On the largest prime divisor of polynomial and related problem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916648561147904 |
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| author | Cung, Thanh Nguyen Hong, Son Duong |
| author_facet | Cung, Thanh Nguyen Hong, Son Duong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_07793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the largest prime divisor of polynomial and related problem Cung, Thanh Nguyen Hong, Son Duong Number Theory 11A41, 11A51, 11C08 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. |
| title | On the largest prime divisor of polynomial and related problem |
| topic | Number Theory 11A41, 11A51, 11C08 |
| url | https://arxiv.org/abs/2503.07793 |