We denote $\mathcal{ P}$ = $\{ P(x)|$ $P(n) \mid n!$ for infinitely many $n\}$. This article identifies some polynomials that belong to $\mathcal{ P}$. Additionally, we also denote $P^+(m)$ as the largest prime factor of $m$. Then, a consequence of this work shows that there are infinitely many $n \in \mathbb{ N}$ so that $P^+(f(n)) < n^{ \frac{ 3}{ 4}+\varepsilon}$ if $f(x)$ is cubic polynomial, $P^+(f(n)) < n$ if $f(x)$ is reducible quartic polynomial and $P^+(f(n)) < n^{ \varepsilon}$ if $f(x)$ is Chebyshev polynomial.
Categories: math.NT