Cube (algebra)

A cubic number (from Latin cubus, " dice " ) is a number that is created when you multiply a natural number twice with itself. For example, a cubic number. The first cubes are

For some authors, the zero is not a perfect cube, so that the sequence of numbers begins with the first one.

The term cubic number is derived from the geometric figure of the cube. The number of stones that you need to build a cube is always equal to a cubic number. Thus, for example, lay with the help of 27 stones a cube with sides of length 3.

Because of this relationship with a geometric figure count the cubes to the figurate numbers, which also include the square numbers and tetrahedral numbers belong.


  • From the successive blocks of one, two, three, four, five, ... odd natural numbers in ascending order can be obtained by summing the cubes generate:
  • Based on the result of the centered Sechseckszahlen 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, ... is obtained, the -th cube of the sum of said first sequence of links:
  • The sum of the first cubes is equal to the square of the -th triangular number:
  • Every natural number can be represented as a sum of at most nine cubes (solution of the problem Waringschen for the exponent 3). That 9 summands may be necessary shows the number 23, this has the representation, but obviously no cubic with less summands.
  • The sum of any two cubes can itself never be a perfect cube. In other words, this means that the equation no solution with natural numbers has. This special case of Fermat's theorem was proved in 1753 by Leonhard Euler. If you leave more than two summands to, it can happen that a cubic number is represented as the sum of cubes, as the following example (even with three directly consecutive cubes ) shows:

Sum of the reciprocals

The sum of the reciprocals of all the cubes is called apery constant. It corresponds to the value of the Riemann function at the point 3

Generating function

Each sequence of whole (or real) numbers can be assigned to a formal power series, the so-called generating function. In this context, however, it is usual to have to start the sequence of cubes to 0, that is to consider the sequence. The generating function of the cubes is then