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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2506.19812 |
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| _version_ | 1866917418822008832 |
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| author | Ern, Thang Pang Xi, Malcolm Tan Jun |
| author_facet | Ern, Thang Pang Xi, Malcolm Tan Jun |
| contents | We show that the symmetry of \[f\left(a,b\right)=\frac{\operatorname{gcd}\left(ab,a+b\right)}{\operatorname{gcd}\left(a,b\right)}\] stems from an $\operatorname{SL}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions to $f\left(a,b\right)=n$ admit a uniform three-parameter description -- recovering arithmetic-progression families via the Chinese remainder theorem when $n$ is squarefree. It shows that the density of pairs with $f\left(a,b\right)=1$ tends to $\prod_p\left(1-p^{-2}(p+1)^{-1}\right)\approx0.88151$, and that its higher-order analogue $f_r$ has a limiting density $6/π^2$ for $r\ge2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Asymptotic Density of a GCD-based Map Ern, Thang Pang Xi, Malcolm Tan Jun Number Theory We show that the symmetry of \[f\left(a,b\right)=\frac{\operatorname{gcd}\left(ab,a+b\right)}{\operatorname{gcd}\left(a,b\right)}\] stems from an $\operatorname{SL}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions to $f\left(a,b\right)=n$ admit a uniform three-parameter description -- recovering arithmetic-progression families via the Chinese remainder theorem when $n$ is squarefree. It shows that the density of pairs with $f\left(a,b\right)=1$ tends to $\prod_p\left(1-p^{-2}(p+1)^{-1}\right)\approx0.88151$, and that its higher-order analogue $f_r$ has a limiting density $6/π^2$ for $r\ge2$. |
| title | On the Asymptotic Density of a GCD-based Map |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.19812 |