Difference of Two Odd Powers

Theorem
Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on.

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

Then:

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