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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2606.01633 |
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| _version_ | 1866913178012614656 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | Büyükaşik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $φ(n)$, by setting $φ_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $φ_0(n) = φ(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ φ_s(n) \mid φ_k(n) \}$, Büyükaşik et al. proved that $\mathcal{D}_{s}$ is finite for each $s \geq 0$, and conjectured that $\mathcal{D}_{1} = \{ 1, 3, 15 \}$ and provided computations to support this conjecture. We succeed in proving this conjecture, using an argument based on our extensive interactions with GPT-5.5 Pro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01633 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a problem on a generalization of Euler's totient function Campbell, John M. Number Theory 11A25 Büyükaşik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $φ(n)$, by setting $φ_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $φ_0(n) = φ(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ φ_s(n) \mid φ_k(n) \}$, Büyükaşik et al. proved that $\mathcal{D}_{s}$ is finite for each $s \geq 0$, and conjectured that $\mathcal{D}_{1} = \{ 1, 3, 15 \}$ and provided computations to support this conjecture. We succeed in proving this conjecture, using an argument based on our extensive interactions with GPT-5.5 Pro. |
| title | On a problem on a generalization of Euler's totient function |
| topic | Number Theory 11A25 |
| url | https://arxiv.org/abs/2606.01633 |