On the largest prime divisor of polynomial and related problem

TL;DR

This work studies the largest prime factor of polynomial values and identifies a broad class of polynomials whose values divide factorials for infinitely many inputs, i.e., polynomials in $\mathcal{P}$. The authors develop a general divisibility framework centered on Lemma 3.1 and Pell-type parametrizations to construct infinitely many $n$ with $P(n)\mid n!$, then leverage this to obtain strong sublinear bounds on $P^+(f(n))$ for several families of degree at most 4 polynomials. They prove that quadratic, cubic, reducible quartic, cyclotomic, and Chebyshev polynomials (and their products) lie in $\mathcal{P}$, with concrete results such as $P^+(f(n))<n^{3/4+\varepsilon}$ for cubic $f$ and $P^+(f(n))<n^{\varepsilon}$ for Chebyshev-type polynomials. The methods extend prior results by Schinzel and related works, offering new instances where the largest prime factor is tightly controlled and providing a versatile approach for constructing polynomials with factorial divisibility properties.

Abstract

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.

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• Proposition 3.3
• proof
• Theorem 3.4