I want to show that the diophantine equation
$ (x^p - y^p)/(x - y) = z^p $
where the prime $p > 3$, can just have trivial integer solutions $\{x, y, z\}$ like $\{1, -1, 1\}$
I used the theorem $IV$ from this article to resolve the case $(x - y) ≡ 0$ $mod(p)$.
Now, when $(x - y) ≢ 0$ $mod (p)$ we have
$ (x^p - y^p)/(x - y) = P[p] $
where $P[p]$ is the $arithmetic$ $primitive$ $factor$ of $ (x^p - y^p) $ defined in that article (p. $175$). With other considerations we can say that each one of those primitive prime factors is of the form $(2k_i p+1)$ where $k_i$ $∈ N$.
And so the question is: when $p > 3$, $P[p]$ can ever have this form?
$[(2k_i p+1)^i(2k_q p+1)^j...(2k_n p+1)^r]^p$
I guess I need Zsigmondy's theorem, but I'm stuck.
