Reference to a theorem of T. Nagell

Question

I have the following problem from USAMO 2006:

"For an integer $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p(\pm 1)=1$ and $p(0)=\infty$. Find all polynomials $f$ with integer coefficients such that the sequence $\{ p(f(n^2))-2n) \}_{n\ge 0}$ is bounded above. (In particular, this requires $f(n^2)\neq 0$ for $n\ge 0$.)"

One solution for the above question is given here:

https://artofproblemsolving.com/wiki/index.php?title=2006_USAMO_Problems/Problem_3"

In this solution, the author makes the following statement:

"A theorem of T. Nagell implies that...." and has continued proving the given statement. Can anybody help me with what theorem of Nagel being referred to here and how it is used in the above solution?

PS: The only place where I have encountered the name before is the Nagell point, which for sure is not at all related to this question.

Answer 1

The theorem of T. Nagell referred to is:

Theorem (T. Nagell, Généralisation d'un théorème de Tchebycheff, J. Math. Pures Appl. (8) 4 (1921), 343-356) Let $f$ be an irreducible integral nonlinear polynomial, and let $P_x$ be the greatest prime factor of $$ \prod_{n\leq x}f(n). $$ Then, for any $a<1$ and all sufficiently large $x$, $$ \frac{P_x}{x}>(\log x)^a. $$

This bound is further improved upon by Erdős and others later but we don't need that here. You should see immediately how Nagell's result is applied in the solution.

And no, Nagel (the 19th century German geometer Christian Heinrich von Nagel) $\neq$ Nagell (Trygve Nagell, the 20th century Norwegian number theorist).