Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum\_{p \le x}p^2$, $a(x) = \sum\_{p \le x}ab$ and $r(x) = \sum\_{p \le x}(a+b)^2$. Is it true that:


$$
\lim\_{x \to \infty}\frac{h(x)}{r(x)} = \frac{\pi}{2+\pi}
$$


$$
\lim\_{x \to \infty}\frac{a(x)}{r(x)} = \frac{1}{2+\pi}
$$



Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number
**Q:** Is $\lim\_{x \to \infty}\frac{h(x)}{r(x)} = \frac{\pi}{2+\pi} $ ?  

**A:** use that $r(x)=h(x)+2a(x)$, hence
$$\frac{h(x)}{r(x)} = \frac{h(x)/a(x)}{2+h(x)/a(x)}$$
and
$$\lim\_{x\rightarrow\infty}\frac{h(x)}{a(x)}=\pi$$
in view of <https://math.stackexchange.com/a/3481801/87355>


