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arXiv · 2503.07793

On the largest prime divisor of polynomial and related problem

Thanh Nguyen Cung·Son Duong Hong
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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.

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Thanh Nguyen Cung, Son Duong Hong. 2025-03-10. On the largest prime divisor of polynomial and related problem. https://arxiv.org/abs/2503.07793

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