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

No time to properly test it, but seemed to yield the same results as your code (at 100 -> 6 results and at 1000 -> 55 results).

With N=1000 a time of 2ms vs your 144ms also without List

and N=10000 a time of 28ms

var N = 1000;
var c = 0;

for (int x = 2; x < N; x+=2)
{
    for (int y = x; y < (N - x); y+=2)
    {
        long z2 = x * x + y * y + 1;
        int z = (int) Math.Sqrt(z2);
        if (x + y + z > N)
            break;
        if (z * z == z2)
            c++;
    }
}

Console.WriteLine(c);