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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.11256 |
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| _version_ | 1866913418821238784 |
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| author | Luca, Florian Manape, Makoko Campbell |
| author_facet | Luca, Florian Manape, Makoko Campbell |
| contents | In this paper, we show that if $(U_n)_{n\ge 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$, then the inequality $ϕ(|U_n|)\ge |U_{ϕ(n)}|$ holds on a set of positive integers $n$ of density $1$, where $ϕ$ is the Euler function. In fact, we show that the set of $n\le x$ for which the above inequality fails has counting function $O_U(x/\log x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11256 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Euler function of linearly recurrence sequences Luca, Florian Manape, Makoko Campbell Number Theory 11B39 In this paper, we show that if $(U_n)_{n\ge 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$, then the inequality $ϕ(|U_n|)\ge |U_{ϕ(n)}|$ holds on a set of positive integers $n$ of density $1$, where $ϕ$ is the Euler function. In fact, we show that the set of $n\le x$ for which the above inequality fails has counting function $O_U(x/\log x)$. |
| title | On the Euler function of linearly recurrence sequences |
| topic | Number Theory 11B39 |
| url | https://arxiv.org/abs/2405.11256 |