Well it's a question about the pi day :
If we have the pi day we need a place to it :
Consider the earth as ball such that :
$$x^2+y^2+z^2=40000$$
Now can we find all the rational point on the ball such that $\gcd(\lfloor x \rfloor,\lfloor y \rfloor,\lfloor z \rfloor,314)=1$
To find it I have used Bezout's theorem without reach the goal. I have tried also three square theorem due to Legendre .
How to find it ?
Without computing it, prove that, if $$I=\large\int_{0}^{\infty}\dfrac{\left(\frac{x^{2}}{\pi^{2}}\right)^{-x}}{\left(x!\right)^{2}}dx $$ $$\frac{1}{\pi }\binom{\pi !}{\pi } < I <\sqrt{\pi } \binom{\pi !}{\log (\pi )}$$