Prove that $n^k+1$ has a prime divisor $>2k$

Question

Let $n\ge3$ & $k\ge2$. Show that $n^k +1$ has a prime divisor $>2k$.

Answer 1

By Zsigmondy's Theorem (see here), $n^{2k} - 1$ has a prime divisor $p$ such that $p$ is not a divisor of $n^j-1$, for any $j < 2k$. In particular, $p$ does not divide $n^k-1$, but it does divide $n^{2k} - 1 = (n^k - 1)(n^k + 1)$, so $p$ divides $n^k + 1$. We also have $n^{j} \neq 1 \pmod{p}$ for $j < 2k$, so $\varphi(p) = p-1 \ge 2k$.