The largest prime factor of an irreducible cubic polynomial

TL;DR

This work proves that every monic irreducible cubic $f\in\mathbb{Z}[X]$ attains a positive density of integers $n$ for which $f(n)$ has a prime factor exceeding $n^{1+c_p}$, with $c_p>0$ depending on $f$, thereby giving $P(x,f)\gg x^{1+c_p}$. The authors adapt Heath-Brown’s cubic-method to general cubics by embedding the problem in a number-field setting $K=\mathbb{Q}(r)$, counting ideals via a sieved structure, and decomposing $\log f(n)$ into small- and large-norm factors. Central to the argument are the construction of a suitable subset $\mathcal{A}_1$ with large $\log^{(1)}(f(n))$, a careful Rosser–Iwaniec-type lower bound sieve, and a $q$-Vander Corput estimate for short exponential sums to bound the error term $S_1$; together these yield $S_1=o(X)$ and $S_0\gg 1$, implying the desired density result. The approach combines algebraic number theory (prime ideals, unit groups, and discriminants) with analytic techniques (exponential-sum bounds and lattice-point counting) to extend Heath-Brown’s cubic bound to all monic irreducible cubics, with explicit dependence of the exponent on the polynomial. This advances understanding of large prime factors in polynomial values and contributes to the broader program around Bunyakovsky-type questions for low-degree polynomials.

Abstract

Heath-Brown proved that for a positive proportion of integers $n$, $n^3+2$ has a prime factor larger than $n^{1+c}$ with $c=10^{-303}$. We generalize this result to arbitrary monic irreducible cubic polynomial of $\mathbb{Z}[x]$ with $c$ replaced by an exponent $c_p$ dependent on the polynomial.

Theorems & Definitions (29)

• Theorem
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem
• Lemma 4
• Remark
• Lemma 5
• proof
• Lemma 6