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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.22494 |
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| _version_ | 1866918454293954560 |
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| author | Ern, Thang Pang Xi, Malcolm Tan Jun Ryan, Loh Wei Xuan |
| author_facet | Ern, Thang Pang Xi, Malcolm Tan Jun Ryan, Loh Wei Xuan |
| contents | The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $ρ=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/ζ(2)=6/π^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Limiting Density of a gcd Map Ern, Thang Pang Xi, Malcolm Tan Jun Ryan, Loh Wei Xuan Number Theory Rings and Algebras The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $ρ=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/ζ(2)=6/π^2$. |
| title | On the Limiting Density of a gcd Map |
| topic | Number Theory Rings and Algebras |
| url | https://arxiv.org/abs/2512.22494 |