How to enumerate x^2 + y^2 = z^2 - 1 (with additional constraints)

Question

Lets N be a number (10<=N<=10^5).

I have to break it into 3 numbers (x,y,z) such that it validates the following conditions.

Answer 1

#include
#include
int main()
{
    int N = 10000;
    int c = 0;
    for (int x = 2; x < N; x+=2)
    {
        for (int y = x; y < (N - x); y+=2)
        {
            auto z = sqrt(x * x + y * y + 1);
            if(x+y+z>N){
                break;
            }
            if (z - (int) z == 0)
            {
                c++;
            }
        }
    }
    std::cout<

This is my solution. On testing the previous solutions for this problem I found that x,y are always even and z is odd. I dont know the mathematical nature behind this, I am currently trying to figure that out.