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| {{redirect|Third power|the band|Third Power}}
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| {{redirect|Cubed|other uses|Cube (disambiguation)}}
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| {{redirect3|³|Its literal meaning is the numeral "[[3 (number)|3]]" in [[superscript]]}}
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| [[Image:CubeChart.svg|thumb|280px|{{math|1=''y'' = ''x''<sup>3</sup>}} for values of {{math|0 ≤ ''x'' ≤ 25}}.]]
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| 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:
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| :{{math|size=120%|1=''n''<sup>3</sup> = ''n'' × ''n'' × ''n''}}.
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| It is also the number multiplied by its [[square (algebra)|square]]:
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| :{{math|size=120%|1=''n''<sup>3</sup> = ''n'' × ''n''<sup>2</sup>}}.
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| 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.
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| Both cube and cube root are [[odd function]]s:
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| :{{math|size=120%|1=(−''n'')<sup>3</sup> = −(''n''<sup>3</sup>)}}.
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| The cube of a number or any other [[expression (mathematics)|mathematical expression]] is denoted by a [[superscript]] 3, for example 2<sup>3</sup> = 8 or {{math|(''x'' + 1)<sup>3</sup>}}.
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| ==In integers==
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| A '''cube number''', or a '''perfect cube''', or sometimes just a '''cube''' is a number which is the cube of an [[integer]].
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| The positive perfect cubes up to 60<sup>3</sup> are {{OEIS|id=A000578}}:
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| {| width="65%"
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| | 1<sup>3</sup> = 1 || 11<sup>3</sup> = 1331 || 21<sup>3</sup> = 9261 || 31<sup>3</sup> = 29791
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| | 41<sup>3</sup> = 68921 || 51<sup>3</sup> = 132651
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| |-
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| | 2<sup>3</sup> = 8 || 12<sup>3</sup> = 1728 || 22<sup>3</sup> = 10648 || 32<sup>3</sup> = 32768
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| | 42<sup>3</sup> = 74088 || 52<sup>3</sup> = 140608
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| |-
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| | 3<sup>3</sup> = 27 || 13<sup>3</sup> = 2197 || 23<sup>3</sup> = 12167 || 33<sup>3</sup> = 35937
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| | 43<sup>3</sup> = 79507 || 53<sup>3</sup> = 148877
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| |-
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| | 4<sup>3</sup> = 64 || 14<sup>3</sup> = 2744 || 24<sup>3</sup> = 13824 || 34<sup>3</sup> = 39304
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| | 44<sup>3</sup> = 85184 || 54<sup>3</sup> = 157464
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| |-
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| | 5<sup>3</sup> = 125 || 15<sup>3</sup> = 3375 || 25<sup>3</sup> = 15625 || 35<sup>3</sup> = 42875
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| | 45<sup>3</sup> = 91125 || 55<sup>3</sup> = 166375
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| |-
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| | 6<sup>3</sup> = 216 || 16<sup>3</sup> = 4096 || 26<sup>3</sup> = 17576 || 36<sup>3</sup> = 46656
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| | 46<sup>3</sup> = 97336 || 56<sup>3</sup> = 175616
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| |-
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| | 7<sup>3</sup> = 343 || 17<sup>3</sup> = 4913 || 27<sup>3</sup> = 19683 || 37<sup>3</sup> = 50653
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| | 47<sup>3</sup> = 103823 || 57<sup>3</sup> = 185193
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| |-
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| | 8<sup>3</sup> = 512 || 18<sup>3</sup> = 5832 || 28<sup>3</sup> = 21952 || 38<sup>3</sup> = 54872
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| | 48<sup>3</sup> = 110592 || 58<sup>3</sup> = 195112
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| |-
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| | 9<sup>3</sup> = 729 || 19<sup>3</sup> = 6859 || 29<sup>3</sup> = 24389 || 39<sup>3</sup> = 59319
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| | 49<sup>3</sup> = 117649 || 59<sup>3</sup> = 205379
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| |-
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| | 10<sup>3</sup> = 1000 || 20<sup>3</sup> = 8000 || 30<sup>3</sup> = 27000 || 40<sup>3</sup> = 64000
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| | 50<sup>3</sup> = 125000 || 60<sup>3</sup> = 216000
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| |}
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| Geometrically speaking, a positive number {{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 3 × 3 × 3 = 27.
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| The pattern between every perfect cube from negative infinity to positive infinity is as follows,
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| :{{math|size=120%|1=''n''<sup>3</sup> = (''n'' − 1)<sup>3</sup> + 3(''n'' − 1)''n'' + 1}}.
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| or
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| :{{math|size=120%|1=''n''<sup>3</sup> = (''n'' + 1)<sup>3</sup> − 3(''n'' + 1)''n'' − 1}}.
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| There is no smallest perfect cube, since negative integers are included. For example, (−4) × (−4) × (−4) = −64.
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| ===Base ten===
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| 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''', ''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.
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| 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:
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| * If the number is divisible by 3, its cube has digital root 9;
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| * If it has a remainder of 1 when divided by 3, its cube has digital root 1;
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| * If it has a remainder of 2 when divided by 3, its cube has digital root 8.
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| ===Waring's problem for cubes===
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| {{main|Waring's problem}}
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| 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:
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| :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>.
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| ===Fermat's last theorem for cubes===
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| {{main|Fermat's last theorem}}
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| 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>
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| 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>}}.
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| ===Sum of first ''n'' cubes===
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| The sum of the first {{mvar|n}} cubes is the {{mvar|n}}th [[triangle number]] squared:
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| :<math>1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2=\left(\frac{n(n+1)}{2}\right)^2.</math>
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| For example, the sum of the first 5 cubes is the square of the 5th triangular number,
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| :<math>1^3+2^3+3^3+4^3+5^3 = 15^2 \,</math>
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| A similar result can be given for the sum of the first {{mvar|y}} [[odd number|odd]] cubes,
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| :<math>1^3+3^3+\dots+(2y-1)^3 = (xy)^2</math>
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| but {{mvar|x}}, {{mvar|y}} must satisfy the negative [[Pell equation]] <math>x^2-2y^2 = -1</math>. For example, for {{math|1=''y'' = 5}} and {{math|29}}, then,
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| :<math>1^3+3^3+\dots+9^3 = (7\cdot 5)^2 \,</math>
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| :<math>1^3+3^3+\dots+57^3 = (41\cdot 29)^2</math>
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| and so on. Also, every [[even number|even]] [[perfect number]], except the first one, is the sum of the first [[power of two|{{math|2<sup>(''p''−1)/2</sup>}}]] odd cubes,
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| :<math>28 = 2^2(2^3-1) = 1^3+3^3</math>
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| :<math>496 = 2^4(2^5-1) = 1^3+3^3+5^3+7^3</math>
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| :<math>8128 = 2^6(2^7-1) = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3</math>
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| ===Sum of cubes in arithmetic progression===
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| There are examples of cubes in [[arithmetic progression]] whose sum is a cube,
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| :<math>3^3+4^3+5^3 = 6^3</math>
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| :<math>11^3+12^3+13^3+14^3 = 20^3</math>
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| :<math>31^3+33^3+35^3+37^3+39^3+41^3 = 66^3</math>
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| with the first one also known as [[Plato's number]]. The formula {{mvar|F}} for finding the sum of an {{mvar|n}} number of cubes in arithmetic progression with common difference {{mvar|d}} and initial cube {{math|''a''<sup>3</sup>}},
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| :<math>F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+\cdots+(a+dn-d)^3</math>
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| is given by
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| :<math>F(d,a,n) = (n/4)(2a-d+dn)(2a^2-2ad+2adn-d^2n+d^2n^2)</math>
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| A parametric solution to
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| :<math>F(d,a,n) = y^3</math>
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| 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}} = 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}}</ref>
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| ==In rational numbers==
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| 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>
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| ==In real numbers, other fields, and rings==
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| {{more information|cubic function}}
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| [[Image:X cubed plot.svg|thumb|right]]
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| 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 equal to the 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>}}, …) of real numbers. Equalities and [[inequality (mathematics)|inequalities]] are also true in any [[ordered ring]].
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| Volumes of [[similarity (geometry)|similar]] Euclidean [[solid geometry|solids]] are related as cubes of their linear sizes.
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| 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''}}.
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| The [[derivative]] of {{math|''x''<sup>3</sup>}} equals to {{math|3''x''<sup>2</sup>}}.
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| 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)]]}}.
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| ==History==
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| Determination of the cubes of large numbers was very common in [[History_of_mathematics|many ancient civilizations]]. [[Aryabhata]], the ancient [[India]]n mathematician in his famous work [[Aryabhatiya]] explains about the mathematical meaning of cube (Aryabhatiya, 2–3), as
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| "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 [[Sanskrit]], where the word ''Ghhana'' means a factor of power with the number, multiplied by itself three times and also a cubical structure. In 2010 Alberto Zanoni found a [http://bodrato.it/papers/zanoni/AnotherSugarCube.pdf new algorithm]<ref>http://www.springerlink.com/content/q1k57pr4853g1513/</ref> to compute the cube of a long integer in a certain range, faster than squaring-and-multiplying.
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| ==Notes==
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| <references/>
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| ==See also==
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| * [[Perfect power]]
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| * [[Euler's sum of powers conjecture]]
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| * [[Taxicab number]]
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| * [[Cabtaxi number]]
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| * [[Doubling the cube]]
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| * [[Kepler's laws of planetary motion #Third law]]
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| * [[Monkey saddle]]
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| ==References==
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| *{{Cite document
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| | last1 = Hardy | first1 = G. H.
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| | last2 = Wright | first2 = E. M.
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| | title = An Introduction to the Theory of Numbers (Fifth edition)
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| | publisher = [[Oxford University Press]]
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| | location = Oxford
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| | date = 1980
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| | isbn = 978-0-19-853171-5
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| | postscript = <!--None-->}}
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| ==External links==
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| *[http://www.alpertron.com.ar/FCUBES.HTM A Java applet that decomposes an integer number not congruent to 4 or 5 (mod 9) into a sum of four cubes.]
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| {{Classes of natural numbers}}
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| [[Category:Integers]]
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| [[Category:Number theory]]
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| [[Category:Elementary arithmetic]]
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| [[Category:Integer sequences]]
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| [[Category:Figurate numbers]]
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