Cube (algebra)

y = x 3

for values of

0 \leq x \leq 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 \times n \times n

It is also the number multiplied by its square:

n 3 = n \times 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 one-third 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 2³ = 8 or $(x + 1) 3$ .

The graph of the cube function f: x → x³ (or the equation y = x³) 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.

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):

0³ =	0
1³ =	1	11³ =	1331	21³ =	9261	31³ =	29,791	41³ =	68,921	51³ =	132,651
2³ =	8	12³ =	1728	22³ =	10,648	32³ =	32,768	42³ =	74,088	52³ =	140,608
3³ =	27	13³ =	2197	23³ =	12,167	33³ =	35,937	43³ =	79,507	53³ =	148,877
4³ =	64	14³ =	2744	24³ =	13,824	34³ =	39,304	44³ =	85,184	54³ =	157,464
5³ =	125	15³ =	3375	25³ =	15,625	35³ =	42,875	45³ =	91,125	55³ =	166,375
6³ =	216	16³ =	4096	26³ =	17,576	36³ =	46,656	46³ =	97,336	56³ =	175,616
7³ =	343	17³ =	4913	27³ =	19,683	37³ =	50,653	47³ =	103,823	57³ =	185,193
8³ =	512	18³ =	5832	28³ =	21,952	38³ =	54,872	48³ =	110,592	58³ =	195,112
9³ =	729	19³ =	6859	29³ =	24,389	39³ =	59,319	49³ =	117,649	59³ =	205,379
10³ =	1000	20³ =	8000	30³ =	27,000	40³ =	64,000	50³ =	125,000	60³ =	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

(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, $o$ 2, $e$ 4, $o$ 6 and $e$ 8 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,

{\text{if}}\quad x\equiv 0{\pmod {3}}\quad {\text{then}}\quad x^{3}\equiv 0{\pmod {9}};

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

{\text{if}}\quad x\equiv 1{\pmod {3}}\quad {\text{then}}\quad x^{3}\equiv 1{\pmod {9}};

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

{\text{if}}\quad x\equiv -1{\pmod {3}}\quad {\text{then}}\quad x^{3}\equiv -1{\pmod {9}}.

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 $x 3 + y 3 = z 3$ has no non-trivial (i.e. $xyz \neq 0$ ) solutions in integers. In fact, it has none in Eisenstein integers.^[1]

Both of these statements are also true for the equation^[2] $x 3 + y 3 = 3 z 3$ .

Sum of first n cubes

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

1^{3}+2^{3}+\dots +n^{3}=(1+2+\dots +n)^{2}=\left({\frac {n(n+1)}{2}}\right)^{2}.

Visual proof that 1³ + 2³ + 3³ + 4³ + 5³ = (1 + 2 + 3 + 4 + 5)².

Proofs Charles Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity

n^{3}=\underbrace {\left(n^{2}-n+1\right)+\left(n^{2}-n+1+2\right)+\left(n^{2}-n+1+4\right)+\cdots +\left(n^{2}+n-1\right)} _{n{\text{ consecutive odd numbers}}}.

That identity is related to triangular numbers $T_{n}$ in the following way:

n^{3}=\sum _{k=T_{n-1}+1}^{T_{n}}(2k-1),

and thus the summands forming $n^{3}$ start off just after those forming all previous values $1^{3}$ up to $(n-1)^{3}$ . Applying this property, along with another well-known identity:

n^{2}=\sum _{k=1}^{n}(2k-1),

we obtain the following derivation:

{\begin{aligned}\sum _{k=1}^{n}k^{3}&=1+8+27+64+\cdots +n^{3}\\&=\underbrace {1} _{1^{3}}+\underbrace {3+5} _{2^{3}}+\underbrace {7+9+11} _{3^{3}}+\underbrace {13+15+17+19} _{4^{3}}+\cdots +\underbrace {\left(n^{2}-n+1\right)+\cdots +\left(n^{2}+n-1\right)} _{n^{3}}\\&=\underbrace {\underbrace {\underbrace {\underbrace {1} _{1^{2}}+3} _{2^{2}}+5} _{3^{2}}+\cdots +\left(n^{2}+n-1\right)} _{\left({\frac {n^{2}+n}{2}}\right)^{2}}\\&=(1+2+\cdots +n)^{2}\\&={\bigg (}\sum _{k=1}^{n}k{\bigg )}^{2}.\end{aligned}}

Visual demonstration that the square of a triangular number equals a sum of cubes.

In the more recent mathematical literature, Stein (1971) uses the rectangle-counting 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,

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 +(2y-1)^{3}=(xy)^{2}

but $x$ , $y$ must satisfy the negative Pell equation $x 2 - 2 y 2 = -1$ . 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 lowest, is the sum of the first $2 p -1 / 2$ 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 $a 3$ ,

F(d,a,n)=a^{3}+(a+d)^{3}+(a+2d)^{3}+\cdots +(a+dn-d)^{3}

is given by

F(d,a,n)=(n/4)(2a-d+dn)(2a^{2}-2ad+2adn-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.^[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 = 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,^[4] and there are rationals that are not the sum of two rational cubes.^[5]

In real numbers, other fields, and rings

y = x 3

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 $x \mapsto x 3 : R \to R$ 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 $3 x 2$ .

Cubes occasionally have the surjective property in other fields, such as in $F p$ for such prime $p$ that $p \neq 1 (mod 3)$ ,^[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 squaring-and-multiplying.

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. ISBN 978-1-118-46029-0.
↑ Nemet-Nejat, Karen Rhea (1998). Daily Life in Ancient Mesopotamia. Greenwood Publishing Group. p. 306. ISBN 978-0-313-29497-6.
↑ Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0-387-12159-5
↑ 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 978-0-19-853936-0.
↑ http://www.springerlink.com/content/q1k57pr4853g1513/

References

Hardy, G. H.; Wright, E. M. (1980). "An Introduction to the Theory of Numbers" (Fifth ed.). Oxford: Oxford University Press. ISBN 978-0-19-853171-5.

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular
non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect
By properties of Ω(n)	Almost prime Semiprime
By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

Sequences and series

Integer
sequences

Basic	Arithmetic progression Geometric progression Square number Cubic number Factorial Powers of two Powers of 10
Advanced	Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number Full list

Properties of sequences

Properties of series

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/1^s + 1/2^s+ 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric
series

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