Number is Sum of Five Cubes

Theorem
Let $n \in \Z$ be an integer.

Then $n$ can be expressed as the sum of $5$ cubes (either positive or negative) in an infinite number of ways.

Proof
We have for any $m, n \in \Z$:

By definition of modulo arithmetic:
 * $\exists k \in \Z: \paren {6 m + n}^3 = n + 6 k$

We also have:

Thus $n = \paren {6 m + n}^3 + k^3 + k^3 - \paren {k + 1}^3 - \paren {k - 1}^3$ is an expression of $n$ as a sum of $5$ cubes.

In the equation above, $m$ is arbitrary and $k$ depends on both $m$ and $n$.

As there is an infinite number of choices for $m$, there is an infinite number of such expressions.

Remark
From the result above we can also write each term of the sum explicitly:

However, because of the complexity of this expression it is recommended to first calculate $k$.