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.
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 .
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