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 $$\prod _{\begin{array}{c}n\in {\mathbb{P}}_2\\ n\le x\end{array}}(n+a)$$ exceeds $x^{\theta }$, where $\theta =0.5-\epsilon ,$ $0<\epsilon \le 0.01$ is arbitrarily small.