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This is a Quantitative Study study.

September 28, 2025Open Access

On the largest prime divisor of polynomial and related problem

Key Points

Identified polynomials in the set of those dividing factorials for infinitely many natural numbers.
For cubic polynomials, the largest prime factor is less than n raised to 3/4 plus epsilon infinitely often.
For reducible quartic polynomials, the largest prime factor is less than n for infinitely many n.
Chebyshev polynomials also have their largest prime factors bounded by n raised to epsilon infinitely often.

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+} if f (x) is cubic polynomial, P^+ (f (n) ) < n if f (x) is reducible quartic polynomial and P^+ (f (n) ) < n^ if f (x) is Chebyshev polynomial.

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Cite This Study

Cung et al. (Mon,) studied this question.

synapsesocial.com/papers/68d90a0f41e1c178a14f6a1f https://doi.org/https://doi.org/10.48550/arxiv.2503.07793

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