Prove that there exist infinitely many positive integers n such that all prime divisors of $n^2 + n + 1$ are not greater than $\sqrt{n}$
This is a problem related to cyclotomic polynomial. It is trivial that the polynomial in the question is $\Phi_3(n)$, but I cannot figure anything out.
What I have tried is I claim that there exists infinitely many prime $p$ and integer $k \ge 2$ such that $$p^k = \Phi_3 (n) = n^2 + n + 1$$
I tried using proof by contradiction that there is an upper bound of such prime, but no luck.
Is there a proof or even the generalization for $\Phi_{p}$? Or is my approach wrong?
I got this idea because of the relation between primitive roots and cyclotomic polynomial.
Source: https://drive.google.com/file/d/1JC3insT6UZYz3jc8u6L1vdYMdeI8Kk1l/view?usp=share_link