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

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.

1. x<=y<=z 
2. x^2+y^2=z^2-1;
3. x+y+z<=N

I have to find how many combinations I can get from the given numbers in a method.

I have tried as follows but it's taking so much time for a higher number and resulting in a timeout..

int N= Int32.Parse(Console.ReadLine());
List<String> res = new List<string>();
//x<=y<=z

int mxSqrt = N - 2;

int a = 0, b = 0;

for (int z = 1; z <= mxSqrt; z++)

{

a = z * z;

for (int y = 1; y <= z; y++)

{

b = y * y;

for (int x = 1; x <= y; x++)

{

int x1 = b + x * x;

int y1 = a - 1;

if (x1 == y1 && ((x + y + z) <= N))

{

res.Add(x + “,” + y + “,” + z);

}

}

}

}

Console.WriteLine(res.Count());

My question:

My solution is taking time for a bigger number (I think it’s the for loops), how can I improve it?

Is there any better approach for the same?

#c-sharp