On the largest prime factors of shifted semiprime numbers

Abstract

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the largest prime factor of number $$ \prod_{\substack{n\in \mathbb{P}_2\\n\leq x}}(n+a) $$ exceeds $x^{\theta}$, where $\theta= 0.5-\varepsilon,$ $0<\varepsilon\leq 0.01$ is arbitrarily small.