Almost-Prime Polynomials with Prime Arguments
Pin-Hung Kao

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 satisfying necessary local conditions are infinitely often.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Identities · Coding theory and cryptography
