How to cut the $\pi$ ?

It can be seen that $314159265358979323$ is odd. Hence, for each of the solutions $(x,y)$ , one of $x,y$ is odd and the other is even.

For the purpose of analysis, let us assume that $x$ is even and $y$ is odd.

This would mean that $x^2 \mod 4 \equiv 0$ and $y^2 \mod 4 \equiv 1$

This in turn would lead to $(x^2+y^2) \mod 4 \equiv 1$

However, $314159265358979323 \mod 4 \equiv 3$ .

Hence, the number of solutions is $\boxed{0}$

A very valid solution is Janardhanan's one, but I want to show that this problem could be solved with only logic. First we can notice that $(x,x)$ can't be a solution, because $314159265358979323$ is odd: hence all solutions are $(x,y)$ with $x \neq y$ . But if $(x,y)$ is a solution, also $(y,x)$ would be one: the total number of solutions would be even. But choices are all odd, apart from $0$ : the answer is really $\boxed{0}$ .