Cube (algebra)

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y = x for values of 1 <= x <= 25. CubeChart.svg
y = x for values of 1 ≤ x ≤ 25.

In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x + 1)3.

Contents

The cube is also the number multiplied by its square:

n3 = n × n2 = n × n × n.

The cube function is the function xx3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as

(−n)3 = −(n3).

The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists 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.

The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at 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 =1113 =1331213 =9261313 =29,791413 =68,921513 =132,651
23 =8123 =1728223 =10,648323 =32,768423 =74,088523 =140,608
33 =27133 =2197233 =12,167333 =35,937433 =79,507533 =148,877
43 =64143 =2744243 =13,824343 =39,304443 =85,184543 =157,464
53 =125153 =3375253 =15,625353 =42,875453 =91,125553 =166,375
63 =216163 =4096263 =17,576363 =46,656463 =97,336563 =175,616
73 =343173 =4913273 =19,683373 =50,653473 =103,823573 =185,193
83 =512183 =5832283 =21,952383 =54,872483 =110,592583 =195,112
93 =729193 =6859293 =24,389393 =59,319493 =117,649593 =205,379
103 =1000203 =8000303 =27,000403 =64,000503 =125,000603 =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:

n3 − (n − 1)3 = 3(n − 1)n + 1.

or

(n + 1)3n3 = 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 occur as the last digits of 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 (in this case 26).

The last digits of each 3rd power are:

0187456329

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:

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.

Sums of three cubes

It is conjectured that every integer (positive or negative) not congruent to ±4 modulo 9 can be written as a sum of three (positive or negative) cubes with infinitely many ways. [1] For example, . (note that the integer 0 cannot be written in this way, by Fermat’s Last Theorem) Integers congruent to ±4 modulo 9 are excluded because they cannot be written as the sum of three cubes, since cubes are congruent to 0 or ±1 mod 9.

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: [2] [3]

One solution to is given in the table below for n ≤ 160, and n not congruent to 4 or 5 modulo 9. The selected solution is the one that is primitive (gcd(x, y, z) = 1), is not of the form or (since they are infinite families of solutions, however, for n3, ALL solutions are of the form a=q[1−(x−3y)(x^2+3y^2)], b=−q[1−(x+3y)(x^2+3y^2)], c=q[(x^2 +3y^2)^2−(x+3y)], n=q[(x^2+3y^2)^2−(x−3y)], [4] i.e. all solutions are in infinitely families, there is no solution which is not in infinitely families, but for 2n3, the only infinity family is , i.e. all solutions with none of x+y, x+z, y+z, is 2n, are not in infinitely families, numbers of the form n3 or 2n3 are the only numbers with representations that can be parameterized by quartic polynomials), satisfies 0 ≤ |x||y||z|, and has minimal values for |z| and |y| (tested in this order). [5] [6] [7]

Only primitive solutions are selected since the non-primitive ones can be trivially deduced from solutions for a smaller value of n. For example, for n = 24, the solution results from the solution by multiplying everything by Therefore, this is another solution that is selected. Similarly, for n = 48, the solution (x, y, z) = (-2, -2, 4) is excluded, and this is the solution (x, y, z) = (-23, -26, 31) that is selected.

The only remaining unsolved cases up to 1000 are 114, 390, 627, 633, 732, 921, and 975, and there are no known primitive solutions (i.e. ) for 192, 375, and 600 [8] [9]

Note that there is only one known solution for 3 besides (1, 1, 1) and (−4, −4, 5):

nxyznxyznxyznxyz
1910−12421260212329733563180435758145817515−80538738812075974811017−181209461531−1643
212149283480205−35288754322382−11−1114123−1−15
311144−5−7883−2341240−15
6−1−12452−3484−8241191−4153172641639611125−3−46
70−1246−23387−1972−41264271126015
8915−164767−8883−45127−144
901248−23−26318966−71285531152−1193
1011251602659−79690−134129144
11−2−23522396129245460702901317−6192271286591034132−125
12710−1153−13392134133025
15−12254−7−111293−5−57134125
16−511−1609162655133961085313139−152501352−67
1712256−11−212297−1−35136225582−593
18−1−23571−24980−35137−9−1113
190−2360−1−4599234138−77−86103
201−23610−45100−3−67141225
21−11−141662233101−344142−3−78
24−2901096694−1555055555515584139827630−14102118229−239143702384942−84958
25−1−1364−3−56105−4−78144−235
260−13650141062−35145−7−810
27−4−5666114107−28−4851146−5−910
28013692−45108−948−11651345147−50−5667
29113701120−21109−2−25150260317−367
30−283059965−2218888517222042293271−12411010993891916540290030−16540291649151−135
33−2736111468807040−877840544286223988661289752875287279−10111−296−881892152035
34−12373124114???153135
350237466229832190556283450105697727−284650292555885115−6−1011154−4−57
36123754381159435203083−435203231116−1−25155344
370−34782653−551170−25156688446456252232194323−68845427846
381−3479−19−333511833415980119−130
39117367134476−1593808069241103532−112969119−2−67160235

Fermat's Last Theorem for cubes

The equation x3 + y3 = z3 has no non-trivial (i.e. xyz ≠ 0) solutions in integers. In fact, it has none in Eisenstein integers. [10]

Both of these statements are also true for the equation [11] x3 + y3 = nz3 for n = 3, 4, 5, 10, 11, 14, 18, 21, 23, 24, 25, 29, 32, 36, 38, 39, 40, 41, 44, 45, 46, 47, 52, 55, 57, 59, 60, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, ... (sequence A185345 in the OEIS ), also for n = k3 and 2k3, there are no nontrivial solutions, i.e. solutions other than 03 + (kx)3 = (k3)x3 and (kx)3 + (kx)3 = (2k3)x3, for all other positive integers values of k, there are infinitely many primitive solutions, see next section.

Sum of first n cubes

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

Visual proof that 1 + 2 + 3 + 4 + 5 = (1 + 2 + 3 + 4 + 5). Nicomachus theorem 3D.svg
Visual proof that 1 + 2 + 3 + 4 + 5 = (1 + 2 + 3 + 4 + 5).

Proofs. CharlesWheatstone  ( 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

That identity is related to triangular numbers in the following way:

and thus the summands forming start off just after those forming all previous values up to . Applying this property, along with another well-known identity:

we obtain the following derivation:

Visual demonstration that the square of a triangular number equals a sum of cubes. Sum of cubes2.png
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,

A similar result can be given for the sum of the first y odd cubes,

but x, y must satisfy the negative Pell equation x2 − 2y2 = −1. For example, for y = 5 and 29, then,

and so on. Also, every even perfect number, except the lowest, is the sum of the first 2p−1/2
odd cubes (p = 3, 5, 7, ...):

Sum of cubes of numbers in arithmetic progression

One interpretation of Plato's number, 3 + 4 + 5 = 6 Plato number.svg
One interpretation of Plato's number, 3 + 4 + 5 = 6

There are examples of cubes of numbers in arithmetic progression whose sum is a cube:

with the first one sometimes identified as the mysterious 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 a3,

is given by

A parametric solution to

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. [12]

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, [13] and there are rationals that are not the sum of two rational cubes. [14]

However, there are rational numbers that are not the sum of two rational cubes. [15] For positive integers, the positive integers that is the sum of two positive noninteger rational cubes (also the positive integer that can be written as sum of two rational cubes in infinitely many different ways) are

6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117, 120, 123, 124, 126, 127, 130, 132, 133, 134, 136, 139, 140, 141, 142, 143, 151, 152, 153, 156, 157, 159, 160, ... (sequence A228499 in the OEIS ) (also see OEIS:  A159843 and OEIS:  A020898 )

The following is a list of the integer solution to (i.e. ) with the smallest c > 1. (we let a < b) [16] [17] [18] (For the sequence of the solutions of a and b, see OEIS:  A254326 and OEIS:  A254324 , and for the same sequences of allowing negative a-values, see OEIS:  A190580 , OEIS:  A190356 and OEIS:  A190581 for the a, b and c)

nabc
6173721
7453
9415280564497676702467503348671682660
12198939
13273
15397683294
17???
19352
201197
2217299254699954
26537528
28???
3010716357
31???
335231853582
34???
35???
3718197
42???
43172
48347421
495308144654017
50???

In real numbers, other fields, and rings

y = x plotted on a Cartesian plane X cubed plot.svg
y = x 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 xx3 : RR 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 x3 > x. If x < −1 or 0 < x < 1, then x3 < x. All aforementioned properties pertain also to any higher odd power (x5, x7, ...) 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 x3 equals 3x2.

Cubes occasionally have the surjective property in other fields, such as in Fp for such prime p that p ≠ 1 (mod 3), [19] but not necessarily: see the counterexample with rationals above. Also in F7 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: x3x = 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). [20] [21] Cubic equations were known to the ancient Greek mathematician Diophantus. [22] Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. [23] 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. [24]

See also

Related Research Articles

In mathematics, the Bernoulli numbersBn are a sequence of rational numbers which occur frequently in number theory. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

In number theory, a Carmichael number is a composite number which satisfies the modular arithmetic congruence relation:

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:

Fibonacci number Integer in the infinite Fibonacci sequence

In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,

Pythagorean triple Three positive integers, the squares of two of which sum to the square of the third

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

Factorization (Mathematical) decomposition into a product

In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x2 – 4.

Root of unity Number that has an integer power equal to 1

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four.

Square root of 2 Unique positive real number which when multiplied by itself gives 2

The square root of 2 is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or , and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.

Powerful number Numbers whose prime factors all divide the number more than once

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

126 is the natural number following 125 and preceding 127.

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Pell number

In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. More generally, if a polynomial factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of p.

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

In mathematics, poly-Bernoulli numbers, denoted as , were defined by M. Kaneko as

Squared triangular number Square of a triangular number

In number theory, the sum of the first n cubes is the square of the nth triangular number. That is,

In mathematics, particularly in the area of number theory, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as

In mathematics and statistics, sums of powers occur in a number of contexts:

References

Citations

  1. Huisman, Sander G. (27 Apr 2016). "Newer sums of three cubes". arXiv: 1604.07746 [math.NT].
  2. "NEWS: The Mystery of 42 is Solved - Numberphile" https://www.youtube.com/watch?v=zyG8Vlw5aAw
  3. Mathematicians Solve '42' Problem With Planetary Supercomputer
  4. A NEW METHOD IN THE PROBLEM OF THREE CUBES
  5. Sequences A060465, A060466 and A060467 in OEIS
  6. Threecubes
  7. n=x^3+y^3+z^3
  8. n=x^3+y^3+z^3
  9. threecubes
  10. Hardy & Wright, Thm. 227
  11. Hardy & Wright, Thm. 232
  12. "A Collection of Algebraic Identities".[ permanent dead link ]
  13. Hardy & Wright, Thm. 234
  14. Hardy & Wright, Thm. 233
  15. Hardy & Wright, Thm. 233
  16. C-sequence for n=3
  17. Rational Points on Elliptic Curves: x^3+y^3=n (n \in [1..10000)]
  18. Solutions of Diophantine equation x^3+y^3=A.z^3
  19. The multiplicative group of Fp is cyclic of order p − 1, and if it is not divisible by 3, then cubes define a group automorphism.
  20. Cooke, Roger (8 November 2012). The History of Mathematics. John Wiley & Sons. p. 63. ISBN   978-1-118-46029-0.
  21. Nemet-Nejat, Karen Rhea (1998). Daily Life in Ancient Mesopotamia . Greenwood Publishing Group. p.  306. ISBN   978-0-313-29497-6.
  22. Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN   0-387-12159-5
  23. Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root". Hermathena. Trinity College Dublin. 19 (42): 64–67. JSTOR   23037103.
  24. 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.

Bibliography