Solution of the Generalized Pythagorean Equation; x2+y2=n! and Formula ? by Number Theory

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
$$\pi=\lim_{N\rightarrow +\infty}\frac{4}{N}\bigg(\sum^N_{n=1}\sum_{d|n}\chi(d)\bigg)=4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}\cdot\cdot\cdot\cdot).$$