then x^{3} < x. All aforementioned properties pertain also to any higher odd power (x^{5}, x^{7}, …) 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, i^{3} = −i.
The derivative of x^{3} equals to 3x^{2}.
Cubes occasionally have the surjective property in other fields, such as in F_{p} for such prime p that p ≠ 1 (mod 3),^{[6]} but not necessarily: see the counterexample with rationals above. Also in F_{7} 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: x^{3} − 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).^{[7]}^{[8]} Cubic equations were known to the ancient Greek mathematician Diophantus.^{[9]} Hero of Alexandria devised a method for calculating cube roots in the 1st century CE.^{[10]} 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.^{[11]} The Indian mathematician Aryabhata wrote an explanation of cubes in his work Aryabhatiya. In 2010 Alberto Zanoni found a new algorithm^{[12]} to compute the cube of a long integer in a certain range, faster than squaringandmultiplying.
Notes

^ Hardy & Wright, Thm. 227

^ Hardy & Wright, Thm. 232

^ "A Collection of Algebraic Identities".

^ Hardy & Wright, Thm. 234

^ Hardy & Wright, Thm. 233

^ The multiplicative group of F_{p} is cyclic of order p − 1, and if it is not divisible by 3, then cubes define a group automorphism.

^ Cooke, Roger (8 November 2012). The History of Mathematics. John Wiley & Sons. p. 63.

^ NemetNejat, Karen Rhea (1998). Daily Life in Ancient Mesopotamia. Greenwood Publishing Group. p. 306.

^ Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0387121595

^ Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root". Hermathena (Trinity College Dublin) 19 (42): 64–67.

^ Crossley, John; W.C. Lun, Anthony (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. pp. 176, 213.

^ http://www.springerlink.com/content/q1k57pr4853g1513/
See also
References

Hardy, G. H.; Wright, E. M. (1980). "An Introduction to the Theory of Numbers (Fifth edition)". Oxford:
External links

A Java applet that decomposes an integer number not congruent to 4 or 5 (mod 9) into a sum of four cubes.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.