x^n - y^n Factorization via Roots of Unity and Cyclotomics

Cyclotomic Polynomials.

The $n$th cyclotomic polynomial, for any positive integer $n$, is the unique irreducible polynomial with integer coefficinets, which is a divisor of $x^n-1$ and is not a divisor of $x^k-1$ for any $k < n$. It is easy to find that the $n$th cyclotomic polynomial can be represented as $$\Phi_n (x) = \displaystyle\prod_{1 \le k \le n, \ \text{gcd}(k,n)=1} \left( x - e^{2i\pi \cdot \frac{k}{n}} \right).$$

Special cases. If $n$ is a prime number, then $$\Phi_n(x) = \displaystyle\sum_{i=0}^{n-1} x^i.$$ If $n=2p$, where $p$ is an odd prime number, then $$\Phi_n(x) = \Phi_{2p}(x) = \displaystyle\sum_{i=0}^{p-1} \left( -x \right)^i.$$