Can someone help me find out what this curve's equation is. The number of spirals does not matter, but it has to pass through the points (0, 0,...

There are several different approaches to this problem. It all depends what kind of spiral do you want. I decided to start with conical spiral and then build up from there.

Parametric equation of conical spiral is

where  is the angular frequency and  is the radius.

Since the shape of the spiral doesn't matter we'll put

Thus, we get 

See the first image.

The spiral starts at  but then starts to spread out to infinity. We want to make it fall back to the axis at  To do that we will replace  in front of sine and cosine with sine function making the radius oscillate. Thus we get

  is here to make sure that the period is  and 10 is here just to make the spiral prettier. 

For this spiral see the second image.

Now comes the tricky part. we want to rotate the spiral so that it passes not through  but through   It may be easier to think in terms of unit vectors. We want to rotate vector  so that we get vector  (unit vector in direction of ). So we basically need to rotate around vector  by angle  (angle between  and ). To do that we'll find the rotation matrix (see the link below).

The rotation matrix is  

              

Now we just multiply that matrix with

  vector from our previous spiral but with extended period by a factor of  because the distance between (0,0,0) and (100,100,100) si greater than distance between (0,0,0) and (100,0,0).

The final equation is

 

The spiral can be seen in the third image.

There is probably some easier way of constructing the required spiral, but i couldn't think of any.