For every positive integer $n>1$ , let $f(n)$ denote the largest prime factor of $n$. How fast does $f(1+n^k)$ grow with respect to $k$ ? Is it true that $f(1+n^k) > 2k, \forall n >2, \forall k >1$ ?



On the largest prime factor of $1+n^k$
I do not attempt to answer the first question. The last question is closely related to Zsygmondy's Theorem: Given your hypotheses, and Zygmondy's theorem, there will be a prime $p$ which divides $n^{2k}-1,$ but does not divide $n^{j}-1$ for any $j < 2k.$ Note then that $p$ must be a prime divisor of $n^{k}+1.$ 
But also , the element $n+p\mathbb{Z}$ has multiplicative order $2k$ in the group of units of $\mathbb{Z}/p\mathbb{Z}.$ Hence $2k$ divides $p-1,$ so that $p >2k.$


