On coprimality of consecutive elements in certain sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911023216197632 |
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| author | Deshouillers, Jean-Marc Naik, Sunil |
| author_facet | Deshouillers, Jean-Marc Naik, Sunil |
| contents | The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this context, we show that if $f$ is a twice continuously differentiable real-valued function on $[1, \infty)$ such that $f''(x) \to 0$ as $x \to \infty$ and $\limsup_{x \to \infty} f'(x) = \infty$, then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$. This result refines the qualitative part of a recent result by the first author, Drmota and Müllner.
We also prove that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that for any two distinct integers $m, n \in \mathcal{A}$, the integers $\lfloor f(m) \rfloor$ and $\lfloor f(n) \rfloor$ are pairwise coprime. Further, we show that there exist arbitrarily long blocks of consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ such that no two of them are pairwise coprime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20948 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On coprimality of consecutive elements in certain sequences Deshouillers, Jean-Marc Naik, Sunil Number Theory 11B05, 11B50, 11K31, 11N56 The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this context, we show that if $f$ is a twice continuously differentiable real-valued function on $[1, \infty)$ such that $f''(x) \to 0$ as $x \to \infty$ and $\limsup_{x \to \infty} f'(x) = \infty$, then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$. This result refines the qualitative part of a recent result by the first author, Drmota and Müllner. We also prove that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that for any two distinct integers $m, n \in \mathcal{A}$, the integers $\lfloor f(m) \rfloor$ and $\lfloor f(n) \rfloor$ are pairwise coprime. Further, we show that there exist arbitrarily long blocks of consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ such that no two of them are pairwise coprime. |
| title | On coprimality of consecutive elements in certain sequences |
| topic | Number Theory 11B05, 11B50, 11K31, 11N56 |
| url | https://arxiv.org/abs/2506.20948 |