Q:

Is $ x^n-y^n$ is a product of coprime factors?

In the expression: $x^n-y^n$, if $n>2$ and $x,y$ are relatively prime, are the factors $x-y$ and $ x^{n-1}+x^{n-2}y+.....$ always coprime? Why? Please exclude the cases where $x-y=\pm 1$ and $\pm 2$. I know it is for $x=2$. Please any input will be greatly appreciated. Thanks.

A:

If $x\equiv y\pmod k$ and $k\mid n$, then they're not coprime.
Proof:
$x\equiv y\pmod k$, so $k\mid x-y$.
Since $x\equiv y\pmod k$, we have $x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1}\equiv nx^{n-1}\equiv 0\pmod k$, since $k\mid n$. $\:\square$

