On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases

2022/12/07 by Cécile Dartyge, Dartyge, Cécile, James E. Maynard +1
Mathematics · #11N05 #11N32 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2212.03381

openalex publication_date 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P(X)∈ℤ[X] be an irreducible, monic, quartic polynomial with cyclic or dihedral Galois group. We prove that there exists a constant c_P>0 such that for a positive proportion of integers n, P(n) has a prime factor ≥ n^1+c_P.

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