Theorems by Thue, Pólya, and Mahler show that if a polynomial with integer coefficients $f(x)$ has at least two distinct complex roots, the greatest prime factor of $f(n)$ tends to infinity as $n$ approaches infinity. Consequentially, for every finite set of prime $P$, the intersection of $f(\mathbb{Z})$ and the $S$-unit of $P$ is finite. An extension on $f(\mathbb{Q})$ will be explored, by mimicking the technique used by Pólya, and applying a corollary of Falting’s theorem for quartic binary forms.

References:

1. Axel Thue. Om en generel i store hele tal uløsbar ligning. Skrifter Udgivne af Videnskabs-Selskabet i Christiana, 1908.
2. George Pólya. Zur arithmetischen Untersuchung der Polynome. Mathematische Zeitschrift, 1918.
3. Kurt Mahler. Zur Approximation algebraischer Zahlen. I. (Über den größten Primteiler binärer Formen). Mathematische Annalen, 1933.