Difference between 2 Consecutive Cubes is Odd/Proof 2

Theorem

Let $a$ and $b$ be consecutive integers.

Then $b^3 - a^3$ is odd.

Proof

Let $a, b \in \Z$ such that $b = a + 1$.

Either:

$a$ is even and $b$ is odd

or:

$b$ is even and $a$ is odd.

Hence from Parity of Integer equals Parity of Positive Power either:

$a^3$ is even and $b^3$ is odd

or:

$b^3$ is even and $a^3$ is odd.

The result follows.

$\blacksquare$