Cube number

n³ = n⋅n⋅n

A cube number (from latin cubus , "cube") which arises when a number is a natural number two with himself multiplied . For example, is a cube number. The first cube numbers are ${\ displaystyle 27 = 3 \ times 3 \ times 3}$

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ... (sequence A000578 in OEIS )

For some authors, the zero is not a cube number, so the sequence of numbers only starts with the one.

The term cubic number is derived from the geometric shape of the cube . The number of bricks that you need to build a cube always corresponds to a cube number. For example, a cube with side length 3 can be placed with the help of 27 stones.

Because of this relationship with a geometric figure, the cube numbers are among the figured numbers , which also include the square numbers and tetrahedral numbers .

properties

From the successive blocks of one, two, three, four, five, ... odd natural numbers in ascending order, the cube numbers can be generated by summation :

{\ displaystyle \ underbrace {1} _ {1} \ \ underbrace {3 \ 5} _ {8} \ \ underbrace {7 \ 9 \ 11} _ {27} \ \ underbrace {13 \ 15 \ 17 \ 19} _ {64} \ \ underbrace {21 \ 23 \ 25 \ 27 \ 29} _ {125} \ \ ldots}

Starting from the sequence of the centered hexagonal numbers 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, ... one obtains the -th cube number as the sum of the first terms in the sequence:

{\ displaystyle n}

{\ displaystyle n}

{\ displaystyle {\ begin {aligned} 1 & = 1 \\ 8 & = 1 + 7 \\ 27 & = 1 + 7 + 19 \\ 64 & = 1 + 7 + 19 + 37 \\ 125 & = 1 + 7 + 19 + 37 +61 \\\ ldots & = \ ldots \ end {aligned}}}

The sum of the first cube numbers is equal to the square of the -th triangular number : ${\ displaystyle n}$ ${\ displaystyle n}$
${\ displaystyle \ sum _ {i = 1} ^ {n} i ^ {3} = 1 ^ {3} + 2 ^ {3} + \ ldots + n ^ {3} = \ left ({\ frac {n (n + 1)} {2}} \ right) ^ {2}}$

Every natural number can be represented as a sum of a maximum of nine cubic numbers (solution of Waring's problem for the exponent 3). The number 23 shows that 9 summands can be necessary. This has the representation , but obviously none with less cubic summands.

{\ displaystyle 23 = 8 + 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1 \,}

The sum of any two cubic numbers can never be a cube number itself. In other words, this means that the equation has no solution with natural numbers . This special case of Fermat's conjecture was proven by Leonhard Euler in 1753 . Allowed more than two terms to, it can happen that a cube number is represented as the sum of cubes, as the following example (even with three directly consecutive cubes) shows .
${\ displaystyle a ^ {3} + b ^ {3} = c ^ {3} \,}$
${\ displaystyle a, b, c}$
${\ displaystyle 3 ^ {3} + 4 ^ {3} + 5 ^ {3} = 6 ^ {3} \,}$

Sum of the reciprocal values

The sum of the reciprocal values of all cube numbers is called the Apéry constant . It corresponds to the value of the Riemannian function ${\ displaystyle \ zeta}$ at position 3.

{\ displaystyle \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {3}}} = \ zeta {(3)} = 1 {,} 2020569 \ ldots}

Generating function

Each sequence of whole (or real) numbers can be assigned a formal power series , the so-called generating function . In this context, however, it is common to start the sequence of cube numbers with 0, i.e. to consider the sequence . The generating function of the cube numbers is then ${\ displaystyle (a_ {i}) _ {i \ geq 0}}$ ${\ displaystyle \ sum _ {i \ geq 0} a_ {i} x ^ {i}}$ ${\ displaystyle 0,1,8,27,64, \ ldots}$

{\ displaystyle \ sum _ {i \ geq 0} i ^ {3} x ^ {i} = x + 8x ^ {2} + 27x ^ {3} + 64x ^ {4} + \ ldots = {\ frac { x (x ^ {2} + 4x + 1)} {(x-1) ^ {4}}}}

keyboard

On the German PC keyboard , the ³ character is the third assignment on the 3 key and can be entered using the Alt-Gr key. You can often use the two keys Ctrl and Alt instead of Alt Gr . In an Apple keyboard , however, there is no defined key combination for the ³ mark. The ³-character and the code number (hexadecimal ) are part of the character coding ISO 8859-1 (or ISO 8859-15 ) and thus also the Unicode block Latin-1, supplement . 179B3

Web links

Wiktionary: cube number - explanations of meanings, word origins, synonyms, translations

Eric W. Weisstein : Cubic Number . In: MathWorld (English).