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 $m \in \Z$.


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