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{{redirect|Third power|the band|Third Power}}
{{redirect|Cubed|other uses|Cube (disambiguation)}}
{{redirect|³|text=Its literal meaning is the numeral "[[3 (number)|3]]" in [[superscript]]}}
[[Image:CubeChart.svg|thumb|280px|{{math|1=''y'' = ''x''<sup>3</sup>}} for values of {{math|1 ≤ ''x'' ≤ 25}}.]]
In [[arithmetic]] and [[algebra]], the '''cube''' of a number {{mvar|n}} is its third [[exponentiation|power]]: the result of the number multiplied by itself twice:

:{{math|size=120%|1=''n''<sup>3</sup> = ''n'' × ''n'' × ''n''}}.

It is also the number multiplied by its [[square (algebra)|square]]:

:{{math|size=120%|1=''n''<sup>3</sup> = ''n'' × ''n''<sup>2</sup>}}.

This is also the [[volume]] formula for a [[Cube (geometry)|geometric cube]] with sides of length {{mvar|n}}, giving rise to the name. The [[Inverse function|inverse]] operation of finding a number whose cube is {{mvar|n}} is called extracting the [[cube root]] of {{mvar|n}}. It determines the side of the cube of a given volume. It is also {{mvar|n}} raised to the one-third power.

Both cube and cube root are [[odd function]]s:

:{{math|size=120%|1=(−''n'')<sup>3</sup> = −(''n''<sup>3</sup>)}}.

The cube of a number or any other [[expression (mathematics)|mathematical expression]] is denoted by a [[superscript]] 3, for example {{nowrap|2<sup>3</sup> {{=}} 8}} or {{math|(''x'' + 1)<sup>3</sup>}}.

The [[graph of a function|graph]] of the cube function f: ''x'' → ''x''<sup>3</sup> (or the equation ''y'' = ''x''<sup>3</sup>) is known as the '''cubic parabola'''. Because cube is an odd function, this curve has a [[Point reflection|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 60<sup>3</sup> are {{OEIS|id=A000578}}:

{| style="text-align:right;"
| style="padding-left:2em;"| 0<sup>3</sup> = || 0
|-
| style="padding-left:2em;"| 1<sup>3</sup> = || 1
| style="padding-left:2em;"|11<sup>3</sup> = || 1331
| style="padding-left:2em;"|21<sup>3</sup> = || 9261
| style="padding-left:2em;"|31<sup>3</sup> = || 29,791
| style="padding-left:2em;"|41<sup>3</sup> = || 68,921
| style="padding-left:2em;"|51<sup>3</sup> = || 132,651
|-
| style="padding-left:2em;"| 2<sup>3</sup> = || 8
| style="padding-left:2em;"|12<sup>3</sup> = || 1728
| style="padding-left:2em;"|22<sup>3</sup> = || 10,648
| style="padding-left:2em;"|32<sup>3</sup> = || 32,768
| style="padding-left:2em;"|42<sup>3</sup> = || 74,088
| style="padding-left:2em;"|52<sup>3</sup> = || 140,608
|-
| style="padding-left:2em;"| 3<sup>3</sup> = || 27
| style="padding-left:2em;"|13<sup>3</sup> = || 2197
| style="padding-left:2em;"|23<sup>3</sup> = || 12,167
| style="padding-left:2em;"|33<sup>3</sup> = || 35,937
| style="padding-left:2em;"|43<sup>3</sup> = || 79,507
| style="padding-left:2em;"|53<sup>3</sup> = || 148,877
|-
| style="padding-left:2em;"| 4<sup>3</sup> = || 64
| style="padding-left:2em;"|14<sup>3</sup> = || 2744
| style="padding-left:2em;"|24<sup>3</sup> = || 13,824
| style="padding-left:2em;"|34<sup>3</sup> = || 39,304
| style="padding-left:2em;"|44<sup>3</sup> = || 85,184
| style="padding-left:2em;"|54<sup>3</sup> = || 157,464
|-
| style="padding-left:2em;"| 5<sup>3</sup> = || 125
| style="padding-left:2em;"|15<sup>3</sup> = || 3375
| style="padding-left:2em;"|25<sup>3</sup> = || 15,625
| style="padding-left:2em;"|35<sup>3</sup> = || 42,875
| style="padding-left:2em;"|45<sup>3</sup> = || 91,125
| style="padding-left:2em;"|55<sup>3</sup> = || 166,375
|-
| style="padding-left:2em;"| 6<sup>3</sup> = || 216
| style="padding-left:2em;"|16<sup>3</sup> = || 4096
| style="padding-left:2em;"|26<sup>3</sup> = || 17,576
| style="padding-left:2em;"|36<sup>3</sup> = || 46,656
| style="padding-left:2em;"|46<sup>3</sup> = || 97,336
| style="padding-left:2em;"|56<sup>3</sup> = || 175,616
|-
| style="padding-left:2em;"| 7<sup>3</sup> = || 343
| style="padding-left:2em;"|17<sup>3</sup> = || 4913
| style="padding-left:2em;"|27<sup>3</sup> = || 19,683
| style="padding-left:2em;"|37<sup>3</sup> = || 50,653
| style="padding-left:2em;"|47<sup>3</sup> = || 103,823
| style="padding-left:2em;"|57<sup>3</sup> = || 185,193
|-
| style="padding-left:2em;"| 8<sup>3</sup> = || 512
| style="padding-left:2em;"|18<sup>3</sup> = || 5832
| style="padding-left:2em;"|28<sup>3</sup> = || 21,952
| style="padding-left:2em;"|38<sup>3</sup> = || 54,872
| style="padding-left:2em;"|48<sup>3</sup> = || 110,592
| style="padding-left:2em;"|58<sup>3</sup> = || 195,112
|-
| style="padding-left:2em;"| 9<sup>3</sup> = || 729
| style="padding-left:2em;"|19<sup>3</sup> = || 6859
| style="padding-left:2em;"|29<sup>3</sup> = || 24,389
| style="padding-left:2em;"|39<sup>3</sup> = || 59,319
| style="padding-left:2em;"|49<sup>3</sup> = || 117,649
| style="padding-left:2em;"|59<sup>3</sup> = || 205,379
|-
| style="padding-left:2em;"|10<sup>3</sup> = || 1000
| style="padding-left:2em;"|20<sup>3</sup> = || 8000
| style="padding-left:2em;"|30<sup>3</sup> = || 27,000
| style="padding-left:2em;"|40<sup>3</sup> = || 64,000
| style="padding-left:2em;"|50<sup>3</sup> = || 125,000
| style="padding-left:2em;"|60<sup>3</sup> = || 216,000
|}

Geometrically speaking, a positive integer {{mvar|m}} is a perfect cube [[if and only if]] one can arrange {{mvar|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 {{nowrap|3 × 3 × 3 {{=}} 27}}.

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

:{{math|size=120%|1=''n''<sup>3</sup> − (''n'' − 1)<sup>3</sup> = 3(''n'' − 1)''n'' + 1}}.

or

:{{math|size=120%|1=(''n'' + 1)<sup>3</sup> − ''n''<sup>3</sup> =  3(''n'' + 1)''n'' + 1}}.

There is no minimum perfect cube, since the cube of a negative integer is negative. For example, {{nowrap|(−4) × (−4) × (−4) {{=}} −64}}.

===Base ten===

Unlike [[square number|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 and odd numbers|even]] cubes, there is considerable restriction, for only '''00''', {{math|''o''}}'''2''', {{math|''e''}}'''4''', {{math|''o''}}'''6''' and {{math|''e''}}'''8''' can be the last two digits of a perfect cube (where {{math|''o''}} stands for any odd digit and {{math|''e''}} for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number {{nowrap|(8 × 8)}} and a cube number {{nowrap|(4 × 4 × 4)}}. This happens if and only if the number is a perfect sixth power (in this case 2{{sup|6}}).

The last digits of each 3rd power are:

{| width="45%"
|| 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 [[modular arithmetic|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, 
:: <math>\text{if}\quad x \equiv 0 \pmod 3 \quad \text{then} \quad x^3\equiv 0 \pmod 9;</math>
* If it has a remainder of 1 when divided by 3, its cube has digital root 1; that is, 
:: <math>\text{if}\quad x \equiv 1 \pmod 3 \quad \text{then} \quad x^3\equiv 1 \pmod 9;</math>
* If it has a remainder of 2 when divided by 3, its cube has digital root 8; that is, 
:: <math>\text{if}\quad x \equiv -1 \pmod 3 \quad \text{then} \quad x^3\equiv -1 \pmod 9.</math>

===Waring's problem for cubes===
{{main article|Waring's problem}}

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<sup>3</sup> + 2<sup>3</sup> + 1<sup>3</sup> + 1<sup>3</sup> + 1<sup>3</sup> + 1<sup>3</sup> + 1<sup>3</sup> + 1<sup>3</sup> + 1<sup>3</sup>.

===Sums of three cubes===

{{main|Sums of three cubes}}

It is conjectured that every integer (positive or negative) not [[modular arithmetic|congruent]] to {{math|±4}} modulo {{math|9}} can be written as a sum of three (positive or negative) cubes with infinitely many ways,<ref>{{cite arXiv |last=Huisman |first=Sander G. |author-link= |eprint=1604.07746 |title=Newer sums of three cubes |class=math.NT |date=27 Apr 2016 }}</ref> for example, <math> 6 = 2^3+(-1)^3+(-1)^3</math>. (Integers congruent to {{math|±4}} modulo {{math|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 3-cube sum, 42, was found to satisfy this equation: <ref>"NEWS: The Mystery of 42 is Solved - Numberphile" https://www.youtube.com/watch?v=zyG8Vlw5aAw</ref>
:<math> 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3.</math>

The solutions to <math>x^3 + y^3 + z^3 = n</math> 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''<sup>3</sup>, all (''x'', −''x'', ''z'') are solutions, and these solutions are all trivial) are given below:<ref>[https://arxiv.org/src/1604.07746v1/anc/sumofthreecubes_20160426.txt List of solutions of x^3 + y^3 + z^3 = k for k < 1000]</ref><ref>[http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm About n=x^3+y^3+z^3]</ref><ref>[http://cr.yp.to/threecubes.html Three cubes]</ref> (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 {{OEIS link|A060465}}, {{OEIS link|A060466}} and {{OEIS link|A060467}} in [[OEIS]])

{|class="wikitable"
|''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===

{{main article|Fermat's last theorem}}

The equation {{math|1=''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>3</sup>}} has no non-trivial (i.e. {{math|''xyz'' ≠ 0}}) solutions in integers. In fact, it has none in [[Eisenstein integers]].<ref>Hardy & Wright, Thm. 227</ref>

Both of these statements are also true for the equation<ref>Hardy & Wright, Thm. 232</ref> {{math|1=''x''<sup>3</sup> + ''y''<sup>3</sup> = 3''z''<sup>3</sup>}}.

===Sum of first ''n'' cubes===

The sum of the first {{mvar|n}} cubes is the {{mvar|n}}th [[triangle number]] squared:

:<math>1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2=\left(\frac{n(n+1)}{2}\right)^2.</math>

[[File:Nicomachus_theorem_3D.svg|thumb|Visual proof that {{nowrap|1{{sup|3}} + 2{{sup|3}} + 3{{sup|3}} + 4{{sup|3}} + 5{{sup|3}}}} = {{nowrap|(1 + 2 + 3 + 4 + 5){{sup|2}}}}.]]

'''Proofs.'''
{{harvs|txt|first=Charles|last=Wheatstone|authorlink=Charles Wheatstone|year=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

:<math>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}}.</math>

That identity is related to [[triangular numbers]] <math>T_n</math> in the following way:

:<math>n^3 =\sum _{k=T_{n-1}+1}^{T_{n}} (2 k-1),</math>

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

:<math>n^2 = \sum_{k=1}^n (2k-1),</math>

we obtain the following derivation:

:<math>
\begin{align}
\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{align}</math>

[[File:Sum of cubes2.png|thumb|right|upright=1.6|Visual demonstration that the square of a triangular number equals a sum of cubes.]]
In the more recent mathematical literature, {{harvtxt|Stein|1971}} uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also {{harvnb|Benjamin|Quinn|Wurtz|2006}}); he observes that it may also be proved easily (but uninformatively) by induction, and states that {{harvtxt|Toeplitz|1963}} provides "an interesting old Arabic proof". {{harvtxt|Kanim|2004}} provides a purely visual proof, {{harvtxt|Benjamin|Orrison|2002}} provide two additional proofs, and {{harvtxt|Nelsen|1993}} gives seven geometric proofs.

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

:<math>1^3+2^3+3^3+4^3+5^3 = 15^2 </math>

A similar result can be given for the sum of the first {{mvar|y}} [[odd number|odd]] cubes,

:<math>1^3+3^3+\dots+(2y-1)^3 = (xy)^2</math>

but {{mvar|x}}, {{mvar|y}} must satisfy the negative [[Pell equation]] {{math|''x''{{sup|2}} − 2''y''{{sup|2}} {{=}} −1}}. For example, for {{math|1=''y'' = 5}} and {{math|29}}, then,

:<math>1^3+3^3+\dots+9^3 = (7\cdot 5)^2 </math>

:<math>1^3+3^3+\dots+57^3 = (41\cdot 29)^2</math>

and so on.  Also, every [[even number|even]] [[perfect number]], except the lowest, is the sum of the first [[power of two|{{math|2{{su|p={{sfrac|''p''−1|2}}}}}}]] odd cubes (''p'' = 3, 5, 7, ...):

:<math>28 = 2^2(2^3-1) = 1^3+3^3</math>
:<math>496 = 2^4(2^5-1) = 1^3+3^3+5^3+7^3</math>
:<math>8128 = 2^6(2^7-1) = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3</math>

===Sum of cubes of numbers in arithmetic progression===
[[File:Plato_number.svg|thumb|One interpretation of Plato's number, {{nowrap|3&#179; + 4&#179; + 5&#179; {{=}} 6&#179;}}]]
There are examples of cubes of numbers in [[arithmetic progression]] whose sum is a cube:

:<math>3^3+4^3+5^3 = 6^3</math>
:<math>11^3+12^3+13^3+14^3 = 20^3</math>
:<math>31^3+33^3+35^3+37^3+39^3+41^3 = 66^3</math>

with the first one sometimes identified as the mysterious [[Plato's number]].  The formula {{mvar|F}} for finding the sum of  {{mvar|n}}
cubes of numbers in arithmetic progression with common difference {{mvar|d}} and initial cube {{math|''a''<sup>3</sup>}},

:<math>F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+\cdots+(a+dn-d)^3</math>

is given by

:<math>F(d,a,n) = (n/4)(2a-d+dn)(2a^2-2ad+2adn-d^2n+d^2n^2)</math>

A parametric solution to

:<math>F(d,a,n) = y^3</math>

is known for the special case of {{math|1=''d'' = 1}}, or consecutive cubes, but only sporadic solutions are known for integer {{math|''d'' > 1}}, such as {{mvar|d}}&nbsp;=&nbsp;2, 3, 5, 7, 11, 13, 37, 39, etc.<ref>{{cite web|url=http://sites.google.com/site/tpiezas/update06|title=A Collection of Algebraic Identities}}{{Dead link|date=July 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

===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 ({{nowrap|1 {{=}} 1<sup>3</sup>}}); the sum of the next ''two'' is the next cube ({{nowrap|3 + 5 {{=}} 2<sup>3</sup>}}); the sum of the next ''three'' is the next cube ({{nowrap|7 + 9 + 11 {{=}} 3<sup>3</sup>}}); and so forth.

==In rational numbers==

Every positive [[rational number]] is the sum of three positive rational cubes,<ref>Hardy & Wright, Thm. 234</ref> and there are rationals that are not the sum of two rational cubes.<ref>Hardy & Wright, Thm. 233</ref>

==In real numbers, other fields, and rings==
{{more information|cubic function}}
[[Image:X cubed plot.svg|thumb|right|{{math|''y'' {{=}} ''x''{{sup|3}}}} plotted on a Cartesian plane]]
In [[real number]]s, the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) [[monotonic function|monotonically increase]]. Also, its [[codomain]] is the entire [[real line]]: the function {{math|''x'' ↦ ''x''<sup>3</sup> : '''R''' → '''R'''}} is a [[surjection]] (takes all possible values). Only three numbers are equal to their own cubes: {{num|−1}}, {{num|0}}, and {{num|1}}. If {{math|−1 < ''x'' < 0}} or {{math|1 < ''x''}}, then {{math|''x''<sup>3</sup> > ''x''}}. If {{math|''x'' < −1}} or {{math|0 < ''x'' < 1}}, then {{math|''x''<sup>3</sup> < ''x''}}. All aforementioned properties pertain also to any higher odd power ({{math|''x''<sup>5</sup>}}, {{math|''x''<sup>7</sup>}},&nbsp;…) of real numbers. Equalities and [[inequality (mathematics)|inequalities]] are also true in any [[ordered ring]].

Volumes of [[similarity (geometry)|similar]] Euclidean [[solid geometry|solids]] are related as cubes of their linear sizes.

In [[complex number]]s, the cube of a [[imaginary number|purely imaginary]] number is also purely imaginary. For example, {{math|1=[[imaginary unit|''i'']]<sup>3</sup> = −''i''}}.

The [[derivative]] of {{math|''x''<sup>3</sup>}} equals {{math|3''x''<sup>2</sup>}}.

Cubes ''occasionally'' have the surjective property in other [[field (mathematics)|fields]], such as in [[finite field|{{math|'''F'''<sub>''p''</sub>}}]] for such prime {{mvar|p}} that {{math|''p'' ≠ 1 (mod 3)}},<ref>The [[multiplicative group]] of {{math|'''F'''<sub>''p''</sub>}} is [[cyclic group|cyclic]] of order {{math|''p'' − 1}}, and if it is not divisible by 3, then cubes define a [[group automorphism]].</ref> but not necessarily: see the counterexample with rationals [[#In rational numbers|above]]. Also in {{math|'''F'''<sub>7</sub>}} 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: {{math|1=''x''<sup>3</sup> − ''x'' = [[factorization of polynomials|''x''(''x'' − 1)(''x'' + 1)]]}}.

==History==

Determination of the cubes of large numbers was very common in [[History of mathematics|many ancient civilizations]]. Mesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the [[First Babylonian dynasty|Old Babylonian]] period (20th to 16th centuries BC).<ref>{{cite book|last=Cooke|first=Roger|title=The History of Mathematics|url=https://books.google.com/books?id=CFDaj0WUvM8C&pg=PT63|date=8 November 2012|publisher=John Wiley & Sons|isbn=978-1-118-46029-0|page=63}}</ref><ref name="nen">{{cite book|last= Nemet-Nejat|first=Karen Rhea|title=Daily Life in Ancient Mesopotamia|url=https://books.google.com/books?id=lbmXsaTGNKUC&pg=PA306|year=1998|publisher=Greenwood Publishing Group|isbn=978-0-313-29497-6|page=306}}</ref> Cubic equations were known to the [[Greek mathematics|ancient Greek]] mathematician [[Diophantus]].<ref>Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich  1983 {{ISBN|0-387-12159-5}}</ref> [[Hero of Alexandria]] devised a method for calculating cube roots in the 1st century CE.<ref>{{cite journal|last=Smyly|first=J. Gilbart|title=Heron's Formula for Cube Root|journal=Hermathena|year=1920|volume=19|issue=42|pages=64–67|publisher=Trinity College Dublin|jstor=23037103}}</ref> Methods for solving cubic equations and extracting cube roots appear in ''[[The Nine Chapters on the Mathematical Art]]'', a [[Chinese mathematics|Chinese mathematical]] text compiled around the 2nd century BCE and commented on by [[Liu Hui]] in the 3rd century CE.<ref name="oxf">{{cite book|last=Crossley|first=John|last2=W.-C. Lun|first2=Anthony|title=The Nine Chapters on the Mathematical Art: Companion and Commentary|url=https://books.google.com/books?id=eiTJHRGTG6YC&pg=PA213|year=1999|publisher=Oxford University Press|isbn=978-0-19-853936-0|pages=176, 213}}</ref> The [[India]]n mathematician [[Aryabhata]] wrote an explanation of cubes in his work [[Aryabhatiya]]. In 2010 Alberto Zanoni found a [http://bodrato.it/papers/zanoni/AnotherSugarCube.pdf new algorithm]<ref>{{cite journal | url = http://www.springerlink.com/content/q1k57pr4853g1513/ | doi=10.1007/978-3-642-32973-9_4 | title=A New Algorithm for Long Integer Cube Computation with Some Insight into Higher Powers | year=2012 | journal=Lecture Notes in Computer Science | pages=34–46 | last1 = Bodrato | first1 = Marco | last2 = Zanoni | first2 = Alberto}}</ref> to compute the cube of a long integer in a certain range, faster than squaring-and-multiplying.

==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==
<references/>

==References==
*{{Cite journal
  | last1 = Hardy
  | first1 = G. H.
  | last2 = Wright
  | first2 = E. M.
  | title = An Introduction to the Theory of Numbers
  | edition = Fifth
  | publisher = [[Oxford University Press]]
  | location = Oxford
  | date = 1980
  | isbn = 978-0-19-853171-5
  | postscript = <!--None-->
  | url-access = registration
  | url = https://archive.org/details/introductiontoth00hard
  }}
*{{citation
 | last = Wheatstone | first = C.
 | authorlink = Charles Wheatstone
 | title = On the formation of powers from arithmetical progressions
 | journal = [[Proceedings of the Royal Society of London]]
 | volume = 7
 | pages = 145–151
 | year = 1854
 | doi = 10.1098/rspl.1854.0036}}.
{{refend}}

{{Classes of natural numbers}}
{{Series_(mathematics)}}

[[Category:Elementary arithmetic]]
[[Category:Figurate numbers]]
[[Category:Integer sequences]]
[[Category:Integers]]
[[Category:Number theory]]
[[Category:Unary operations]]