Year: 2026
Type: article
Abstract: In this paper, we are interested in studying the generalized Pythagorean equation: $$x^2+y^2=n!\quad \quad n\in \mathbb{N}\quad (x,y) \in \mathbb{Z}^2.$$ Furthermore, the non-principe Dirichlet character module 4, defined as follows:$$\chi(d)=\left\{\begin{array}{c}1 \quad if \quad d\equiv 1(mod 4)\\-1 \quad if \quad d\equiv 3(mod 4)\\0 \quad if \quad d \>is \>even\end{array}\right.$$ Using Jacobi's Two-Square Theorem, we show that the total number of lattices in the circle of radius $r=\sqrt{N}$ is equal $\sum^N_{n=1}\sum_{d|n}\chi(d)$.\\ Finally, Application: the formula for $\pi$ by number theory is given by$$\pi=\lim_{N\rightarrow +\infty}\frac{4}{N}\bigg(\sum^N_{n=1}\sum_{d|n}\chi(d)\bigg)=4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}\cdot\cdot\cdot\cdot).$$
Author: Deval Sidi Mohamed
Institution: Université de Nouakchott
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7214810486