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arXiv · 2512.09245

On the largest prime factors of shifted semiprime numbers

Do Duc Tam
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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.

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Do Duc Tam. 2025-12-10. On the largest prime factors of shifted semiprime numbers. https://arxiv.org/abs/2512.09245

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