What function can produce a perfect saddleback plot and fulfil the following requirement?

Question

I need to find a function that produce a good saddleback plot.

The function has the following requirements:

1. Having 2 arguments: x and y
2. Both x and y are natural numbers
3. The result of the function is natural number
4. The function is increasing in each argument

In order to plot a saddleback (in 3D), I tried $ x^2 - y^2 $, which gives such plot:

It looks like a saddleback, but it doesn't fit to the requirements.

I then tried $ 3x+27y+y^2 $, it gives plot like this:

It is not that saddleback, right?

So can anyone supply me a good one on this?

Answer 1

What function can produce a perfect saddle shape?

I propose function $f(x,y)=x y $. It fulfills all the arithmetic conditions, and its plot has the shape of a saddle (‘hyperbolic paraboloid’) because, as a quadratic form it has signature (1,1), i.e. is the sum of a definite positive and a definite negative form:

$$ xy=\frac14(x+y)^2-\frac14(x-y)^2. $$

So it may be the same or some rotated shape.

Please suggest any equations to other such saddle shapes.