Английская Википедия:Cube (algebra)
Шаблон:Short description Шаблон:Redirect Шаблон:Redirect Шаблон:Redirect Шаблон:Pp-sock
In arithmetic and algebra, the cube of a number Шаблон:Mvar is its third power, that is, the result of multiplying three instances of Шаблон:Mvar together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example Шаблон:Nowrap or Шаблон:Math.
The cube is also the number multiplied by its square:
The cube function is the function Шаблон:Math (often denoted Шаблон:Math) that maps a number to its cube. It is an odd function, as
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 Шаблон:Mvar is called extracting the cube root of Шаблон:Mvar. It determines the side of the cube of a given volume. It is also Шаблон:Mvar 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
Шаблон:See also A cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The non-negative perfect cubes up to 603 are Шаблон: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 Шаблон:Mvar is a perfect cube if and only if one can arrange Шаблон:Mvar 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 Шаблон:Nowrap.
The difference between the cubes of consecutive integers can be expressed as follows:
or
There is no minimum perfect cube, since the cube of a negative integer is negative. For example, Шаблон:Nowrap.
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, Шаблон:Math2, Шаблон:Math4, Шаблон:Math6 and Шаблон:Math8 can be the last two digits of a perfect cube (where Шаблон:Math stands for any odd digit and Шаблон:Math for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number Шаблон:Nowrap and a cube number Шаблон:Nowrap. This happens if and only if the number is a perfect sixth power (in this case 2Шаблон:Sup).
The last digits of each 3rd power are:
| 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 modulo 9 may be only 0, 1, and 8. 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 \text{ (actually} \quad 0 \pmod {27}\text{)};</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 2 \pmod 3 \quad \text{then} \quad x^3\equiv 8 \pmod 9.</math>
Sums of two cubes
Sums of three cubes
It is conjectured that every integer (positive or negative) not congruent to Шаблон:Math modulo Шаблон:Math can be written as a sum of three (positive or negative) cubes with infinitely many ways.[1] For example, <math> 6 = 2^3+(-1)^3+(-1)^3</math>. Integers congruent to Шаблон:Math modulo Шаблон:Math are excluded because they cannot be written as the sum of three cubes.
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]Шаблон:Better source needed
- <math> 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3.</math>
One solution to <math>x^3 + y^3 + z^3 = n</math> is given in the table below for Шаблон:Math, and Шаблон:Math not congruent to Шаблон:Math or Шаблон:Math modulo Шаблон:Math. The selected solution is the one that is primitive (Шаблон:Math), is not of the form <math>c^3+(-c)^3+n^3=n^3</math> or <math>(n+6nc^3)^3+(n-6nc^3)^3+(-6nc^2)^3=2n^3</math> (since they are infinite families of solutions), satisfies Шаблон:Math, and has minimal values for Шаблон:Math and Шаблон:Math (tested in this order).[3][4][5]
Only primitive solutions are selected since the non-primitive ones can be trivially deduced from solutions for a smaller value of Шаблон:Mvar. For example, for Шаблон:Math, the solution <math>2^3+2^3+2^3 =24</math> results from the solution <math>1^3+1^3+1^3=3</math> by multiplying everything by <math>8=2^3.</math> Therefore, this is another solution that is selected. Similarly, for Шаблон:Math, the solution Шаблон:Math is excluded, and this is the solution Шаблон:Math that is selected.
Шаблон:Sums of three cubes table
Fermat's Last Theorem for cubes
The equation Шаблон:Math has no non-trivial (i.e. Шаблон:Math) solutions in integers. In fact, it has none in Eisenstein integers.[6]
Both of these statements are also true for the equation[7] Шаблон:Math.
Sum of first n cubes
The sum of the first Шаблон:Mvar cubes is the Шаблон:Mvarth 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>
Proofs. Шаблон:Harvs 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>
In the more recent mathematical literature, Шаблон:Harvtxt uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also Шаблон:Harvnb); he observes that it may also be proved easily (but uninformatively) by induction, and states that Шаблон:Harvtxt provides "an interesting old Arabic proof". Шаблон:Harvtxt provides a purely visual proof, Шаблон:Harvtxt provide two additional proofs, and Шаблон:Harvtxt 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 Шаблон:Mvar odd cubes,
- <math>1^3+3^3+\dots+(2y-1)^3 = (xy)^2</math>
but Шаблон:Mvar, Шаблон:Mvar must satisfy the negative Pell equation Шаблон:Math. For example, for Шаблон:Math and Шаблон:Math, 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 perfect number, except the lowest, is the sum of the first Шаблон:Math 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
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 Шаблон:Mvar for finding the sum of Шаблон:Mvar cubes of numbers in arithmetic progression with common difference Шаблон:Mvar and initial cube Шаблон:Math,
- <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 Шаблон:Math, or consecutive cubes, but only sporadic solutions are known for integer Шаблон:Math, such as Шаблон:Mvar = 2, 3, 5, 7, 11, 13, 37, 39, etc.[8]
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 (Шаблон:Nowrap); the sum of the next two is the next cube (Шаблон:Nowrap); the sum of the next three is the next cube (Шаблон:Nowrap); and so forth.
Waring's problem for cubes
Шаблон:Main 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.
In rational numbers
Every positive rational number is the sum of three positive rational cubes,[9] and there are rationals that are not the sum of two rational cubes.[10]
In real numbers, other fields, and rings
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 Шаблон:Math is a surjection (takes all possible values). Only three numbers are equal to their own cubes: Шаблон:Num, Шаблон:Num, and Шаблон:Num. If Шаблон:Math or Шаблон:Math, then Шаблон:Math. If Шаблон:Math or Шаблон:Math, then Шаблон:Math. All aforementioned properties pertain also to any higher odd power (Шаблон:Math, Шаблон:Math, ...) 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, Шаблон:Math.
The derivative of Шаблон:Math equals Шаблон:Math.
Cubes occasionally have the surjective property in other fields, such as in Шаблон:Math for such prime Шаблон:Mvar that Шаблон:Math,[11] but not necessarily: see the counterexample with rationals above. Also in Шаблон:Math 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: Шаблон:Math.
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).[12][13] Cubic equations were known to the ancient Greek mathematician Diophantus.[14] Hero of Alexandria devised a method for calculating cube roots in the 1st century CE.[15] 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.[16]
See also
- Cabtaxi number
- Cubic equation
- Doubling the cube
- Eighth power
- Euler's sum of powers conjecture
- Fifth power
- Fourth power
- Kepler's laws of planetary motion#Third law
- Monkey saddle
- Perfect power
- Seventh power
- Sixth power
- Square
- Taxicab number
References
Sources
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
Шаблон:Figurate numbers Шаблон:Classes of natural numbers Шаблон:Series (mathematics)
- ↑ Шаблон:Cite arXiv
- ↑ "NEWS: The Mystery of 42 is Solved - Numberphile" https://www.youtube.com/watch?v=zyG8Vlw5aAw
- ↑ Sequences Шаблон:OEIS link, Шаблон:OEIS link and Шаблон:OEIS link in OEIS
- ↑ Threecubes
- ↑ n=x^3+y^3+z^3
- ↑ Hardy & Wright, Thm. 227
- ↑ Hardy & Wright, Thm. 232
- ↑ Шаблон:Cite webШаблон:Dead link
- ↑ Hardy & Wright, Thm. 234
- ↑ Hardy & Wright, Thm. 233
- ↑ The multiplicative group of Шаблон:Math is cyclic of order Шаблон:Math, and if it is not divisible by 3, then cubes define a group automorphism.
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 Шаблон:ISBN
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
