{"history":[],"article":{"id":"a32b0e79-b0fe-4c73-a93d-5a51ea2669e1","title":"Cube (algebra)","slug":"cube-algebra","content":":::summary\n:::\n\n# \n[[Image:CubeChart.svg|thumb|280px|\ny = x3\nfor values of\n1 ≤ x ≤ 25\n.]] In [arithmetic](https://bliptext.com/articles/arithmetic) and [algebra](https://bliptext.com/articles/algebra), the **cube** of a number n is its third [power](https://bliptext.com/articles/exponentiation), that is, the result of multiplying three instances of n together.\nThe cube of a number or any other [mathematical expression](https://bliptext.com/articles/expression-mathematics) is denoted by a [superscript](https://bliptext.com/articles/superscript) 3, for example or\n(x + 1)3\n.\nThe cube is also the number multiplied by its [square](https://bliptext.com/articles/square-algebra):\nn3 = n × n2 = n × n × n\n.\nThe cube [function](https://bliptext.com/articles/function-mathematics) is the [function](https://bliptext.com/articles/function-mathematics)\nx ↦ x3\n(often denoted\ny = x3\n) that maps a number to its cube. It is an [odd function](https://bliptext.com/articles/odd-function), as\n(−n)3 = −(n3)\n.\nThe [volume](https://bliptext.com/articles/volume) of a [geometric cube](https://bliptext.com/articles/cube-geometry) is the cube of its side length, giving rise to the name.\nThe [inverse](https://bliptext.com/articles/inverse-function) operation that consists of finding a number whose cube is n is called extracting the [cube root](https://bliptext.com/articles/cube-root) of n. It determines the side of the cube of a given volume.\nIt is also n raised to the one-third power.\nThe [graph](https://bliptext.com/articles/graph-of-a-function) of the cube function is known as the **cubic parabola**.\nBecause the cube function is an odd function, this curve has a [center of symmetry](https://bliptext.com/articles/center-of-symmetry) at the origin, but no [axis of symmetry](https://bliptext.com/articles/axis-of-symmetry).\n\n# In integers\nA ****cube** number**, or a **perfect **cube****, or sometimes just a **cube**, is a number which is the **cube** of an [integer](https://bliptext.com/articles/integer).\nThe non-negative perfect cubes up to 603 are :\nGeometrically speaking, a positive integer m is a perfect cube [if and only if](https://bliptext.com/articles/if-and-only-if) one can arrange m solid unit cubes into a larger, solid cube.\nFor example, 27 small cubes can be arranged into one larger one with the appearance of a [Rubik's Cube](https://bliptext.com/articles/rubik-s-cube), since 3 × 3 × 3 = 27.\nThe difference between the cubes of consecutive integers can be expressed as follows:\nn3 − (n − 1)3 = 3(n − 1)n + 1\n.\nor\n(n + 1)3 − n3 = 3(n + 1)n + 1\n.\nThere is no minimum perfect cube, since the cube of a negative integer is negative.\nFor example, (−4) × (−4) × (−4) = −64.\n\n## Base ten\nUnlike [perfect squares](https://bliptext.com/articles/square-number), perfect cubes do not have a small number of possibilities for the last two digits.\nExcept 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 occur as the last digits of a perfect cube.\nWith [even](https://bliptext.com/articles/even-and-odd-numbers) cubes, there is considerable restriction, for only **00**,\no\n**2**,\ne\n**4**,\no\n**6** and\ne\n**8** can be the last two digits of a perfect cube (where\no\nstands for any odd digit and\ne\nfor 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).\nThe last digits of each 3rd power are:\nIt is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have [digital root](https://bliptext.com/articles/digital-root) **1**, **8** or **9**.\nThat is their values [modulo](https://bliptext.com/articles/modular-arithmetic) 9 may be only 0, 1, and 8.\nMoreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:\n\n## Sums of two cubes\n\n## Sums of three cubes\nIt is conjectured that every integer (positive or negative) not [congruent](https://bliptext.com/articles/modular-arithmetic) to\n±4\nmodulo\n9\ncan be written as a sum of three (positive or negative) cubes with infinitely many ways.\nFor example,.\nIntegers congruent to\n±4\nmodulo\n9\nare excluded because they cannot be written as the sum of three cubes.\nThe smallest such integer for which such a sum is not known is 114.\nIn September 2019, the previous smallest such integer with no known 3-cube sum, 42, was found to satisfy this equation:\nOne solution to is given in the table below for\nn ≤ 78\n, and\nn\nnot congruent to\n4\nor\n5\nmodulo\n9\n. The selected solution is the one that is primitive (\ngcd(x, y, z) = 1\n), is not of the form or (since they are infinite families of solutions), satisfies\n0 ≤\n, and has minimal values for\nand\n(tested in this order).\nOnly primitive solutions are selected since the non-primitive ones can be trivially deduced from solutions for a smaller value of n. For example, for\nn = 24\n, the solution results from the solution by multiplying everything by 8=2^3.\nTherefore, this is another solution that is selected.\nSimilarly, for\nn = 48\n, the solution\n(x, y, z) = (-2, -2, 4)\nis excluded, and this is the solution\n(x, y, z) = (-23, -26, 31)\nthat is selected.\n\n## Fermat's Last Theorem for cubes\nThe equation\nx3 + y3 = z3\nhas no non-trivial (i.e.\nxyz ≠ 0\n) solutions in integers. In fact, it has none in [Eisenstein integers](https://bliptext.com/articles/eisenstein-integers).\nBoth of these statements are also true for the equation\nx3 + y3 = 3z3\n.\n\n## Sum of first n cubes\nThe sum of the first n cubes is the nth [triangle number](https://bliptext.com/articles/triangle-number) squared:\n**Proofs.**\ngives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers.\nHe begins by giving the identity\nThat identity is related to [triangular numbers](https://bliptext.com/articles/triangular-numbers) T_n in the following way:\nand thus the summands forming n^3 start off just after those forming all previous values 1^3 up to (n-1)^3.\nApplying this property, along with another well-known identity:\nwe obtain the following derivation:\nIn the more recent mathematical literature, uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also ); he observes that it may also be proved easily (but uninformatively) by induction, and states that provides \"an interesting old Arabic proof\".\nprovides a purely visual proof, provide two additional proofs, and gives seven geometric proofs.\nFor example, the sum of the first 5 cubes is the square of the 5th triangular number,\nA similar result can be given for the sum of the first y [odd](https://bliptext.com/articles/odd-number) cubes,\nbut x, y must satisfy the negative [Pell equation](https://bliptext.com/articles/pell-equation)\nx2 − 2y2 = −1\n. For example, for\ny = 5\nand\n29\n, then,\nand so on.\nAlso, every [even](https://bliptext.com/articles/even-number) [perfect number](https://bliptext.com/articles/perfect-number), except the lowest, is the sum of the first\n2p−1⁄2\nodd cubes (p = 3, 5, 7, ...):\n# [Visual proof that 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2. | upload.wikimedia.org/wikipedia/commons/2/26/Nicomachus///theorem///3D.svg]\n# [Visual demonstration that the square of a triangular number equals a sum of cubes. | upload.wikimedia.org/wikipedia/commons/2/2f/Sum///of///cubes2.png]\n\n## Sum of cubes of numbers in arithmetic progression\nThere are examples of cubes of numbers in [arithmetic progression](https://bliptext.com/articles/arithmetic-progression) whose sum is a cube:\nwith the first one sometimes identified as the mysterious [Plato's number](https://bliptext.com/articles/plato-s-number).\nThe formula F for finding the sum of n cubes of numbers in arithmetic progression with common difference d and initial cube\na3\n,\nis given by\nA parametric solution to\nis known for the special case of\nd = 1\n, or consecutive cubes, as found by Pagliani in 1829.\n# [One interpretation of Plato's number, 33 + 43 + 53 = 63 | upload.wikimedia.org/wikipedia/commons/2/2a/Plato///number.svg]\n\n## Cubes as sums of successive odd integers\nIn the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ..., the first one is a cube (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.\n\n## Waring's problem for cubes\nEvery positive integer can be written as the sum of nine (or fewer) positive cubes.\nThis upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:\n\n# In rational numbers\nEvery positive [rational number](https://bliptext.com/articles/rational-number) is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes.\n\n# In real numbers, other fields, and rings\n[[Image:X cubed plot.svg|thumb|right|\nplotted on a Cartesian plane]]\nIn [real numbers](https://bliptext.com/articles/real-number), the cube function preserves the order: larger numbers have larger cubes.\nIn other words, cubes (strictly) [monotonically increase](https://bliptext.com/articles/monotonic-function).\nAlso, its [codomain](https://bliptext.com/articles/codomain) is the entire [real line](https://bliptext.com/articles/real-line): the function\nx ↦ x3 : R → R\nis a [surjection](https://bliptext.com/articles/surjection) (takes all possible values).\nOnly three numbers are equal to their own cubes:, , and.\nIf\n−1 < x < 0\nor\n1 < x\n, then\nx3 > x\n. If\nx < −1\nor\n0 < x < 1\n, then\nx3 < x\n. All aforementioned properties pertain also to any higher odd power (\nx5\n,\nx7\n, ...) of real numbers. Equalities and [inequalities](https://bliptext.com/articles/inequality-mathematics) are also true in any [ordered ring](https://bliptext.com/articles/ordered-ring).\nVolumes of [similar](https://bliptext.com/articles/similarity-geometry) Euclidean [solids](https://bliptext.com/articles/solid-geometry) are related as cubes of their linear sizes.\nIn [complex numbers](https://bliptext.com/articles/complex-number), the cube of a [purely imaginary](https://bliptext.com/articles/imaginary-number) number is also [purely imaginary](https://bliptext.com/articles/imaginary-number).\nFor example,\ni3 = −i\n.\nThe [derivative](https://bliptext.com/articles/derivative) of\nx3\nequals\n3x2\n.\nCubes occasionally have the surjective property in other [fields](https://bliptext.com/articles/field-mathematics), such as in\nFp\nfor such prime p that\np ≠ 1 (mod 3)\n, but not necessarily: see the counterexample with rationals [above](https://bliptext.com/articles/).\nAlso in\nF7\nonly three elements 0, ±1 are perfect cubes, of seven total.\n−1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to their own cubes:\nx3 − x = x(x − 1)(x + 1)\n.\n\n# History\nDetermination of the cubes of large numbers was very common in [many ancient civilizations](https://bliptext.com/articles/history-of-mathematics).\nMesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the [Old Babylonian](https://bliptext.com/articles/first-babylonian-dynasty) period (20th to 16th centuries BC).\nCubic equations were known to the [ancient Greek](https://bliptext.com/articles/greek-mathematics) mathematician [Diophantus](https://bliptext.com/articles/diophantus).\n[Hero of Alexandria](https://bliptext.com/articles/hero-of-alexandria) devised a method for calculating cube roots in the 1st century CE.\nMethods for solving cubic equations and extracting cube roots appear in [The Nine Chapters on the Mathematical Art](https://bliptext.com/articles/the-nine-chapters-on-the-mathematical-art), a [Chinese mathematical](https://bliptext.com/articles/chinese-mathematics) text compiled around the 2nd century BCE and commented on by [Liu Hui](https://bliptext.com/articles/liu-hui) in the 3rd century CE.\n\n## Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:40:15.857Z","updated_at":"2025-01-25T19:40:15.857Z","search_vector":"'algebra':2 'cube':1","search_content":" \n \n\n# \n[[Image:CubeChart.svg|thumb|280px|\ny = x3\nfor values of\n1 ≤ x ≤ 25\n.]] In [arithmetic](https://bliptext.com/articles/arithmetic) and [algebra](https://bliptext.com/articles/algebra), the **cube** of a number n is its third [power](https://bliptext.com/articles/exponentiation), that is, the result of multiplying three instances of n together.\nThe cube of a number or any other [mathematical expression](https://bliptext.com/articles/expression-mathematics) is denoted by a [superscript](https://bliptext.com/articles/superscript) 3, for example or\n(x + 1)3\n.\nThe cube is also the number multiplied by its [square](https://bliptext.com/articles/square-algebra):\nn3 = n × n2 = n × n × n\n.\nThe cube [function](https://bliptext.com/articles/function-mathematics) is the [function](https://bliptext.com/articles/function-mathematics)\nx ↦ x3\n(often denoted\ny = x3\n) that maps a number to its cube. It is an [odd function](https://bliptext.com/articles/odd-function), as\n(−n)3 = −(n3)\n.\nThe [volume](https://bliptext.com/articles/volume) of a [geometric cube](https://bliptext.com/articles/cube-geometry) is the cube of its side length, giving rise to the name.\nThe [inverse](https://bliptext.com/articles/inverse-function) operation that consists of finding a number whose cube is n is called extracting the [cube root](https://bliptext.com/articles/cube-root) of n. It determines the side of the cube of a given volume.\nIt is also n raised to the one-third power.\nThe [graph](https://bliptext.com/articles/graph-of-a-function) of the cube function is known as the **cubic parabola**.\nBecause the cube function is an odd function, this curve has a [center of symmetry](https://bliptext.com/articles/center-of-symmetry) at the origin, but no [axis of symmetry](https://bliptext.com/articles/axis-of-symmetry).\n\n# In integers\nA ****cube** number**, or a **perfect **cube****, or sometimes just a **cube**, is a number which is the **cube** of an [integer](https://bliptext.com/articles/integer).\nThe non-negative perfect cubes up to 603 are :\nGeometrically speaking, a positive integer m is a perfect cube [if and only if](https://bliptext.com/articles/if-and-only-if) one can arrange m solid unit cubes into a larger, solid cube.\nFor example, 27 small cubes can be arranged into one larger one with the appearance of a [Rubik's Cube](https://bliptext.com/articles/rubik-s-cube), since 3 × 3 × 3 = 27.\nThe difference between the cubes of consecutive integers can be expressed as follows:\nn3 − (n − 1)3 = 3(n − 1)n + 1\n.\nor\n(n + 1)3 − n3 = 3(n + 1)n + 1\n.\nThere is no minimum perfect cube, since the cube of a negative integer is negative.\nFor example, (−4) × (−4) × (−4) = −64.\n\n## Base ten\nUnlike [perfect squares](https://bliptext.com/articles/square-number), perfect cubes do not have a small number of possibilities for the last two digits.\nExcept 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 occur as the last digits of a perfect cube.\nWith [even](https://bliptext.com/articles/even-and-odd-numbers) cubes, there is considerable restriction, for only **00**,\no\n**2**,\ne\n**4**,\no\n**6** and\ne\n**8** can be the last two digits of a perfect cube (where\no\nstands for any odd digit and\ne\nfor 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).\nThe last digits of each 3rd power are:\nIt is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have [digital root](https://bliptext.com/articles/digital-root) **1**, **8** or **9**.\nThat is their values [modulo](https://bliptext.com/articles/modular-arithmetic) 9 may be only 0, 1, and 8.\nMoreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:\n\n## Sums of two cubes\n\n## Sums of three cubes\nIt is conjectured that every integer (positive or negative) not [congruent](https://bliptext.com/articles/modular-arithmetic) to\n±4\nmodulo\n9\ncan be written as a sum of three (positive or negative) cubes with infinitely many ways.\nFor example,.\nIntegers congruent to\n±4\nmodulo\n9\nare excluded because they cannot be written as the sum of three cubes.\nThe smallest such integer for which such a sum is not known is 114.\nIn September 2019, the previous smallest such integer with no known 3-cube sum, 42, was found to satisfy this equation:\nOne solution to is given in the table below for\nn ≤ 78\n, and\nn\nnot congruent to\n4\nor\n5\nmodulo\n9\n. The selected solution is the one that is primitive (\ngcd(x, y, z) = 1\n), is not of the form or (since they are infinite families of solutions), satisfies\n0 ≤\n, and has minimal values for\nand\n(tested in this order).\nOnly primitive solutions are selected since the non-primitive ones can be trivially deduced from solutions for a smaller value of n. For example, for\nn = 24\n, the solution results from the solution by multiplying everything by 8=2^3.\nTherefore, this is another solution that is selected.\nSimilarly, for\nn = 48\n, the solution\n(x, y, z) = (-2, -2, 4)\nis excluded, and this is the solution\n(x, y, z) = (-23, -26, 31)\nthat is selected.\n\n## Fermat's Last Theorem for cubes\nThe equation\nx3 + y3 = z3\nhas no non-trivial (i.e.\nxyz ≠ 0\n) solutions in integers. In fact, it has none in [Eisenstein integers](https://bliptext.com/articles/eisenstein-integers).\nBoth of these statements are also true for the equation\nx3 + y3 = 3z3\n.\n\n## Sum of first n cubes\nThe sum of the first n cubes is the nth [triangle number](https://bliptext.com/articles/triangle-number) squared:\n**Proofs.**\ngives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers.\nHe begins by giving the identity\nThat identity is related to [triangular numbers](https://bliptext.com/articles/triangular-numbers) T_n in the following way:\nand thus the summands forming n^3 start off just after those forming all previous values 1^3 up to (n-1)^3.\nApplying this property, along with another well-known identity:\nwe obtain the following derivation:\nIn the more recent mathematical literature, uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also ); he observes that it may also be proved easily (but uninformatively) by induction, and states that provides \"an interesting old Arabic proof\".\nprovides a purely visual proof, provide two additional proofs, and gives seven geometric proofs.\nFor example, the sum of the first 5 cubes is the square of the 5th triangular number,\nA similar result can be given for the sum of the first y [odd](https://bliptext.com/articles/odd-number) cubes,\nbut x, y must satisfy the negative [Pell equation](https://bliptext.com/articles/pell-equation)\nx2 − 2y2 = −1\n. For example, for\ny = 5\nand\n29\n, then,\nand so on.\nAlso, every [even](https://bliptext.com/articles/even-number) [perfect number](https://bliptext.com/articles/perfect-number), except the lowest, is the sum of the first\n2p−1⁄2\nodd cubes (p = 3, 5, 7, ...):\n# [Visual proof that 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2. | upload.wikimedia.org/wikipedia/commons/2/26/Nicomachus///theorem///3D.svg]\n# [Visual demonstration that the square of a triangular number equals a sum of cubes. | upload.wikimedia.org/wikipedia/commons/2/2f/Sum///of///cubes2.png]\n\n## Sum of cubes of numbers in arithmetic progression\nThere are examples of cubes of numbers in [arithmetic progression](https://bliptext.com/articles/arithmetic-progression) whose sum is a cube:\nwith the first one sometimes identified as the mysterious [Plato's number](https://bliptext.com/articles/plato-s-number).\nThe formula F for finding the sum of n cubes of numbers in arithmetic progression with common difference d and initial cube\na3\n,\nis given by\nA parametric solution to\nis known for the special case of\nd = 1\n, or consecutive cubes, as found by Pagliani in 1829.\n# [One interpretation of Plato's number, 33 + 43 + 53 = 63 | upload.wikimedia.org/wikipedia/commons/2/2a/Plato///number.svg]\n\n## Cubes as sums of successive odd integers\nIn the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ..., the first one is a cube (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.\n\n## Waring's problem for cubes\nEvery positive integer can be written as the sum of nine (or fewer) positive cubes.\nThis upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:\n\n# In rational numbers\nEvery positive [rational number](https://bliptext.com/articles/rational-number) is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes.\n\n# In real numbers, other fields, and rings\n[[Image:X cubed plot.svg|thumb|right|\nplotted on a Cartesian plane]]\nIn [real numbers](https://bliptext.com/articles/real-number), the cube function preserves the order: larger numbers have larger cubes.\nIn other words, cubes (strictly) [monotonically increase](https://bliptext.com/articles/monotonic-function).\nAlso, its [codomain](https://bliptext.com/articles/codomain) is the entire [real line](https://bliptext.com/articles/real-line): the function\nx ↦ x3 : R → R\nis a [surjection](https://bliptext.com/articles/surjection) (takes all possible values).\nOnly three numbers are equal to their own cubes:, , and.\nIf\n−1 x\n. If\nx < −1\nor\n0 < x < 1\n, then\nx3 < x\n. All aforementioned properties pertain also to any higher odd power (\nx5\n,\nx7\n, ...) of real numbers. Equalities and [inequalities](https://bliptext.com/articles/inequality-mathematics) are also true in any [ordered ring](https://bliptext.com/articles/ordered-ring).\nVolumes of [similar](https://bliptext.com/articles/similarity-geometry) Euclidean [solids](https://bliptext.com/articles/solid-geometry) are related as cubes of their linear sizes.\nIn [complex numbers](https://bliptext.com/articles/complex-number), the cube of a [purely imaginary](https://bliptext.com/articles/imaginary-number) number is also [purely imaginary](https://bliptext.com/articles/imaginary-number).\nFor example,\ni3 = −i\n.\nThe [derivative](https://bliptext.com/articles/derivative) of\nx3\nequals\n3x2\n.\nCubes occasionally have the surjective property in other [fields](https://bliptext.com/articles/field-mathematics), such as in\nFp\nfor such prime p that\np ≠ 1 (mod 3)\n, but not necessarily: see the counterexample with rationals [above](https://bliptext.com/articles/).\nAlso in\nF7\nonly three elements 0, ±1 are perfect cubes, of seven total.\n−1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to their own cubes:\nx3 − x = x(x − 1)(x + 1)\n.\n\n# History\nDetermination of the cubes of large numbers was very common in [many ancient civilizations](https://bliptext.com/articles/history-of-mathematics).\nMesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the [Old Babylonian](https://bliptext.com/articles/first-babylonian-dynasty) period (20th to 16th centuries BC).\nCubic equations were known to the [ancient Greek](https://bliptext.com/articles/greek-mathematics) mathematician [Diophantus](https://bliptext.com/articles/diophantus).\n[Hero of Alexandria](https://bliptext.com/articles/hero-of-alexandria) devised a method for calculating cube roots in the 1st century CE.\nMethods for solving cubic equations and extracting cube roots appear in [The Nine Chapters on the Mathematical Art](https://bliptext.com/articles/the-nine-chapters-on-the-mathematical-art), a [Chinese mathematical](https://bliptext.com/articles/chinese-mathematics) text compiled around the 2nd century BCE and commented on by [Liu Hui](https://bliptext.com/articles/liu-hui) in the 3rd century CE.\n\n## Sources","revisionCount":0},"hasMore":false,"nextCursor":null}