Almost-Prime Polynomials with Prime Arguments

2016/06/11 by Pin-Hung Kao, Kao, Pin-Hung
Computer Science · Mathematics · #11N35 #11N36 #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N35 #msc:11N36

paper · pdf · doi:10.48550/arxiv.1606.03505

17 pages, 4 tables

arxiv created 2016/06/11 · openalex publication_date 2016/06/11 · arxiv updated 2016/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℤ satisfying necessary local conditions are P₄ infinitely often.

Related

• Almost-Primes Represented by Quadratic Polynomials
• Variants of the Selberg sieve, and almost prime k-tuples
• Irreducible Polynomials with Varying Constraints on Coefficients
• Prime numbers and factorization of polynomials
• Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials