Cube number

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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

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 .


  • From the successive blocks of one, two, three, four, five, ... odd natural numbers in ascending order, the cube numbers can be generated by summation :
  • 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:
  • The sum of the first cube numbers is equal to the square of the -th triangular number :
  • 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.
  • 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 .


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 at position 3.

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


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

See also

Web links

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