"Third power" redirects here. For the band, see Third Power.
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.
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.
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}.
A perfect cube (also called a cube number, 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 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.
Cubes in number theory
There is no smallest perfect cube, since negative integers are included. For example, (−4) × (−4) × (−4) = −64. For any n, (−n)^{3} = −(n^{3}).
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}.
Sums of rational cubes
Every positive rational number is the sum of three positive rational cubes,^{[3]} and there are rationals that are not the sum of two rational cubes.^{[4]}
Sum of first n cubes
The sum of the first n cubes is the n^{th} triangle number squared:
 $1^3+2^3+\backslash dots+n^3\; =\; (1+2+\backslash dots+n)^2=\backslash left(\backslash frac\{n(n+1)\}\{2\}\backslash right)^2.$
For example, the sum of the first 5 cubes is the square of the 5th triangular number,
 $1^3+2^3+3^3+4^3+5^3\; =\; 15^2\; \backslash ,$
A similar result can be given for the sum of the first y odd cubes,
 $1^3+3^3+\backslash dots+(2y1)^3\; =\; (xy)^2$
but {x,y} must satisfy the negative Pell equation $x^22y^2\; =\; 1$. For example, for y = 5 and 29, then,
 $1^3+3^3+\backslash dots+9^3\; =\; (7\backslash cdot\; 5)^2\; \backslash ,$
 $1^3+3^3+\backslash dots+57^3\; =\; (41\backslash cdot\; 29)^2$
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 in arithmetic progression
There are examples of cubes 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 an n number of cubes in arithmetic progression with common difference d and initial cube a^{3},
 $F(d,a,n)\; =\; a^3+(a+d)^3+(a+2d)^3+...+(a+dnd)^3$
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.^{[5]}
History
Determination of the cubes of large numbers was very common in many ancient civilizations. Aryabhata, the ancient Indian mathematician in his famous work Aryabhatiya explains about the mathematical meaning of cube (Aryabhatiya, 23), as
"the continuous product of three equals as also the (rectangular) solid having 12 equal edges are called cube". Similar definitions can be seen in ancient texts such as Brahmasphuta Siddhanta (XVIII. 42), Ganitha sara sangraha (II. 43) and Siddhanta sekhara (XIII. 4). It is interesting that in modern mathematics too, the term "Cube" stands for two mathematical meanings just like in to compute the cube of a long integer in a certain range, faster than squaringandmultiplying.
Notes
See also
References
External links
 A Java applet that decomposes an integer number not congruent to 4 or 5 (mod 9) into a sum of four cubes.
Template:Classes of natural numbers
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