Reducing the number of prime factors of long $\kappa$-tuples

TL;DR

This paper proves that for large enough ppa, there are infinitely many integers where the product of ppa shifted terms has at most about half ppa times log ppa prime factors, improving understanding of prime factorization in ppa-tuples.

Contribution

It establishes a new bound on the minimal total number of prime factors in products of ppa shifted integers for large ppa, advancing prime factorization research.

Findings

Infinitely many integers with bounded prime factors in ppa-tuples

Asymptotic bound of ppa/2 ppa log ppa + O(ppa) on prime factors

Results hold for sufficiently large ppa

Abstract

We prove that there are infinitely many integers $n$ such that the total number of prime factors of $(n+h_{1})(n+h_{2})...(n+h_{\kappa})$ is at most $(1/2)\kappa\log\kappa+O(\kappa)$, provided $\kappa$ is sufficiently large.