Conjecture on sum of powers Let $n$ be an odd number, $x,y$ integers and $p$ a prime number. Now, suppose that $p\ne n$ and
$$
p|\frac{x^n+y^n}{x+y}
$$
Then, I have been observed that $p \equiv 1 \pmod{n}$.
This is, all of the prime factors of the previous expression are of the form $p=nq+1$
An easy way to see this with numerical examples is to choose $n=5$, then the last digit of each prime number will be $1$.
Examples of this case are: 
$$
\frac{5^5+8^5}{5+8}=11\cdot 251
$$
$$
\frac{19^5+26^5}{19+26}=5\cdot 11\cdot 1801
$$
$$
\frac{113^5+257^5}{113+257}=5\cdot 615988781\
$$
I post it as a conjecture, although I ignore if it's a know theorem. I have been searching on the Internet, where I haven't found any interesting about it.
Note: this conjecture seems to be true also with:
 $$
p|\frac{x^n-y^n}{x-y}
$$

The answer to your conjecture and much, much more about the form(s)
$$
  \frac{a^n \pm b^n}{a \pm b}, \qquad a,b,n \in \mathbb{Z}
$$
and its factors can be found in Ribenboim’s wonderful book, Fermat’s Last Theorem for Amateurs, specifically Chapter II.
