arXiv subjects

Thang Pang Ern

Publications and source records attributed to Thang Pang Ern.

4 recordsLinked to original sources

arXiv2026

On the Asymptotic Density of a GCD-based Map

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 $r\ge2$.

math.NT↗

arXiv2026

On the Limiting Density of a gcd Map

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 $1/ζ(2)=6/π^2$.

math.NT↗

arXiv2026

A Proof of Ramanujan's Classic $π$ Formula

In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $π$. Among these, one of the most celebrated is the following series: \[\frac{1}π=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{26390n+1103}{\left(n!\right)^4} \frac{\left(4n\right)!}{396^{4n}}\] In this paper, we give a full proof of this classic formula using hypergeometric series and a special type of lattice sums due to Zucker and Robertson. We will also use some results by Dirichlet and Edwards in algebraic number theory.

math.NT↗

arXiv2024

Finding Squares in a Product of Squares

We wish to discuss positive integer solutions to the Diophantine equation $$\prod_{k=1}^n(k^2+1)=b^2.$$ Some methods in analytic number theory will be used to tackle this problem.

math.NT↗