Difference of Two Odd Powers

Theorem
Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Then for all $a, b \in \mathbb F$:

Proof
A direct application of Difference of Two Powers:

and setting $n \to 2 n + 1$.

Also see

 * Factors of Difference of Two Odd Powers
 * Difference of Two Even Powers