On the largest prime factors of shifted semiprime numbers

TL;DR

This paper investigates the largest prime factors of shifted semi-prime numbers, proving that for large x, these factors exceed x to a certain power close to 0.5, advancing understanding of prime distribution in semi-primes.

Contribution

It establishes a new lower bound on the largest prime factors of shifted semi-primes for large x, a significant step in prime factorization research.

Findings

Largest prime factor exceeds x^{0.5 - ε} for large x

Results hold for semi-primes shifted by any fixed integer a

Advances understanding of prime factors in semi-prime products

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.