On the largest prime factors of shifted semiprime numbers
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909953479933952 |
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| author | Tam, Do Duc |
| author_facet | Tam, Do Duc |
| contents | 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 $x^θ$, where $θ= 0.5-\varepsilon,$ $0<\varepsilon\leq 0.01$ is arbitrarily small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the largest prime factors of shifted semiprime numbers Tam, Do Duc Number Theory 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 $x^θ$, where $θ= 0.5-\varepsilon,$ $0<\varepsilon\leq 0.01$ is arbitrarily small. |
| title | On the largest prime factors of shifted semiprime numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.09245 |