Almost-prime values of polynomials at prime arguments

TL;DR

This paper investigates the distribution of almost-prime values of irreducible polynomials evaluated at prime arguments, improving bounds on the number of prime factors for large-degree polynomials.

Contribution

It extends previous results by providing tighter bounds on the number of prime factors of polynomial values at primes, especially for polynomials of degree at least 3.

Findings

Infinitely many primes p with f(p) having at most degree f + O(log degree f) prime factors

Improved bounds over previous results for polynomials of degree ≥ 3

Enhanced understanding of almost-prime values of polynomials at prime inputs

Abstract

We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the degree is large, there are infinitely many primes $p$ for which $f(p)$ has at most $\deg f+O(\log\deg f)$ prime factors.