Five Cube Theorem

Theorem
Every integer can be expressed as a sum of five cube numbers.

Proof
Let $r$ be an integer.

Then $r$ can be expressed in exactly one the following forms:
 * $\paren {6 m + 0}$
 * $\paren {6 m + 1}$
 * $\paren {6 m + 2}$
 * $\paren {6 m + 3}$
 * $\paren {6 m + 4}$
 * $\paren {6 m + 5}$

for some $m \in \Z$.


 * $\paren {6 m + 0} = \paren {m + 1}^3 + \paren {m - 1}^3 + \paren {- m}^3 + \paren {- m}^3 + 0^3$
 * $\paren {6 m + 1} = \paren {m + 1}^3 + \paren {m - 1}^3 + \paren {- m}^3 + \paren {- m}^3 + 1^3$
 * $\paren {6 m + 2} = \paren {m}^3 + \paren {m - 2}^3 + \paren {1 - m}^3 + \paren {1 - m}^3 + 2^3$
 * $\paren {6 m + 3} = \paren {m - 3}^3 + \paren {m - 5}^3 + \paren {- m + 4}^3 + \paren {- m + 4}^3 + 3^3$
 * $\paren {6 m + 4} = \paren {m + 3}^3 + \paren {m + 1}^3 + \paren {- m - 2}^3 + \paren {- m - 2}^3 + \paren {-2}^3$
 * $\paren {6 m + 5} = \paren {m + 2}^3 + \paren {m}^3 + \paren {- m - 1}^3 + \paren {- m - 1}^3 + \paren {-1}^3$

Also see

 * Number is Sum of Five Cubes, which shows that there is in fact an infinite number of ways to represent any integer as a sum of five cubes.