3-tuples have at most 7 prime factors infinitely often

TL;DR

This paper proves that there are infinitely many integers for which the product of three linear functions has at most seven prime factors, using advanced sieve techniques to improve previous bounds.

Contribution

It introduces a weighted multidimensional sieve approach to significantly reduce the number of prime factors in such products, advancing the understanding of prime factorization in polynomial values.

Findings

Infinitely many n with L_1(n)L_2(n)L_3(n) having ≤7 prime factors

Application of Diamond-Halberstam-Richert multidimensional sieve

Improves previous bounds established by Porter

Abstract

Let $L_1$, $L_2$ $L_3$ be integer linear functions with no fixed prime divisor. We show there are infinitely many $n$ for which the product $L_1(n)L_2(n)L_3(n)$ has at most 7 prime factors, improving a result of Porter. We do this by means of a weighted sieve based upon the Diamond-Halberstam-Richert multidimensional sieve.