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

TL;DR

This paper proves that for certain quartic polynomials with cyclic or dihedral Galois groups, a positive proportion of their integer values have large prime factors exceeding a power of the input.

Contribution

It establishes a lower bound on the size of prime factors of polynomial values for a broad class of quartic polynomials, extending understanding of prime factorization in polynomial sequences.

Findings

A positive proportion of polynomial values have prime factors at least n^{1+c_P}.

Existence of a constant c_P > 0 depending on the polynomial.

Results apply specifically to irreducible, monic quartic polynomials with cyclic or dihedral Galois groups.

Abstract

Let $P(X)\in\mathbb{Z}[X]$ 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 $\ge n^{1+c_P}$.