On the Asymptotic Density of a GCD-based Map

2025/06/24 by Thang Pang Ern, Ern, Thang Pang, Malcolm Tan Jun Xi +1
Computer Science · #Digital Image Processing Techniques #FOS: Mathematics #Image Retrieval and Classification Techniques #Medical Image Segmentation Techniques #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.19812

openalex publication_date 2025/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We show that the symmetry of f(a,b)=(gcd(ab,a+b))/(gcd(a,b)) stems from an SL₂(ℤ) action on primitive pairs and that all solutions to f(a,b)=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(a,b)=1 tends to ∏_p(1-p^-2(p+1)^-1)≈0.88151, and that its higher-order analogue f_r has a limiting density 6/π² for r≥2.

Citations

• A Strengthening of Theorems of Halász and Wirsing

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