cube of an integer

Theorem.  Any cube of integer is a difference of two squares, which in the case of a positive cube are the squares of two successive triangular numbersMathworldPlanetmath.

For proving the assertion, one needs only to check the identity


For example we have  (-2)3=12-32  and  43=64=102-62.

Summing the first n positive cubes, the identity allows between consecutive brackets,

13+23+33+43++n3 =[12-02]+[32-12]+[62-32]+[102-62]++[((n+1)n2)2-((n-1)n2)2]

saving only the square ((n+1)n2)2.  Thus we have this expression presenting the sum of the first n positive cubes (cf. the Nicomachus theoremMathworldPlanetmath).

Title cube of an integer
Canonical name CubeOfAnInteger
Date of creation 2013-03-22 19:34:33
Last modified on 2013-03-22 19:34:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 11B37
Classification msc 11A25
Related topic NicomachusTheorem
Related topic TriangularNumbers
Related topic DifferenceOfSquares