Almost-Prime Polynomials with Prime Arguments

TL;DR

This paper advances the understanding of prime values of irreducible polynomials by improving sieve methods, demonstrating that certain quadratic polynomials take on prime values with at most four prime factors infinitely often.

Contribution

It introduces an enhanced sieve technique extending the DHR sieve to non-elementary ranges, improving bounds on prime factors of polynomial values at prime arguments.

Findings

Proves irreducible quadratics satisfying local conditions are P4 infinitely often.

Improves bounds on the number of prime factors for polynomial values.

Extends sieve methods to broader ranges for better results.

Abstract

We improve Irving's method of the double-sieve by using the DHR sieve. By extending the upper and lower sieve functions into their respective non-elementary ranges, we are able to make improvements on the previous records on the number of prime factors of irreducible polynomials at prime arguments. In particular, we prove that irreducible quadratics over $\mathbb{Z}$ satisfying necessary local conditions are $P_4$ infinitely often.