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}.
The graph of the cube function f: x → x^{3} (or the equation y = x^{3}) is known as the cubic parabola. Because cube is an odd function, this curve has a point of symmetry in the origin, but no axis of symmetry.
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Transcription
Contents
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 60^{3} are (sequence A000578 in the OEIS):
0^{3} =  0  
1^{3} =  1  11^{3} =  1331  21^{3} =  9261  31^{3} =  29,791  41^{3} =  68,921  51^{3} =  132,651 
2^{3} =  8  12^{3} =  1728  22^{3} =  10,648  32^{3} =  32,768  42^{3} =  74,088  52^{3} =  140,608 
3^{3} =  27  13^{3} =  2197  23^{3} =  12,167  33^{3} =  35,937  43^{3} =  79,507  53^{3} =  148,877 
4^{3} =  64  14^{3} =  2744  24^{3} =  13,824  34^{3} =  39,304  44^{3} =  85,184  54^{3} =  157,464 
5^{3} =  125  15^{3} =  3375  25^{3} =  15,625  35^{3} =  42,875  45^{3} =  91,125  55^{3} =  166,375 
6^{3} =  216  16^{3} =  4096  26^{3} =  17,576  36^{3} =  46,656  46^{3} =  97,336  56^{3} =  175,616 
7^{3} =  343  17^{3} =  4913  27^{3} =  19,683  37^{3} =  50,653  47^{3} =  103,823  57^{3} =  185,193 
8^{3} =  512  18^{3} =  5832  28^{3} =  21,952  38^{3} =  54,872  48^{3} =  110,592  58^{3} =  195,112 
9^{3} =  729  19^{3} =  6859  29^{3} =  24,389  39^{3} =  59,319  49^{3} =  117,649  59^{3} =  205,379 
10^{3} =  1000  20^{3} =  8000  30^{3} =  27,000  40^{3} =  64,000  50^{3} =  125,000  60^{3} =  216,000 
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:
 n^{3} − (n − 1)^{3} = 3(n − 1)n + 1.
or
 (n + 1)^{3} − n^{3} = 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 2^{6}).
The last digits of each 3rd power are:
0  1  8  7  4  5  6  3  2  9 
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 = 2^{3} + 2^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3}.
Sums of three cubes
It is conjectured that every integer (positive or negative) not congruent to ±4 modulo 9 can be written as a sum of three (positive or negative) cubes with infinitely many ways,^{[1]} for example, . (Integers congruent to ±4 modulo 9 cannot be so written.) The smallest such integer for which such a sum is not known is 114. In September 2019, the previous smallest such integer with no known 3cube sum, 42, was found to satisfy this equation: ^{[2]}
The solutions to for n ≤ 78 (where n is not = 4 or 5 mod 9) with smallest z and smallest y, 0 ≤ x ≤ y ≤ z, gcd(x,y,z) = 1 (i.e. only primitive solutions are selected), and none of x+y, y+z, z+x is 0 (if no this condition, then for every positive cube z^{3}, all (x, −x, z) are solutions, and these solutions are all trivial) are given below:^{[3]}^{[4]}^{[5]} (We only select "primitive solutions", i.e. gcd(x,y,z) must be 1, e.g. for n=24, the solution (x,y,z) = (2, 2, 2) is not allowed, since its gcd is 2 (not 1), thus we select (x,y,z) = (2901096694, 15550555555, 15584139827), another example is for n=48, the solution (x,y,z) = (2, 2, 4) is also not allowed, and we select (x,y,z) = (23, 26, 31)) (sequences A060465, A060466 and A060467 in OEIS)
n  x  y  z  n  x  y  z 
1  9  10  −12  39  117367  134476  −159380 
2  0  1  1  42  12602123297335631  80435758145817515  −80538738812075974 
3  1  1  1  43  2  2  3 
6  −1  −1  2  44  −5  −7  8 
7  0  −1  2  45  2  −3  4 
8  9  15  −16  46  −2  3  3 
9  0  1  2  47  6  7  −8 
10  1  1  2  48  −23  −26  31 
11  −2  −2  3  51  602  659  −796 
12  7  10  −11  52  23961292454  60702901317  −61922712865 
15  −1  2  2  53  −1  3  3 
16  −511  −1609  1626  54  −7  −11  12 
17  1  2  2  55  1  3  3 
18  −1  −2  3  56  −11  −21  22 
19  0  −2  3  57  1  −2  4 
20  1  −2  3  60  −1  −4  5 
21  −11  −14  16  61  0  −4  5 
24  −2901096694  −15550555555  15584139827  62  2  3  3 
25  −1  −1  3  63  0  −1  4 
26  0  −1  3  64  −3  −5  6 
27  −4  −5  6  65  0  1  4 
28  0  1  3  66  1  1  4 
29  1  1  3  69  2  −4  5 
30  −283059965  −2218888517  2220422932  70  11  20  −21 
33  −2736111468807040  −8778405442862239  8866128975287528  71  −1  2  4 
34  −1  2  3  72  7  9  −10 
35  0  2  3  73  1  2  4 
36  1  2  3  74  66229832190556  283450105697727  −284650292555885 
37  0  −3  4  75  4381159  435203083  −435203231 
38  1  −3  4  78  26  53  −55 
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.^{[6]}
Both of these statements are also true for the equation^{[7]} x^{3} + y^{3} = 3z^{3}.
Sum of first n cubes
The sum of the first n cubes is the nth triangle number squared:
Proofs. Charles Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. He begins by giving the identity
That identity is related to triangular numbers in the following way:
and thus the summands forming start off just after those forming all previous values up to . Applying this property, along with another wellknown identity:
we obtain the following derivation:
In the more recent mathematical literature, Stein (1971) uses the rectanglecounting interpretation of these numbers to form a geometric proof of the identity (see also Benjamin, Quinn & Wurtz 2006); he observes that it may also be proved easily (but uninformatively) by induction, and states that Toeplitz (1963) provides "an interesting old Arabic proof". Kanim (2004) provides a purely visual proof, Benjamin & Orrison (2002) provide two additional proofs, and Nelsen (1993) gives seven geometric proofs.
For example, the sum of the first 5 cubes is the square of the 5th triangular number,
A similar result can be given for the sum of the first y odd cubes,
but x, y must satisfy the negative Pell equation x^{2} − 2y^{2} = −1. For example, for y = 5 and 29, then,
and so on. Also, every even perfect number, except the lowest, is the sum of the first 2^{p−1/2}
_{} odd cubes (p = 3, 5, 7, ...):
Sum of cubes of numbers in arithmetic progression
There are examples of cubes of numbers in arithmetic progression whose sum is a cube:
with the first one sometimes identified as the mysterious 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},
is given by
A parametric solution to
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.^{[8]}
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,^{[9]} and there are rationals that are not the sum of two rational cubes.^{[10]}
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 ↦ x^{3} : 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 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 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),^{[11]} 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).^{[12]}^{[13]} Cubic equations were known to the ancient Greek mathematician Diophantus.^{[14]} Hero of Alexandria devised a method for calculating cube roots in the 1st century CE.^{[15]} 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.^{[16]} The Indian mathematician Aryabhata wrote an explanation of cubes in his work Aryabhatiya. In 2010 Alberto Zanoni found a new algorithm^{[17]} to compute the cube of a long integer in a certain range, faster than squaringandmultiplying.
See also
 Cabtaxi number
 Cubic equation
 Doubling the cube
 Euler's sum of powers conjecture
 Fifth power (algebra)
 Fourth power
 Kepler's laws of planetary motion#Third law
 Monkey saddle
 Perfect power
 Taxicab number
Notes
 ^ Huisman, Sander G. (27 Apr 2016). "Newer sums of three cubes". arXiv:1604.07746 [math.NT].
 ^ "NEWS: The Mystery of 42 is Solved  Numberphile" https://www.youtube.com/watch?v=zyG8Vlw5aAw
 ^ List of solutions of x^3 + y^3 + z^3 = k for k < 1000
 ^ About n=x^3+y^3+z^3
 ^ Three cubes
 ^ Hardy & Wright, Thm. 227
 ^ Hardy & Wright, Thm. 232
 ^ "A Collection of Algebraic Identities".^{[permanent dead link]}
 ^ 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. ISBN 9781118460290.
 ^ NemetNejat, Karen Rhea (1998). Daily Life in Ancient Mesopotamia. Greenwood Publishing Group. p. 306. ISBN 9780313294976.
 ^ 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. JSTOR 23037103.
 ^ Crossley, John; W.C. Lun, Anthony (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. pp. 176, 213. ISBN 9780198539360.
 ^ Bodrato, Marco; Zanoni, Alberto (2012). "A New Algorithm for Long Integer Cube Computation with Some Insight into Higher Powers". Lecture Notes in Computer Science: 34–46. doi:10.1007/9783642329739_4.
References
 Hardy, G. H.; Wright, E. M. (1980). "An Introduction to the Theory of Numbers" (Fifth ed.). Oxford: Oxford University Press. ISBN 9780198531715. Cite journal requires
journal=
(help)  Wheatstone, C. (1854), "On the formation of powers from arithmetical progressions", Proceedings of the Royal Society of London, 7: 145–151, doi:10.1098/rspl.1854.0036.