On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916697952223232 |
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| author | Peneva, Temenoujka P. Todorova, Tatiana L. |
| author_facet | Peneva, Temenoujka P. Todorova, Tatiana L. |
| contents | Let $α\in \mathbb{R}\setminus\mathbb{Q}$ and $β\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there exist infinitely many primes $p$ such that the inequality $||αp+β||<p^{-1/10}$ holds and all the numbers $p+a_1,\ldots,p+a_s$ are square-free. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13993 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers Peneva, Temenoujka P. Todorova, Tatiana L. Number Theory Let $α\in \mathbb{R}\setminus\mathbb{Q}$ and $β\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there exist infinitely many primes $p$ such that the inequality $||αp+β||<p^{-1/10}$ holds and all the numbers $p+a_1,\ldots,p+a_s$ are square-free. |
| title | On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.13993 |