The spanning method and the Lehmer totient problem
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912959587942400 |
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| author | Agama, Theophilus |
| author_facet | Agama, Theophilus |
| contents | In this paper, we introduce and develop the notion of spanning of integers along functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems that requires to determine if the equations of the form $tf(n)=n-k$ has a solution $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$ and some $t\in \mathbb{N}$. In particular, we show that \begin{align} \# \{n\leq s~|~tφ(n)+1=n,~\mathbf{for~some}~t\in \mathbb{N}\}\geq \frac{s}{2\log s}\prod \limits_{p | \lfloor s\rfloor }(1-\frac{1}{p})^{-1}-\frac{3}{2}e^γ\nonumber \end{align} as $s\longrightarrow \infty$, where $φ$ is the Euler totient function and $γ=0.5772\cdots$ is the Euler-Macheroni constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_13055 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The spanning method and the Lehmer totient problem Agama, Theophilus General Mathematics In this paper, we introduce and develop the notion of spanning of integers along functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems that requires to determine if the equations of the form $tf(n)=n-k$ has a solution $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$ and some $t\in \mathbb{N}$. In particular, we show that \begin{align} \# \{n\leq s~|~tφ(n)+1=n,~\mathbf{for~some}~t\in \mathbb{N}\}\geq \frac{s}{2\log s}\prod \limits_{p | \lfloor s\rfloor }(1-\frac{1}{p})^{-1}-\frac{3}{2}e^γ\nonumber \end{align} as $s\longrightarrow \infty$, where $φ$ is the Euler totient function and $γ=0.5772\cdots$ is the Euler-Macheroni constant. |
| title | The spanning method and the Lehmer totient problem |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2003.13055 |