Factors of Difference of Two Odd Powers

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:

Proof
From Factorisation of $z^n - a$:


 * $x^{2 n + 1} - y^{2 n + 1} = \displaystyle \prod_{k \mathop = 0}^{2 n} \paren {x - \alpha^k y}$

where $\alpha$ is a primitive complex $2 n + 1$th roots of unity, for example:

From Complex Roots of Unity occur in Conjugate Pairs:
 * $U_{2 n + 1} = \set {1, \tuple {\alpha, \alpha^{2 n} }, \tuple {\alpha^2, \alpha^{2 n - 1} }, \ldots, \tuple {\alpha^k, \alpha^{2 n - k + 1} }, \ldots, \tuple {\alpha^n, \alpha^{n + 1} } }$

where $U_{2 n + 1}$ denotes the complex $2 n + 1$th roots of unity:
 * $U_{2 n + 1} = \set {z \in \C: z^{2 n + 1} = 1}$

The case $k = 0$ is taken care of by setting $\alpha^0 = 1$, from whence we have the factor $x - y$.

Taking the product of each of the remaining factors of $x^{2 n + 1} - y^{2 n + 1}$ in pairs:

Hence the result.