On the largest prime divisor of polynomial and related problem

2025/03/10 by Thanh Nguyen Cung, Cung, Thanh Nguyen, Hong, Son Duong
Mathematics · Computer Science · #Advanced Mathematical Theories #Analytic Number Theory Research #Graph Labeling and Dimension Problems

paper · pdf · doi:10.48550/arxiv.2503.07793

Abstract

We denote P = \P(x)| P(n) | n! for infinitely many n\. This article identifies some polynomials that belong to 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 ∈ ℕ so that P⁺(f(n)) < n^(3)/(4)+ε if f(x) is cubic polynomial, P⁺(f(n)) < n if f(x) is reducible quartic polynomial and P⁺(f(n)) < n^ε if f(x) is Chebyshev polynomial.

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